Subgroups — C₂×#555×D₅₃ #28291109

Order 13568.

8710 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811339	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16811338	C₂	1	2	yes	maximal, complemented
3	C₂×#16811337	C₂	1	4	yes	maximal, complemented
4	C₂×#16811336	C₂	1	4	yes	maximal, complemented
5	C₂×#16811335	C₂	1	4	yes	maximal, complemented
6	#555×D₅₃	C₂	1	32	yes	maximal, direct product factor
7	C₂×C₄⋊Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#124×D₅₃	C₂	1	4	yes	maximal, complemented
9	C₂×#121×D₅₃	C₂	1	4	yes	maximal, complemented
10	C₂×C₄×D₄×D₅₃	C₂	1	2	yes	maximal, complemented
11	C₂×#117×D₅₃	C₂	1	4	yes	maximal, complemented
12	C₂×C₅₃×#555	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Q₈×D₅₃	C₂²	1	2	yes	—
14	C₂²×D₄×D₅₃	C₂²	1	2	yes	—
15	C₂²×D₄×D₅₃	C₂²	1	4	yes	—
16	C₂³×C₄×D₅₃	C₂²	1	4	yes	—
17	#16811339	C₂²	1	16	yes	complemented
18	#16811338	C₂²	1	32	yes	complemented
19	#16811337	C₂²	1	64	yes	complemented
20	#16811336	C₂²	1	64	yes	complemented
21	#16811335	C₂²	1	64	yes	complemented
22	C₂×#13133217	C₂²	1	1	yes	characteristic, complemented
23	C₂×#13133216	C₂²	1	4	yes	complemented
24	C₂×#13133215	C₂²	1	2	yes	complemented
25	C₄⋊Q₈×D₅₃	C₂²	1	16	yes	complemented
26	C₂×#13133206	C₂²	1	4	yes	complemented
27	C₂×#13133205	C₂²	1	4	yes	complemented
28	C₂×#13133204	C₂²	1	4	yes	complemented
29	C₂×#13133203	C₂²	1	8	yes	complemented
30	C₂×#13133203	C₂²	1	8	yes	complemented
31	#124×D₅₃	C₂²	1	64	yes	complemented
32	C₂×#13133192	C₂²	1	4	yes	complemented
33	C₂×#13133191	C₂²	1	4	yes	complemented
34	C₂×#13133190	C₂²	1	8	yes	complemented
35	C₂×#13133189	C₂²	1	8	yes	complemented
36	C₂×#13133188	C₂²	1	4	yes	complemented
37	#121×D₅₃	C₂²	1	64	yes	complemented
38	C₂×C₄×D₂₁₂	C₂²	1	2	yes	complemented
39	C₂×D₄×Dic₅₃	C₂²	1	2	yes	complemented
40	C₂×#13133179	C₂²	1	2	yes	complemented
41	C₂×C₄×C₂²⋊D₅₃	C₂²	1	4	yes	complemented
42	C₂×#13133177	C₂²	1	4	yes	complemented
43	C₄×D₄×D₅₃	C₂²	1	32	yes	complemented
44	C₂×#13133176	C₂²	1	4	yes	complemented
45	C₂×#13133175	C₂²	1	4	yes	complemented
46	C₂×#13133175	C₂²	1	4	yes	complemented
47	C₂×#13133174	C₂²	1	8	yes	complemented
48	C₂×#13133173	C₂²	1	8	yes	complemented
49	#117×D₅₃	C₂²	1	64	yes	complemented
50	C₂×C₄⋊C₄×D₅₃	C₂²	1	2	yes	—
51	C₂×C₄⋊C₄×D₅₃	C₂²	1	4	yes	—
52	C₂×C₂²⋊C₄×D₅₃	C₂²	1	8	yes	—
53	C₂×C₂²⋊C₄×D₅₃	C₂²	1	4	yes	—
54	C₂×C₄²×D₅₃	C₂²	1	4	yes	—
55	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
56	C₅₃×#555	C₂²	1	16	yes	complemented
57	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
58	C₂×#124×C₅₃	C₂²	1	4	yes	complemented
59	C₂×#121×C₅₃	C₂²	1	4	yes	complemented
60	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
61	C₂×#117×C₅₃	C₂²	1	4	yes	complemented
62	C₂⁴×D₅₃	C₂³	1	4	yes	—
63	C₂²×Dic₁₀₆	C₂³	1	2	yes	—
64	C₂²×Dic₁₀₆	C₂³	1	4	yes	—
65	C₂×Q₈×D₅₃	—	4	8	no	—
66	C₂×Q₈×D₅₃	C₂³	1	16	yes	—
67	C₂²×D₂₁₂	C₂³	1	2	yes	—
68	C₂²×D₂₁₂	C₂³	1	4	yes	—
69	C₂²×C₂²⋊D₅₃	C₂³	1	4	yes	—
70	C₂²×C₂²⋊D₅₃	C₂³	1	8	yes	—
71	C₂×D₄×D₅₃	—	2	16	no	—
72	C₂×D₄×D₅₃	—	4	16	no	—
73	C₂×D₄×D₅₃	C₂³	1	16	yes	—
74	C₂×D₄×D₅₃	C₂³	1	32	yes	—
75	C₂³×Dic₅₃	C₂³	1	4	yes	—
76	C₂²×C₄×D₅₃	—	2	32	no	—
77	C₂²×C₄×D₅₃	C₂³	1	4	yes	—
78	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
79	C₂²×C₄×D₅₃	C₂³	1	4	yes	—
80	C₂²×C₄×D₅₃	C₂³	1	32	yes	—
81	C₂²×C₄×D₅₃	C₂³	1	4	yes	—
82	C₂²×C₄×D₅₃	C₂³	1	4	yes	—
83	C₂²×C₄×D₅₃	C₂³	1	2	yes	—
84	#13133217	C₂³	1	8	yes	complemented
85	#13133216	C₂³	1	32	yes	complemented
86	#13133215	C₂³	1	16	yes	complemented
87	#13133206	C₂³	1	32	yes	complemented
88	#13133205	C₂³	1	32	yes	complemented
89	#13133204	C₂³	1	32	yes	complemented
90	#13133203	C₂³	1	64	yes	complemented
91	#13133203	C₂³	1	64	yes	complemented
92	#13133192	C₂³	1	32	yes	complemented
93	#13133191	C₂³	1	32	yes	complemented
94	#13133190	C₂³	1	64	yes	complemented
95	#13133189	C₂³	1	64	yes	complemented
96	#13133188	C₂³	1	32	yes	complemented
97	C₄×D₂₁₂	C₂³	1	16	yes	complemented
98	D₄×Dic₅₃	C₂³	1	16	yes	complemented
99	#13133179	C₂³	1	16	yes	complemented
100	C₄×C₂²⋊D₅₃	C₂³	1	32	yes	complemented
101	#13133177	C₂³	1	32	yes	complemented
102	#13133176	C₂³	1	32	yes	complemented
103	#13133175	C₂³	1	32	yes	complemented
104	#13133175	C₂³	1	32	yes	complemented
105	#13133174	C₂³	1	64	yes	complemented
106	#13133173	C₂³	1	64	yes	complemented
107	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
108	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
109	C₂×#11719919	C₂³	1	4	yes	—
110	C₂×#11719919	C₂³	1	8	yes	—
111	C₄⋊C₄×D₅₃	C₂³	1	16	yes	—
112	C₄⋊C₄×D₅₃	C₂³	1	32	yes	—
113	C₂×C₂²⋊Dic₅₃	C₂³	1	8	yes	—
114	C₂×C₂²⋊Dic₅₃	C₂³	1	4	yes	—
115	C₂×#11719921	C₂³	1	8	yes	—
116	C₂×#11719921	C₂³	1	8	yes	—
117	C₂×#11719921	C₂³	1	4	yes	—
118	C₂×#11719921	C₂³	1	4	yes	—
119	C₂²⋊C₄×D₅₃	C₂³	1	64	yes	—
120	C₂²⋊C₄×D₅₃	C₂³	1	32	yes	—
121	C₂×C₄×Dic₅₃	C₂³	1	4	yes	—
122	C₂×C₄×Dic₅₃	C₂³	1	8	yes	—
123	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
124	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
125	C₄²×D₅₃	C₂³	1	32	yes	—
126	C₄²×D₅₃	C₂³	1	8	yes	—
127	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
128	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
129	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
130	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
131	C₄⋊Q₈×C₅₃	C₂³	1	8	yes	complemented
132	#124×C₅₃	C₂³	1	32	yes	complemented
133	#121×C₅₃	C₂³	1	32	yes	complemented
134	C₄×D₄×C₅₃	C₂³	1	16	yes	complemented
135	#117×C₅₃	C₂³	1	32	yes	complemented
136	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
137	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
138	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
139	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
140	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
141	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
142	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
143	C₂³×D₅₃	—	2	16	no	—
144	C₂³×D₅₃	—	2	16	no	—
145	C₂³×D₅₃	—	2	16	no	—
146	C₂³×D₅₃	C₂⁴	1	16	yes	—
147	C₂³×D₅₃	C₂⁴	1	4	yes	—
148	C₂³×D₅₃	C₂⁴	1	1	yes	characteristic
149	C₂×Q₈×C₅₃	—	4	4	no	—
150	C₂×Q₈×C₅₃	C₂⁴	1	8	yes	—
151	C₂×D₄×C₅₃	—	2	8	no	—
152	C₂×D₄×C₅₃	—	4	8	no	—
153	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
154	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
155	C₂²×C₄×C₅₃	—	2	16	no	—
156	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
157	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
158	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
159	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
160	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
161	C₂×C₂²⋊D₅₃	—	2	16	no	—
162	C₂×C₂²⋊D₅₃	—	4	16	no	—
163	C₂×C₂²⋊D₅₃	C₂⁴	1	16	yes	—
164	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
165	C₂²×Dic₅₃	—	2	16	no	—
166	C₂²×Dic₅₃	C₂⁴	1	4	yes	—
167	C₂²×Dic₅₃	C₂⁴	1	1	yes	characteristic
168	C₂²×Dic₅₃	C₂⁴	1	4	yes	—
169	C₂²×Dic₅₃	C₂⁴	1	16	yes	—
170	C₂²×Dic₅₃	C₂⁴	1	2	yes	—
171	Q₈×D₅₃	—	4	32	no	—
172	D₄×D₅₃	—	2	64	no	—
173	D₄×D₅₃	—	4	64	no	—
174	C₂×D₂₁₂	—	2	8	no	—
175	C₂×D₂₁₂	—	4	8	no	—
176	C₂×D₂₁₂	C₂⁴	1	8	yes	—
177	C₂×D₂₁₂	C₂⁴	1	16	yes	—
178	C₂×C₄×D₅₃	—	2	16	no	—
179	C₂×C₄×D₅₃	—	2	16	no	—
180	C₂×C₄×D₅₃	—	2	4	no	—
181	C₂×C₄×D₅₃	—	2	16	no	—
182	C₂×C₄×D₅₃	—	2	16	no	—
183	C₂×C₄×D₅₃	—	2	128	no	—
184	C₂×C₄×D₅₃	#41	1	16	yes	—
185	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
186	C₂×C₄×D₅₃	C₂⁴	1	4	yes	—
187	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
188	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
189	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
190	C₂×C₄×D₅₃	C₂⁴	1	8	yes	—
191	C₂×Dic₁₀₆	—	4	8	no	—
192	C₂×Dic₁₀₆	—	4	4	no	—
193	C₂×Dic₁₀₆	C₂⁴	1	8	yes	—
194	C₂×Dic₁₀₆	C₂⁴	1	16	yes	—
195	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
196	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
197	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	—
198	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	—
199	C₄²×C₅₃	C₂⁴	1	16	yes	—
200	C₄²×C₅₃	C₂⁴	1	4	yes	—
201	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	—
202	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	—
203	#11719921	C₂⁴	1	32	yes	—
204	#11719921	C₂⁴	1	32	yes	—
205	#11719921	C₂⁴	1	16	yes	—
206	#11719921	C₂⁴	1	16	yes	—
207	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
208	C₄⋊Dic₅₃	C₂⁴	1	16	yes	—
209	#11719919	C₂⁴	1	16	yes	—
210	#11719919	C₂⁴	1	32	yes	—
211	C₄×Dic₅₃	C₂⁴	1	16	yes	—
212	C₄×Dic₅₃	C₂⁴	1	32	yes	—
213	C₄×Dic₅₃	C₂⁴	1	4	yes	—
214	C₄×Dic₅₃	C₂⁴	1	8	yes	—
215	C₂³×C₅₃	—	2	8	no	—
216	C₂³×C₅₃	—	2	8	no	—
217	C₂³×C₅₃	—	2	8	no	—
218	C₂³×C₅₃	C₂⁵	1	8	yes	—
219	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
220	C₂²×D₅₃	—	2	8	no	—
221	C₂²×D₅₃	—	2	8	no	—
222	C₂²×D₅₃	—	2	8	no	—
223	C₂²×D₅₃	—	2	32	no	—
224	C₂²×D₅₃	—	2	32	no	—
225	C₂²×D₅₃	—	2	32	no	—
226	C₂²×D₅₃	—	4	64	no	—
227	C₂²×D₅₃	C₂×#41	1	8	yes	—
228	C₂²×D₅₃	H(5,𝔽₂)	1	4	yes	—
229	C₂²×D₅₃	C₂⁵	1	8	yes	—
230	C₂²×D₅₃	C₂⁵	1	2	yes	—
231	Q₈×C₅₃	—	4	8	no	—
232	D₄×C₅₃	—	2	16	no	—
233	D₄×C₅₃	—	4	16	no	—
234	C₂×C₄×C₅₃	—	2	8	no	—
235	C₂×C₄×C₅₃	—	2	2	no	—
236	C₂×C₄×C₅₃	—	2	8	no	—
237	C₂×C₄×C₅₃	—	2	32	no	—
238	C₂×C₄×C₅₃	C₂×#41	1	8	yes	—
239	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
240	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
241	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
242	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
243	C₂²⋊D₅₃	—	2	32	no	—
244	C₂²⋊D₅₃	—	4	32	no	—
245	C₂×Dic₅₃	—	2	8	no	—
246	C₂×Dic₅₃	—	2	2	no	—
247	C₂×Dic₅₃	—	2	8	no	—
248	C₂×Dic₅₃	—	2	32	no	—
249	C₂×Dic₅₃	C₂×#41	1	8	yes	—
250	C₂×Dic₅₃	C₂⁵	1	2	yes	—
251	C₂×Dic₅₃	C₂⁵	1	8	yes	—
252	C₂×Dic₅₃	C₂⁵	1	8	yes	—
253	C₂×Dic₅₃	C₂⁵	1	4	yes	—
254	D₂₁₂	—	2	16	no	—
255	D₂₁₂	—	4	16	no	—
256	C₄×D₅₃	—	2	32	no	—
257	C₄×D₅₃	—	2	32	no	—
258	C₄×D₅₃	—	2	8	no	—
259	C₄×D₅₃	—	2	32	no	—
260	C₄×D₅₃	—	2	32	no	—
261	C₄×D₅₃	C₂×#41	1	32	yes	—
262	Dic₁₀₆	—	4	16	no	—
263	Dic₁₀₆	—	4	8	no	—
264	C₂²×#555	—	53	1	no	maximal, Sylow
265	C₂²×C₅₃	—	2	8	no	—
266	C₂²×C₅₃	—	2	8	no	—
267	C₂²×C₅₃	—	2	8	no	—
268	C₂²×C₅₃	—	4	16	no	—
269	C₂²×C₅₃	C₂²×#41	1	4	yes	—
270	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
271	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
272	C₂×D₅₃	—	2	8	no	—
273	C₂×D₅₃	—	2	8	no	—
274	C₂×D₅₃	—	2	8	no	—
275	C₂×D₅₃	—	4	16	no	—
276	C₂×D₅₃	—	4	64	no	—
277	C₂×D₅₃	#555	1	16	yes	direct product factor
278	C₂×D₅₃	C₂²×#41	1	8	yes	—
279	C₂×D₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
280	C₄×C₅₃	—	2	8	no	—
281	C₄×C₅₃	—	2	2	no	—
282	C₄×C₅₃	—	2	8	no	—
283	C₄×C₅₃	C₂²×#41	1	8	yes	—
284	Dic₅₃	—	2	8	no	—
285	Dic₅₃	—	2	2	no	—
286	Dic₅₃	—	2	8	no	—
287	Dic₅₃	C₂²×#41	1	8	yes	—
288	C₂×#555	—	53	32	no	—
289	C₂×#555	—	53	1	no	—
290	C₂×#555	—	53	4	no	—
291	C₂×#555	—	53	4	no	—
292	C₂×#555	—	53	2	no	—
293	C₂×#555	—	53	4	no	—
294	C₂×#555	D₅₃	1	1	yes	characteristic, direct product factor
295	C₂²×C₄⋊Q₈	—	53	1	no	—
296	C₂²×#124	—	53	4	no	—
297	C₂²×#121	—	53	4	no	—
298	C₂²×C₄×D₄	—	53	2	no	—
299	C₂²×#117	—	53	4	no	—
300	C₂×C₅₃	—	4	8	no	—
301	C₂×C₅₃	C₂×#555	1	4	yes	complemented
302	C₂×C₅₃	C₂³×#41	1	2	yes	—
303	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
304	D₅₃	—	4	8	no	—
305	D₅₃	C₂×#555	1	8	yes	direct product factor
306	C₂³×Q₈	—	53	2	no	—
307	C₂³×D₄	—	53	2	no	—
308	C₂³×D₄	—	53	4	no	—
309	C₂⁴×C₄	—	53	4	no	—
310	#555	—	53	16	no	—
311	#555	—	53	64	no	—
312	#555	—	53	64	no	—
313	#555	—	53	32	no	—
314	#555	—	53	64	no	—
315	#555	C₂×D₅₃	1	16	yes	direct product factor
316	C₂×C₄⋊Q₈	—	53	2	no	—
317	C₂×C₄⋊Q₈	—	53	16	no	—
318	C₂×C₄⋊Q₈	—	53	4	no	—
319	C₂×C₄⋊Q₈	—	53	1	no	—
320	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
321	C₂×#124	—	53	4	no	—
322	C₂×#124	—	53	4	no	—
323	C₂×#124	—	53	64	no	—
324	C₂×#124	—	53	8	no	—
325	C₂×#124	—	53	8	no	—
326	C₂×#124	—	53	4	no	—
327	C₂×#124	C₂×D₅₃	1	4	yes	complemented
328	C₂×#121	—	53	4	no	—
329	C₂×#121	—	53	8	no	—
330	C₂×#121	—	53	4	no	—
331	C₂×#121	—	53	64	no	—
332	C₂×#121	—	53	4	no	—
333	C₂×#121	—	53	8	no	—
334	C₂×#121	C₂×D₅₃	1	4	yes	complemented
335	C₂×C₄×D₄	—	53	2	no	—
336	C₂×C₄×D₄	—	53	4	no	—
337	C₂×C₄×D₄	—	53	32	no	—
338	C₂×C₄×D₄	—	53	2	no	—
339	C₂×C₄×D₄	—	53	4	no	—
340	C₂×C₄×D₄	—	53	2	no	—
341	C₂×C₄×D₄	C₂×D₅₃	1	2	yes	complemented
342	C₂×#117	—	53	4	no	—
343	C₂×#117	—	53	64	no	—
344	C₂×#117	—	53	4	no	—
345	C₂×#117	—	53	4	no	—
346	C₂×#117	—	53	8	no	—
347	C₂×#117	—	53	8	no	—
348	C₂×#117	C₂×D₅₃	1	4	yes	complemented
349	C₂²×C₄⋊C₄	—	53	2	no	—
350	C₂²×C₄⋊C₄	—	53	4	no	—
351	C₂²×C₂²⋊C₄	—	53	8	no	—
352	C₂²×C₂²⋊C₄	—	53	4	no	—
353	C₂²×C₄²	—	53	4	no	—
354	C₂²×C₄²	—	53	1	no	—
355	C₅₃	C₂²×#555	1	1	yes	characteristic, complemented, Sylow
356	C₂⁵	—	53	4	no	—
357	C₂²×Q₈	—	53	16	no	—
358	C₂²×Q₈	—	53	4	no	—
359	C₂²×Q₈	—	53	2	no	—
360	C₂²×Q₈	—	212	8	no	—
361	C₂²×Q₈	C₂²×D₅₃	1	2	yes	—
362	C₂²×D₄	—	53	4	no	—
363	C₂²×D₄	—	53	8	no	—
364	C₂²×D₄	—	53	16	no	—
365	C₂²×D₄	—	53	32	no	—
366	C₂²×D₄	—	53	2	no	—
367	C₂²×D₄	—	53	4	no	—
368	C₂²×D₄	—	106	16	no	—
369	C₂²×D₄	—	212	16	no	—
370	C₂²×D₄	C₂²×D₅₃	1	2	yes	—
371	C₂²×D₄	C₂²×D₅₃	1	4	yes	—
372	C₂³×C₄	—	53	2	no	—
373	C₂³×C₄	—	53	4	no	—
374	C₂³×C₄	—	53	1	no	—
375	C₂³×C₄	—	53	4	no	—
376	C₂³×C₄	—	53	4	no	—
377	C₂³×C₄	—	53	4	no	—
378	C₂³×C₄	—	53	32	no	—
379	C₂³×C₄	—	53	4	no	—
380	C₂³×C₄	—	106	32	no	—
381	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
382	C₄⋊Q₈	—	53	16	no	—
383	C₄⋊Q₈	—	53	32	no	—
384	C₄⋊Q₈	—	53	8	no	—
385	C₄⋊Q₈	C₂²×D₅₃	1	8	yes	complemented
386	#124	—	53	32	no	—
387	#124	—	53	32	no	—
388	#124	—	53	64	no	—
389	#124	—	53	64	no	—
390	#124	—	53	32	no	—
391	#124	C₂²×D₅₃	1	32	yes	complemented
392	#121	—	53	32	no	—
393	#121	—	53	64	no	—
394	#121	—	53	32	no	—
395	#121	—	53	32	no	—
396	#121	—	53	64	no	—
397	#121	C₂²×D₅₃	1	32	yes	complemented
398	C₄×D₄	—	53	16	no	—
399	C₄×D₄	—	53	32	no	—
400	C₄×D₄	—	53	16	no	—
401	C₄×D₄	—	53	32	no	—
402	C₄×D₄	—	53	16	no	—
403	C₄×D₄	C₂²×D₅₃	1	16	yes	complemented
404	#117	—	53	32	no	—
405	#117	—	53	32	no	—
406	#117	—	53	32	no	—
407	#117	—	53	64	no	—
408	#117	—	53	64	no	—
409	#117	C₂²×D₅₃	1	32	yes	complemented
410	C₂×C₄⋊C₄	—	53	4	no	—
411	C₂×C₄⋊C₄	—	53	2	no	—
412	C₂×C₄⋊C₄	—	53	8	no	—
413	C₂×C₄⋊C₄	—	53	4	no	—
414	C₂×C₄⋊C₄	—	53	16	no	—
415	C₂×C₄⋊C₄	—	53	32	no	—
416	C₂×C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
417	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
418	C₂×C₂²⋊C₄	—	53	8	no	—
419	C₂×C₂²⋊C₄	—	53	4	no	—
420	C₂×C₂²⋊C₄	—	53	8	no	—
421	C₂×C₂²⋊C₄	—	53	8	no	—
422	C₂×C₂²⋊C₄	—	53	4	no	—
423	C₂×C₂²⋊C₄	—	53	4	no	—
424	C₂×C₂²⋊C₄	—	53	64	no	—
425	C₂×C₂²⋊C₄	—	53	32	no	—
426	C₂×C₂²⋊C₄	C₂²×D₅₃	1	8	yes	—
427	C₂×C₂²⋊C₄	C₂²×D₅₃	1	4	yes	—
428	C₂×C₄²	—	53	4	no	—
429	C₂×C₄²	—	53	8	no	—
430	C₂×C₄²	—	53	1	no	—
431	C₂×C₄²	—	53	2	no	—
432	C₂×C₄²	—	53	32	no	—
433	C₂×C₄²	—	53	8	no	—
434	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
435	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
436	C₂⁴	—	53	1	no	—
437	C₂⁴	—	53	16	no	—
438	C₂⁴	—	53	4	no	—
439	C₂⁴	—	106	16	no	—
440	C₂⁴	—	106	16	no	—
441	C₂⁴	—	106	16	no	—
442	C₂⁴	C₂³×D₅₃	1	4	yes	—
443	C₂×Q₈	—	4	4	no	—
444	C₂×Q₈	—	53	16	no	—
445	C₂×Q₈	—	53	8	no	—
446	C₂×Q₈	—	212	8	no	—
447	C₂×Q₈	—	212	4	no	—
448	C₂×Q₈	—	212	32	no	—
449	C₂×Q₈	C₂³×D₅₃	1	8	yes	—
450	C₂×D₄	—	2	8	no	—
451	C₂×D₄	—	4	8	no	—
452	C₂×D₄	—	53	16	no	—
453	C₂×D₄	—	53	32	no	—
454	C₂×D₄	—	53	8	no	—
455	C₂×D₄	—	53	16	no	—
456	C₂×D₄	—	106	64	no	—
457	C₂×D₄	—	106	16	no	—
458	C₂×D₄	—	106	8	no	—
459	C₂×D₄	—	212	8	no	—
460	C₂×D₄	—	212	64	no	—
461	C₂×D₄	—	212	16	no	—
462	C₂×D₄	C₂³×D₅₃	1	8	yes	—
463	C₂×D₄	C₂³×D₅₃	1	16	yes	—
464	C₂²×C₄	—	2	16	no	—
465	C₂²×C₄	—	53	2	no	—
466	C₂²×C₄	—	53	16	no	—
467	C₂²×C₄	—	53	8	no	—
468	C₂²×C₄	—	53	16	no	—
469	C₂²×C₄	—	53	4	no	—
470	C₂²×C₄	—	53	1	no	—
471	C₂²×C₄	—	53	4	no	—
472	C₂²×C₄	—	53	16	no	—
473	C₂²×C₄	—	53	4	no	—
474	C₂²×C₄	—	53	16	no	—
475	C₂²×C₄	—	53	16	no	—
476	C₂²×C₄	—	53	16	no	—
477	C₂²×C₄	—	106	16	no	—
478	C₂²×C₄	—	106	128	no	—
479	C₂²×C₄	—	106	16	no	—
480	C₂²×C₄	—	106	16	no	—
481	C₂²×C₄	—	106	4	no	—
482	C₂²×C₄	—	106	16	no	—
483	C₂²×C₄	—	106	16	no	—
484	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
485	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
486	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
487	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
488	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
489	C₄⋊C₄	—	53	16	no	—
490	C₄⋊C₄	—	53	8	no	—
491	C₄⋊C₄	—	53	32	no	—
492	C₄⋊C₄	—	53	16	no	—
493	C₄⋊C₄	C₂³×D₅₃	1	8	yes	—
494	C₄⋊C₄	C₂³×D₅₃	1	16	yes	—
495	C₂²⋊C₄	—	53	32	no	—
496	C₂²⋊C₄	—	53	16	no	—
497	C₂²⋊C₄	—	53	32	no	—
498	C₂²⋊C₄	—	53	32	no	—
499	C₂²⋊C₄	—	53	16	no	—
500	C₂²⋊C₄	—	53	16	no	—
501	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	—
502	C₂²⋊C₄	C₂³×D₅₃	1	16	yes	—
503	C₄²	—	53	16	no	—
504	C₄²	—	53	32	no	—
505	C₄²	—	53	4	no	—
506	C₄²	—	53	8	no	—
507	C₄²	C₂³×D₅₃	1	16	yes	—
508	C₄²	C₂³×D₅₃	1	4	yes	—
509	C₂³	—	2	8	no	—
510	C₂³	—	2	8	no	—
511	C₂³	—	2	8	no	—
512	C₂³	—	53	8	no	—
513	C₂³	—	53	4	no	—
514	C₂³	—	53	2	no	—
515	C₂³	—	53	8	no	—
516	C₂³	—	106	32	no	—
517	C₂³	—	106	32	no	—
518	C₂³	—	106	32	no	—
519	C₂³	—	106	8	no	—
520	C₂³	—	106	8	no	—
521	C₂³	—	106	8	no	—
522	C₂³	—	212	64	no	—
523	C₂³	C₂⁴×D₅₃	1	8	yes	—
524	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
525	Q₈	—	4	8	no	—
526	Q₈	—	212	16	no	—
527	Q₈	—	212	8	no	—
528	D₄	—	2	16	no	—
529	D₄	—	4	16	no	—
530	D₄	—	106	32	no	—
531	D₄	—	106	16	no	—
532	D₄	—	212	16	no	—
533	D₄	—	212	32	no	—
534	C₂×C₄	—	2	8	no	—
535	C₂×C₄	—	2	8	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	32	no	—
538	C₂×C₄	—	53	8	no	—
539	C₂×C₄	—	53	4	no	—
540	C₂×C₄	—	53	32	no	—
541	C₂×C₄	—	53	8	no	—
542	C₂×C₄	—	53	2	no	—
543	C₂×C₄	—	53	8	no	—
544	C₂×C₄	—	106	32	no	—
545	C₂×C₄	—	106	32	no	—
546	C₂×C₄	—	106	32	no	—
547	C₂×C₄	—	106	8	no	—
548	C₂×C₄	—	106	32	no	—
549	C₂×C₄	—	106	8	no	—
550	C₂×C₄	—	106	8	no	—
551	C₂×C₄	—	106	2	no	—
552	C₂×C₄	—	106	32	no	—
553	C₂×C₄	#41×D₅₃	1	8	yes	—
554	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
555	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
556	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
557	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
558	C₂²	—	2	8	no	—
559	C₂²	—	2	8	no	—
560	C₂²	—	2	8	no	—
561	C₂²	—	4	16	no	—
562	C₂²	—	53	8	no	—
563	C₂²	—	53	4	no	—
564	C₂²	—	53	16	no	—
565	C₂²	—	106	8	no	—
566	C₂²	—	106	8	no	—
567	C₂²	—	106	8	no	—
568	C₂²	—	212	64	no	—
569	C₂²	—	212	16	no	—
570	C₂²	C₂×#41×D₅₃	1	4	yes	—
571	C₂²	H(5,𝔽₂)×D₅₃	1	2	yes	—
572	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
573	C₄	—	2	8	no	—
574	C₄	—	2	8	no	—
575	C₄	—	2	2	no	—
576	C₄	—	53	8	no	—
577	C₄	—	106	8	no	—
578	C₄	—	106	8	no	—
579	C₄	—	106	2	no	—
580	C₄	C₂×#41×D₅₃	1	8	yes	—
581	C₂	—	4	8	no	—
582	C₂	—	53	8	no	—
583	C₂	—	212	8	no	—
584	C₂	#555×D₅₃	1	4	yes	direct product factor
585	C₂	C₂²×#41×D₅₃	1	2	yes	—
586	C₂	C₂×H(5,𝔽₂)×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂×C₅₃×#555	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂×#555	D₅₃
hypocenter	C₅₃	C₂²×#555
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#555

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#16811339	C₂	semidirect
2. C₂×#16811338	C₂	semidirect
3. C₂×#16811337	C₂	semidirect
4. C₂×#16811336	C₂	semidirect
5. C₂×#16811335	C₂	semidirect
6. #555×D₅₃	C₂	direct (N × H)
7. C₂×C₄⋊Q₈×D₅₃	C₂	semidirect
8. C₂×#124×D₅₃	C₂	semidirect
9. C₂×#121×D₅₃	C₂	semidirect
10. C₂×C₄×D₄×D₅₃	C₂	semidirect
11. C₂×#117×D₅₃	C₂	semidirect
12. C₂×C₅₃×#555	C₂	semidirect
17. #16811339	C₂²	semidirect
18. #16811338	C₂²	semidirect
19. #16811337	C₂²	semidirect
20. #16811336	C₂²	semidirect
21. #16811335	C₂²	semidirect
22. C₂×#13133217	C₂²	semidirect
23. C₂×#13133216	C₂²	semidirect
24. C₂×#13133215	C₂²	semidirect
25. C₄⋊Q₈×D₅₃	C₂²	semidirect
26. C₂×#13133206	C₂²	semidirect
27. C₂×#13133205	C₂²	semidirect
28. C₂×#13133204	C₂²	semidirect
29. C₂×#13133203	C₂²	semidirect
30. C₂×#13133203	C₂²	semidirect
31. #124×D₅₃	C₂²	semidirect
32. C₂×#13133192	C₂²	semidirect
33. C₂×#13133191	C₂²	semidirect
34. C₂×#13133190	C₂²	semidirect
35. C₂×#13133189	C₂²	semidirect
36. C₂×#13133188	C₂²	semidirect
37. #121×D₅₃	C₂²	semidirect
38. C₂×C₄×D₂₁₂	C₂²	semidirect
39. C₂×D₄×Dic₅₃	C₂²	semidirect
40. C₂×#13133179	C₂²	semidirect
41. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
42. C₂×#13133177	C₂²	semidirect
43. C₄×D₄×D₅₃	C₂²	semidirect
44. C₂×#13133176	C₂²	semidirect
45. C₂×#13133175	C₂²	semidirect
46. C₂×#13133175	C₂²	semidirect
47. C₂×#13133174	C₂²	semidirect
48. C₂×#13133173	C₂²	semidirect
49. #117×D₅₃	C₂²	semidirect
56. C₅₃×#555	C₂²	semidirect
57. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
58. C₂×#124×C₅₃	C₂²	semidirect
59. C₂×#121×C₅₃	C₂²	semidirect
60. C₂×C₄×D₄×C₅₃	C₂²	semidirect
61. C₂×#117×C₅₃	C₂²	semidirect
84. #13133217	C₂³	semidirect
85. #13133216	C₂³	semidirect
86. #13133215	C₂³	semidirect
87. #13133206	C₂³	semidirect
88. #13133205	C₂³	semidirect
89. #13133204	C₂³	semidirect
90. #13133203	C₂³	semidirect
91. #13133203	C₂³	semidirect
92. #13133192	C₂³	semidirect
93. #13133191	C₂³	semidirect
94. #13133190	C₂³	semidirect
95. #13133189	C₂³	semidirect
96. #13133188	C₂³	semidirect
97. C₄×D₂₁₂	C₂³	semidirect
98. D₄×Dic₅₃	C₂³	semidirect
99. #13133179	C₂³	semidirect
100. C₄×C₂²⋊D₅₃	C₂³	semidirect
101. #13133177	C₂³	semidirect
102. #13133176	C₂³	semidirect
103. #13133175	C₂³	semidirect
104. #13133175	C₂³	semidirect
105. #13133174	C₂³	semidirect
106. #13133173	C₂³	semidirect
131. C₄⋊Q₈×C₅₃	C₂³	semidirect
132. #124×C₅₃	C₂³	semidirect
133. #121×C₅₃	C₂³	semidirect
134. C₄×D₄×C₅₃	C₂³	semidirect
135. #117×C₅₃	C₂³	semidirect
277. C₂×D₅₃	#555	direct (N × H)
294. C₂×#555	D₅₃	direct (N × H)
301. C₂×C₅₃	C₂×#555	semidirect
305. D₅₃	C₂×#555	direct (N × H)
315. #555	C₂×D₅₃	direct (N × H)
320. C₂×C₄⋊Q₈	C₂×D₅₃	semidirect
327. C₂×#124	C₂×D₅₃	semidirect
334. C₂×#121	C₂×D₅₃	semidirect
341. C₂×C₄×D₄	C₂×D₅₃	semidirect
348. C₂×#117	C₂×D₅₃	semidirect
355. C₅₃	C₂²×#555	semidirect
385. C₄⋊Q₈	C₂²×D₅₃	semidirect
391. #124	C₂²×D₅₃	semidirect
397. #121	C₂²×D₅₃	semidirect
403. C₄×D₄	C₂²×D₅₃	semidirect
409. #117	C₂²×D₅₃	semidirect
584. C₂	#555×D₅₃	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16811339	2
C₂×#16811338	2
C₂×#16811337	2
C₂×#16811336	2
C₂×#16811335	2
#555×D₅₃	2
C₂×C₄⋊Q₈×D₅₃	2
C₂×#124×D₅₃	2
C₂×#121×D₅₃	2
C₂×C₄×D₄×D₅₃	2
C₂×#117×D₅₃	2
C₂×C₅₃×#555	2
C₂²×#555	53

Subgroups — C2×#555×D53 #28291109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#555×D₅₃ #28291109