Subgroups — C₂²×C₅₃×#551 #27270215

Order 13568.

9526 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#551	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#126×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₂²×#124×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#123×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#123×C₅₃	C₂	1	2	yes	maximal, complemented
6	C₂²×C₂²⋊Q₈×C₅₃	C₂	1	2	yes	maximal, complemented
7	C₂²×#120×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄×D₄×C₅₃	C₂	1	2	yes	maximal, complemented
9	C₂²×#117×C₅₃	C₂	1	2	yes	maximal, complemented
10	C₂³×C₂²⋊C₄×C₅₃	C₂	1	2	yes	maximal, complemented
11	C₂⁶×C₅₃	C₂²	1	1	yes	characteristic
12	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
13	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
14	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
16	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
17	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
18	C₅₃×#551	C₂²	1	256	yes	direct product factor
19	C₂×#126×C₅₃	C₂²	1	48	yes	complemented
20	C₂×#124×C₅₃	C₂²	1	24	yes	complemented
21	C₂×#123×C₅₃	C₂²	1	24	yes	complemented
22	C₂×#123×C₅₃	C₂²	1	48	yes	complemented
23	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	48	yes	complemented
24	C₂×#120×C₅₃	C₂²	1	24	yes	—
25	C₂×C₄×D₄×C₅₃	C₂²	1	48	yes	complemented
26	C₂×#117×C₅₃	C₂²	1	48	yes	complemented
27	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
28	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
29	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
30	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
31	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
32	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
33	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
34	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
35	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
36	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
37	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	48	yes	complemented
38	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
39	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
40	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
41	C₂²×C₄²×C₅₃	C₂²	1	2	yes	complemented
42	C₂⁵×C₅₃	—	2	16	no	—
43	C₂⁵×C₅₃	—	4	4	no	—
44	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
45	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
46	C₂⁵×C₅₃	C₂³	1	12	yes	—
47	C₂⁵×C₅₃	C₂³	1	2	yes	—
48	C₂²×Q₈×C₅₃	—	4	4	no	—
49	C₂²×Q₈×C₅₃	C₂³	1	12	yes	complemented
50	C₂²×D₄×C₅₃	—	2	16	no	—
51	C₂²×D₄×C₅₃	—	4	4	no	—
52	C₂²×D₄×C₅₃	C₂³	1	24	yes	—
53	C₂²×D₄×C₅₃	C₂³	1	12	yes	—
54	C₂³×C₄×C₅₃	—	2	16	no	—
55	C₂³×C₄×C₅₃	—	4	8	no	—
56	C₂³×C₄×C₅₃	—	4	4	no	—
57	C₂³×C₄×C₅₃	C₂³	1	24	yes	—
58	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
59	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
60	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
61	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
63	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
64	C₂³×C₄×C₅₃	C₂³	1	24	yes	—
65	C₂³×C₄×C₅₃	C₂³	1	12	yes	complemented
66	#126×C₅₃	C₂³	1	128	yes	complemented
67	#124×C₅₃	C₂³	1	64	yes	complemented
68	#123×C₅₃	C₂³	1	128	yes	complemented
69	#123×C₅₃	C₂³	1	64	yes	complemented
70	C₂²⋊Q₈×C₅₃	C₂³	1	128	yes	complemented
71	#120×C₅₃	C₂³	1	64	yes	—
72	C₄×D₄×C₅₃	C₂³	1	128	yes	complemented
73	#117×C₅₃	C₂³	1	128	yes	complemented
74	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	complemented
75	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	complemented
76	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	complemented
77	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	complemented
78	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	—
79	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	—
80	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	—
81	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	—
82	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	—
83	C₂×C₂²⋊C₄×C₅₃	C₂³	1	12	yes	—
84	C₂×C₂²⋊C₄×C₅₃	C₂³	1	128	yes	complemented
85	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	complemented
86	C₂×C₂²⋊C₄×C₅₃	C₂³	1	12	yes	complemented
87	C₂×C₄²×C₅₃	C₂³	1	12	yes	complemented
88	C₂×C₄²×C₅₃	C₂³	1	24	yes	complemented
89	C₂⁴×C₅₃	—	2	8	no	—
90	C₂⁴×C₅₃	—	2	8	no	—
91	C₂⁴×C₅₃	—	2	8	no	—
92	C₂⁴×C₅₃	—	2	96	no	—
93	C₂⁴×C₅₃	—	2	8	no	—
94	C₂⁴×C₅₃	—	2	4	no	—
95	C₂⁴×C₅₃	—	2	4	no	—
96	C₂⁴×C₅₃	—	4	24	no	—
97	C₂⁴×C₅₃	—	4	64	no	—
98	C₂⁴×C₅₃	#41	1	16	yes	—
99	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
100	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
101	C₂⁴×C₅₃	C₂⁴	1	16	yes	—
102	C₂⁴×C₅₃	C₂⁴	1	12	yes	—
103	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
104	C₂×Q₈×C₅₃	—	4	24	no	—
105	C₂×Q₈×C₅₃	C₂⁴	1	16	yes	complemented
106	C₂×D₄×C₅₃	—	2	96	no	—
107	C₂×D₄×C₅₃	—	4	24	no	—
108	C₂×D₄×C₅₃	C₂⁴	1	32	yes	—
109	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
110	C₂²×C₄×C₅₃	—	2	96	no	—
111	C₂²×C₄×C₅₃	—	2	8	no	—
112	C₂²×C₄×C₅₃	—	2	8	no	—
113	C₂²×C₄×C₅₃	—	2	8	no	—
114	C₂²×C₄×C₅₃	—	2	8	no	—
115	C₂²×C₄×C₅₃	—	2	8	no	—
116	C₂²×C₄×C₅₃	—	2	4	no	—
117	C₂²×C₄×C₅₃	—	4	48	no	—
118	C₂²×C₄×C₅₃	—	4	24	no	—
119	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
120	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
121	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
122	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
123	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
124	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
125	C₂²×C₄×C₅₃	C₂⁴	1	32	yes	—
126	C₂²×C₄×C₅₃	C₂⁴	1	32	yes	—
127	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	complemented
128	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
129	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
130	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
131	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
132	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	—
133	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	—
134	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	—
135	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	—
136	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	—
137	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	—
138	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
139	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	complemented
140	C₄²×C₅₃	C₂⁴	1	16	yes	complemented
141	C₄²×C₅₃	C₂⁴	1	32	yes	complemented
142	C₂³×C₅₃	—	2	64	no	—
143	C₂³×C₅₃	—	2	24	no	—
144	C₂³×C₅₃	—	2	24	no	—
145	C₂³×C₅₃	—	2	24	no	—
146	C₂³×C₅₃	—	2	64	no	—
147	C₂³×C₅₃	—	2	24	no	—
148	C₂³×C₅₃	—	2	12	no	—
149	C₂³×C₅₃	—	2	12	no	—
150	C₂³×C₅₃	—	4	16	no	—
151	C₂³×C₅₃	—	4	192	no	—
152	C₂³×C₅₃	—	4	16	no	—
153	C₂³×C₅₃	—	4	16	no	—
154	C₂³×C₅₃	C₂×#41	1	48	yes	—
155	C₂³×C₅₃	C₂×#41	1	8	yes	—
156	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
157	C₂³×C₅₃	C₂⁵	1	4	yes	—
158	C₂³×C₅₃	C₂⁵	1	4	yes	—
159	C₂³×C₅₃	C₂⁵	1	8	yes	—
160	C₂³×C₅₃	C₂⁵	1	3	yes	—
161	Q₈×C₅₃	—	4	16	no	—
162	D₄×C₅₃	—	2	64	no	—
163	D₄×C₅₃	—	4	16	no	—
164	C₂×C₄×C₅₃	—	2	64	no	—
165	C₂×C₄×C₅₃	—	2	24	no	—
166	C₂×C₄×C₅₃	—	2	24	no	—
167	C₂×C₄×C₅₃	—	2	24	no	—
168	C₂×C₄×C₅₃	—	2	24	no	—
169	C₂×C₄×C₅₃	—	2	24	no	—
170	C₂×C₄×C₅₃	—	2	12	no	—
171	C₂×C₄×C₅₃	—	4	32	no	—
172	C₂×C₄×C₅₃	—	4	16	no	—
173	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
174	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
175	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
176	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
177	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
178	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
179	C₂²×#551	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
180	C₂²×C₅₃	—	2	8	no	—
181	C₂²×C₅₃	—	2	8	no	—
182	C₂²×C₅₃	—	2	96	no	—
183	C₂²×C₅₃	—	2	8	no	—
184	C₂²×C₅₃	—	2	8	no	—
185	C₂²×C₅₃	—	2	4	no	—
186	C₂²×C₅₃	—	2	4	no	—
187	C₂²×C₅₃	—	4	64	no	—
188	C₂²×C₅₃	—	4	24	no	—
189	C₂²×C₅₃	—	4	24	no	—
190	C₂²×C₅₃	#551	1	16	yes	direct product factor
191	C₂²×C₅₃	C₂²×#41	1	16	yes	—
192	C₂²×C₅₃	C₂²×#41	1	12	yes	—
193	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
194	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
195	C₄×C₅₃	—	2	8	no	—
196	C₄×C₅₃	—	2	8	no	—
197	C₄×C₅₃	—	2	8	no	—
198	C₄×C₅₃	—	2	8	no	—
199	C₄×C₅₃	—	2	8	no	—
200	C₄×C₅₃	—	2	4	no	—
201	C₂×#551	C₂×C₅₃	1	48	yes	direct product factor
202	C₂²×#126	C₂×C₅₃	1	2	yes	complemented
203	C₂²×#124	C₂×C₅₃	1	1	yes	characteristic, complemented
204	C₂²×#123	C₂×C₅₃	1	1	yes	characteristic, complemented
205	C₂²×#123	C₂×C₅₃	1	2	yes	complemented
206	C₂²×C₂²⋊Q₈	C₂×C₅₃	1	2	yes	complemented
207	C₂²×#120	C₂×C₅₃	1	1	yes	characteristic
208	C₂²×C₄×D₄	C₂×C₅₃	1	2	yes	complemented
209	C₂²×#117	C₂×C₅₃	1	2	yes	complemented
210	C₂³×C₂²⋊C₄	C₂×C₅₃	1	2	yes	complemented
211	C₂×C₅₃	—	2	16	no	—
212	C₂×C₅₃	—	4	4	no	—
213	C₂×C₅₃	—	4	4	no	—
214	C₂×C₅₃	C₂×#551	1	12	yes	direct product factor
215	C₂×C₅₃	C₂³×#41	1	2	yes	—
216	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
217	C₂⁶	C₂²×C₅₃	1	1	yes	characteristic
218	C₂³×Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
219	C₂³×D₄	C₂²×C₅₃	1	2	yes	—
220	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
221	C₂⁴×C₄	C₂²×C₅₃	1	2	yes	—
222	C₂⁴×C₄	C₂²×C₅₃	1	2	yes	—
223	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
224	#551	C₂²×C₅₃	1	256	yes	direct product factor
225	C₂×#126	C₂²×C₅₃	1	48	yes	complemented
226	C₂×#124	C₂²×C₅₃	1	24	yes	complemented
227	C₂×#123	C₂²×C₅₃	1	24	yes	complemented
228	C₂×#123	C₂²×C₅₃	1	48	yes	complemented
229	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	48	yes	complemented
230	C₂×#120	C₂²×C₅₃	1	24	yes	—
231	C₂×C₄×D₄	C₂²×C₅₃	1	48	yes	complemented
232	C₂×#117	C₂²×C₅₃	1	48	yes	complemented
233	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
234	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
235	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
236	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
237	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
238	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
239	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
240	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
241	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
242	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
243	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	48	yes	complemented
244	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	complemented
245	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
246	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic, complemented
247	C₂²×C₄²	C₂²×C₅₃	1	2	yes	complemented
248	C₅₃	C₂²×#551	1	1	yes	characteristic, direct product factor, Sylow
249	C₂⁵	—	2	16	no	—
250	C₂⁵	—	4	4	no	—
251	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
252	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
253	C₂⁵	C₂³×C₅₃	1	12	yes	—
254	C₂⁵	C₂³×C₅₃	1	2	yes	—
255	C₂²×Q₈	—	4	4	no	—
256	C₂²×Q₈	C₂³×C₅₃	1	12	yes	complemented
257	C₂²×D₄	—	2	16	no	—
258	C₂²×D₄	—	4	4	no	—
259	C₂²×D₄	C₂³×C₅₃	1	24	yes	—
260	C₂²×D₄	C₂³×C₅₃	1	12	yes	—
261	C₂³×C₄	—	2	16	no	—
262	C₂³×C₄	—	4	8	no	—
263	C₂³×C₄	—	4	4	no	—
264	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
265	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
266	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
267	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
268	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
269	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
270	C₂³×C₄	C₂³×C₅₃	1	24	yes	—
271	C₂³×C₄	C₂³×C₅₃	1	24	yes	—
272	C₂³×C₄	C₂³×C₅₃	1	12	yes	complemented
273	#126	C₂³×C₅₃	1	128	yes	complemented
274	#124	C₂³×C₅₃	1	64	yes	complemented
275	#123	C₂³×C₅₃	1	64	yes	complemented
276	#123	C₂³×C₅₃	1	128	yes	complemented
277	C₂²⋊Q₈	C₂³×C₅₃	1	128	yes	complemented
278	#120	C₂³×C₅₃	1	64	yes	—
279	C₄×D₄	C₂³×C₅₃	1	128	yes	complemented
280	#117	C₂³×C₅₃	1	128	yes	complemented
281	C₂×C₄⋊C₄	C₂³×C₅₃	1	24	yes	complemented
282	C₂×C₄⋊C₄	C₂³×C₅₃	1	24	yes	complemented
283	C₂×C₄⋊C₄	C₂³×C₅₃	1	24	yes	complemented
284	C₂×C₄⋊C₄	C₂³×C₅₃	1	24	yes	complemented
285	C₂×C₂²⋊C₄	C₂³×C₅₃	1	24	yes	—
286	C₂×C₂²⋊C₄	C₂³×C₅₃	1	24	yes	—
287	C₂×C₂²⋊C₄	C₂³×C₅₃	1	24	yes	—
288	C₂×C₂²⋊C₄	C₂³×C₅₃	1	24	yes	—
289	C₂×C₂²⋊C₄	C₂³×C₅₃	1	24	yes	—
290	C₂×C₂²⋊C₄	C₂³×C₅₃	1	12	yes	—
291	C₂×C₂²⋊C₄	C₂³×C₅₃	1	128	yes	complemented
292	C₂×C₂²⋊C₄	C₂³×C₅₃	1	24	yes	complemented
293	C₂×C₂²⋊C₄	C₂³×C₅₃	1	12	yes	complemented
294	C₂×C₄²	C₂³×C₅₃	1	12	yes	complemented
295	C₂×C₄²	C₂³×C₅₃	1	24	yes	complemented
296	C₂⁴	—	2	8	no	—
297	C₂⁴	—	2	8	no	—
298	C₂⁴	—	2	8	no	—
299	C₂⁴	—	2	96	no	—
300	C₂⁴	—	2	8	no	—
301	C₂⁴	—	2	4	no	—
302	C₂⁴	—	2	4	no	—
303	C₂⁴	—	4	24	no	—
304	C₂⁴	—	4	64	no	—
305	C₂⁴	#41×C₅₃	1	16	yes	—
306	C₂⁴	C₂⁴×C₅₃	1	6	yes	—
307	C₂⁴	C₂⁴×C₅₃	1	6	yes	—
308	C₂⁴	C₂⁴×C₅₃	1	16	yes	—
309	C₂⁴	C₂⁴×C₅₃	1	12	yes	—
310	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
311	C₂×Q₈	—	4	24	no	—
312	C₂×Q₈	C₂⁴×C₅₃	1	16	yes	complemented
313	C₂×D₄	—	2	96	no	—
314	C₂×D₄	—	4	24	no	—
315	C₂×D₄	C₂⁴×C₅₃	1	32	yes	—
316	C₂×D₄	C₂⁴×C₅₃	1	16	yes	—
317	C₂²×C₄	—	2	96	no	—
318	C₂²×C₄	—	2	8	no	—
319	C₂²×C₄	—	2	8	no	—
320	C₂²×C₄	—	2	8	no	—
321	C₂²×C₄	—	2	8	no	—
322	C₂²×C₄	—	2	8	no	—
323	C₂²×C₄	—	2	4	no	—
324	C₂²×C₄	—	4	48	no	—
325	C₂²×C₄	—	4	24	no	—
326	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
327	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
328	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
329	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
330	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
331	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
332	C₂²×C₄	C₂⁴×C₅₃	1	32	yes	—
333	C₂²×C₄	C₂⁴×C₅₃	1	32	yes	—
334	C₂²×C₄	C₂⁴×C₅₃	1	16	yes	complemented
335	C₄⋊C₄	C₂⁴×C₅₃	1	32	yes	complemented
336	C₄⋊C₄	C₂⁴×C₅₃	1	32	yes	complemented
337	C₄⋊C₄	C₂⁴×C₅₃	1	32	yes	complemented
338	C₄⋊C₄	C₂⁴×C₅₃	1	32	yes	complemented
339	C₂²⋊C₄	C₂⁴×C₅₃	1	32	yes	—
340	C₂²⋊C₄	C₂⁴×C₅₃	1	32	yes	—
341	C₂²⋊C₄	C₂⁴×C₅₃	1	32	yes	—
342	C₂²⋊C₄	C₂⁴×C₅₃	1	32	yes	—
343	C₂²⋊C₄	C₂⁴×C₅₃	1	32	yes	—
344	C₂²⋊C₄	C₂⁴×C₅₃	1	16	yes	—
345	C₂²⋊C₄	C₂⁴×C₅₃	1	32	yes	complemented
346	C₂²⋊C₄	C₂⁴×C₅₃	1	16	yes	complemented
347	C₄²	C₂⁴×C₅₃	1	16	yes	complemented
348	C₄²	C₂⁴×C₅₃	1	32	yes	complemented
349	C₂³	—	2	64	no	—
350	C₂³	—	2	24	no	—
351	C₂³	—	2	24	no	—
352	C₂³	—	2	24	no	—
353	C₂³	—	2	64	no	—
354	C₂³	—	2	24	no	—
355	C₂³	—	2	12	no	—
356	C₂³	—	2	12	no	—
357	C₂³	—	4	16	no	—
358	C₂³	—	4	192	no	—
359	C₂³	—	4	16	no	—
360	C₂³	—	4	16	no	—
361	C₂³	C₂×#41×C₅₃	1	48	yes	—
362	C₂³	C₂×#41×C₅₃	1	8	yes	—
363	C₂³	H(5,𝔽₂)×C₅₃	1	4	yes	—
364	C₂³	C₂⁵×C₅₃	1	4	yes	—
365	C₂³	C₂⁵×C₅₃	1	4	yes	—
366	C₂³	C₂⁵×C₅₃	1	8	yes	—
367	C₂³	C₂⁵×C₅₃	1	3	yes	—
368	Q₈	—	4	16	no	—
369	D₄	—	2	64	no	—
370	D₄	—	4	16	no	—
371	C₂×C₄	—	2	64	no	—
372	C₂×C₄	—	2	24	no	—
373	C₂×C₄	—	2	24	no	—
374	C₂×C₄	—	2	24	no	—
375	C₂×C₄	—	2	24	no	—
376	C₂×C₄	—	2	24	no	—
377	C₂×C₄	—	2	12	no	—
378	C₂×C₄	—	4	32	no	—
379	C₂×C₄	—	4	16	no	—
380	C₂×C₄	C₂⁵×C₅₃	1	8	yes	—
381	C₂×C₄	C₂⁵×C₅₃	1	8	yes	—
382	C₂×C₄	C₂⁵×C₅₃	1	8	yes	—
383	C₂×C₄	C₂⁵×C₅₃	1	4	yes	—
384	C₂×C₄	C₂⁵×C₅₃	1	8	yes	—
385	C₂×C₄	C₂⁵×C₅₃	1	8	yes	—
386	C₂²	—	2	96	no	—
387	C₂²	—	2	8	no	—
388	C₂²	—	2	8	no	—
389	C₂²	—	2	8	no	—
390	C₂²	—	2	8	no	—
391	C₂²	—	2	4	no	—
392	C₂²	—	2	4	no	—
393	C₂²	—	4	64	no	—
394	C₂²	—	4	24	no	—
395	C₂²	—	4	24	no	—
396	C₂²	C₅₃×#551	1	16	yes	direct product factor
397	C₂²	C₂²×#41×C₅₃	1	16	yes	—
398	C₂²	C₂²×#41×C₅₃	1	12	yes	—
399	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	6	yes	—
400	C₂²	C₂⁶×C₅₃	1	1	yes	characteristic
401	C₄	—	2	8	no	—
402	C₄	—	2	8	no	—
403	C₄	—	2	8	no	—
404	C₄	—	2	8	no	—
405	C₄	—	2	8	no	—
406	C₄	—	2	4	no	—
407	C₂	—	2	16	no	—
408	C₂	—	4	4	no	—
409	C₂	—	4	4	no	—
410	C₂	C₂×C₅₃×#551	1	12	yes	direct product factor
411	C₂	C₂³×#41×C₅₃	1	2	yes	—
412	C₂	C₂²×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₅₃
prime radical	C₂²×#120×C₅₃	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#551	C₂	direct (N × H)
2. C₂²×#126×C₅₃	C₂	semidirect
3. C₂²×#124×C₅₃	C₂	semidirect
4. C₂²×#123×C₅₃	C₂	semidirect
5. C₂²×#123×C₅₃	C₂	semidirect
6. C₂²×C₂²⋊Q₈×C₅₃	C₂	semidirect
8. C₂²×C₄×D₄×C₅₃	C₂	semidirect
9. C₂²×#117×C₅₃	C₂	semidirect
10. C₂³×C₂²⋊C₄×C₅₃	C₂	semidirect
12. C₂³×Q₈×C₅₃	C₂²	semidirect
17. C₂⁴×C₄×C₅₃	C₂²	semidirect
18. C₅₃×#551	C₂²	direct (N × H)
19. C₂×#126×C₅₃	C₂²	semidirect
20. C₂×#124×C₅₃	C₂²	semidirect
21. C₂×#123×C₅₃	C₂²	semidirect
22. C₂×#123×C₅₃	C₂²	semidirect
23. C₂×C₂²⋊Q₈×C₅₃	C₂²	semidirect
25. C₂×C₄×D₄×C₅₃	C₂²	semidirect
26. C₂×#117×C₅₃	C₂²	semidirect
27. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
28. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
29. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
30. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
37. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
38. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
39. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
40. C₂²×C₄²×C₅₃	C₂²	semidirect
41. C₂²×C₄²×C₅₃	C₂²	semidirect
49. C₂²×Q₈×C₅₃	C₂³	semidirect
65. C₂³×C₄×C₅₃	C₂³	semidirect
66. #126×C₅₃	C₂³	semidirect
67. #124×C₅₃	C₂³	semidirect
68. #123×C₅₃	C₂³	semidirect
69. #123×C₅₃	C₂³	semidirect
70. C₂²⋊Q₈×C₅₃	C₂³	semidirect
72. C₄×D₄×C₅₃	C₂³	semidirect
73. #117×C₅₃	C₂³	semidirect
74. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
75. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
76. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
77. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
84. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
85. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
86. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
87. C₂×C₄²×C₅₃	C₂³	semidirect
88. C₂×C₄²×C₅₃	C₂³	semidirect
105. C₂×Q₈×C₅₃	C₂⁴	semidirect
127. C₂²×C₄×C₅₃	C₂⁴	semidirect
128. C₄⋊C₄×C₅₃	C₂⁴	semidirect
129. C₄⋊C₄×C₅₃	C₂⁴	semidirect
130. C₄⋊C₄×C₅₃	C₂⁴	semidirect
131. C₄⋊C₄×C₅₃	C₂⁴	semidirect
138. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
139. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
140. C₄²×C₅₃	C₂⁴	semidirect
141. C₄²×C₅₃	C₂⁴	semidirect
179. C₂²×#551	C₅₃	direct (N × H)
190. C₂²×C₅₃	#551	direct (N × H)
201. C₂×#551	C₂×C₅₃	direct (N × H)
202. C₂²×#126	C₂×C₅₃	semidirect
203. C₂²×#124	C₂×C₅₃	semidirect
204. C₂²×#123	C₂×C₅₃	semidirect
205. C₂²×#123	C₂×C₅₃	semidirect
206. C₂²×C₂²⋊Q₈	C₂×C₅₃	semidirect
208. C₂²×C₄×D₄	C₂×C₅₃	semidirect
209. C₂²×#117	C₂×C₅₃	semidirect
210. C₂³×C₂²⋊C₄	C₂×C₅₃	semidirect
214. C₂×C₅₃	C₂×#551	direct (N × H)
218. C₂³×Q₈	C₂²×C₅₃	semidirect
223. C₂⁴×C₄	C₂²×C₅₃	semidirect
224. #551	C₂²×C₅₃	direct (N × H)
225. C₂×#126	C₂²×C₅₃	semidirect
226. C₂×#124	C₂²×C₅₃	semidirect
227. C₂×#123	C₂²×C₅₃	semidirect
228. C₂×#123	C₂²×C₅₃	semidirect
229. C₂×C₂²⋊Q₈	C₂²×C₅₃	semidirect
231. C₂×C₄×D₄	C₂²×C₅₃	semidirect
232. C₂×#117	C₂²×C₅₃	semidirect
233. C₂²×C₄⋊C₄	C₂²×C₅₃	semidirect
234. C₂²×C₄⋊C₄	C₂²×C₅₃	semidirect
235. C₂²×C₄⋊C₄	C₂²×C₅₃	semidirect
236. C₂²×C₄⋊C₄	C₂²×C₅₃	semidirect
243. C₂²×C₂²⋊C₄	C₂²×C₅₃	semidirect
244. C₂²×C₂²⋊C₄	C₂²×C₅₃	semidirect
245. C₂²×C₂²⋊C₄	C₂²×C₅₃	semidirect
246. C₂²×C₄²	C₂²×C₅₃	semidirect
247. C₂²×C₄²	C₂²×C₅₃	semidirect
248. C₅₃	C₂²×#551	direct (N × H)
256. C₂²×Q₈	C₂³×C₅₃	semidirect
272. C₂³×C₄	C₂³×C₅₃	semidirect
273. #126	C₂³×C₅₃	semidirect
274. #124	C₂³×C₅₃	semidirect
275. #123	C₂³×C₅₃	semidirect
276. #123	C₂³×C₅₃	semidirect
277. C₂²⋊Q₈	C₂³×C₅₃	semidirect
279. C₄×D₄	C₂³×C₅₃	semidirect
280. #117	C₂³×C₅₃	semidirect
281. C₂×C₄⋊C₄	C₂³×C₅₃	semidirect
282. C₂×C₄⋊C₄	C₂³×C₅₃	semidirect
283. C₂×C₄⋊C₄	C₂³×C₅₃	semidirect
284. C₂×C₄⋊C₄	C₂³×C₅₃	semidirect
291. C₂×C₂²⋊C₄	C₂³×C₅₃	semidirect
292. C₂×C₂²⋊C₄	C₂³×C₅₃	semidirect
293. C₂×C₂²⋊C₄	C₂³×C₅₃	semidirect
294. C₂×C₄²	C₂³×C₅₃	semidirect
295. C₂×C₄²	C₂³×C₅₃	semidirect
312. C₂×Q₈	C₂⁴×C₅₃	semidirect
334. C₂²×C₄	C₂⁴×C₅₃	semidirect
335. C₄⋊C₄	C₂⁴×C₅₃	semidirect
336. C₄⋊C₄	C₂⁴×C₅₃	semidirect
337. C₄⋊C₄	C₂⁴×C₅₃	semidirect
338. C₄⋊C₄	C₂⁴×C₅₃	semidirect
345. C₂²⋊C₄	C₂⁴×C₅₃	semidirect
346. C₂²⋊C₄	C₂⁴×C₅₃	semidirect
347. C₄²	C₂⁴×C₅₃	semidirect
348. C₄²	C₂⁴×C₅₃	semidirect
396. C₂²	C₅₃×#551	direct (N × H)
410. C₂	C₂×C₅₃×#551	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×C₅₃×#551	2
C₂²×#126×C₅₃	2
C₂²×#124×C₅₃	2
C₂²×#123×C₅₃	2
C₂²×C₂²⋊Q₈×C₅₃	2
C₂²×#120×C₅₃	2
C₂²×C₄×D₄×C₅₃	2
C₂²×#117×C₅₃	2
C₂³×C₂²⋊C₄×C₅₃	2
C₂²×#551	53

Subgroups — C22×C53×#551 #27270215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₃×#551 #27270215