Subgroups — C₂²×#16811135 #28290750

Order 13568.

14334 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719945	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₄×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16811135	C₂	1	48	yes	maximal, direct product factor
4	C₂²×#13133201	C₂	1	2	yes	maximal, complemented
5	C₂²×#13133197	C₂	1	2	yes	maximal, complemented
6	C₂²×#13133192	C₂	1	2	yes	maximal, complemented
7	C₂²×#13133187	C₂	1	2	yes	maximal, complemented
8	C₂²×C₄×D₂₁₂	C₂	1	4	yes	maximal, complemented
9	C₂²×#117×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
11	C₂²×#11719945	—	2	16	no	—
12	C₂²×#11719945	C₂²	1	24	yes	complemented
13	C₂³×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
14	C₂³×D₂₁₂	C₂²	1	4	yes	—
15	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
16	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	—
17	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
18	C₂³×C₄×D₅₃	—	2	16	no	—
19	C₂³×C₄×D₅₃	C₂²	1	2	yes	—
20	C₂³×C₄×D₅₃	C₂²	1	2	yes	—
21	C₂³×C₄×D₅₃	C₂²	1	24	yes	complemented
22	C₂³×C₄×D₅₃	C₂²	1	2	yes	complemented
23	#16811135	C₂²	1	256	yes	direct product factor
24	C₂×#13133201	C₂²	1	48	yes	complemented
25	C₂×#13133197	C₂²	1	48	yes	complemented
26	C₂×#13133192	C₂²	1	48	yes	complemented
27	C₂×#13133187	C₂²	1	48	yes	complemented
28	C₂×C₄×D₂₁₂	C₂²	1	96	yes	complemented
29	C₂²×C₄⋊Dic₅₃	C₂²	1	4	yes	complemented
30	C₂²×#11719921	C₂²	1	4	yes	—
31	C₂²×#11719921	C₂²	1	4	yes	complemented
32	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×#117×C₅₃	C₂²	1	24	yes	complemented
34	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
35	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
36	C₂²×C₄²×C₅₃	C₂²	1	2	yes	complemented
37	C₂⁴×D₅₃	—	2	8	no	—
38	C₂⁴×D₅₃	—	2	16	no	—
39	C₂⁴×D₅₃	C₂³	1	2	yes	—
40	C₂⁴×D₅₃	C₂³	1	12	yes	—
41	C₂⁴×D₅₃	C₂³	1	2	yes	—
42	C₂×#11719945	—	2	192	no	—
43	C₂×#11719945	C₂³	1	64	yes	complemented
44	C₂²×Dic₁₀₆	—	2	8	no	—
45	C₂²×Dic₁₀₆	C₂³	1	12	yes	complemented
46	C₂²×D₂₁₂	—	2	32	no	—
47	C₂²×D₂₁₂	—	2	8	no	—
48	C₂²×D₂₁₂	C₂³	1	48	yes	—
49	C₂²×D₂₁₂	C₂³	1	12	yes	complemented
50	C₂²×C₂²⋊D₅₃	—	2	16	no	—
51	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	—
52	C₂³×Dic₅₃	—	2	8	no	—
53	C₂³×Dic₅₃	C₂³	1	2	yes	—
54	C₂³×Dic₅₃	C₂³	1	2	yes	—
55	C₂³×Dic₅₃	C₂³	1	12	yes	—
56	C₂²×C₄×D₅₃	—	2	16	no	—
57	C₂²×C₄×D₅₃	—	2	16	no	—
58	C₂²×C₄×D₅₃	—	2	16	no	—
59	C₂²×C₄×D₅₃	—	2	192	no	—
60	C₂²×C₄×D₅₃	C₂³	1	24	yes	—
61	C₂²×C₄×D₅₃	C₂³	1	24	yes	—
62	C₂²×C₄×D₅₃	C₂³	1	64	yes	complemented
63	C₂²×C₄×D₅₃	C₂³	1	24	yes	complemented
64	#13133201	C₂³	1	128	yes	complemented
65	#13133197	C₂³	1	128	yes	complemented
66	#13133192	C₂³	1	128	yes	complemented
67	#13133187	C₂³	1	128	yes	complemented
68	C₄×D₂₁₂	C₂³	1	256	yes	complemented
69	C₂×C₄⋊Dic₅₃	C₂³	1	48	yes	complemented
70	C₂×#11719921	C₂³	1	48	yes	—
71	C₂×#11719921	C₂³	1	48	yes	complemented
72	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₅₃	D₄	1	16	yes	complemented
74	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
75	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
78	#117×C₅₃	C₂³	1	64	yes	complemented
79	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	complemented
80	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	—
81	C₂×C₄²×C₅₃	C₂³	1	24	yes	complemented
82	C₂⁴×C₅₃	—	2	8	no	—
83	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
84	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
85	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
86	C₂³×D₅₃	—	2	128	no	—
87	C₂³×D₅₃	—	2	48	no	—
88	C₂³×D₅₃	—	2	8	no	—
89	C₂³×D₅₃	—	2	8	no	—
90	C₂³×D₅₃	—	2	96	no	—
91	C₂³×D₅₃	—	2	8	no	—
92	C₂³×D₅₃	—	2	16	no	—
93	C₂³×D₅₃	#41	1	16	yes	complemented
94	C₂³×D₅₃	C₂⁴	1	12	yes	—
95	C₂³×D₅₃	C₂⁴	1	16	yes	—
96	C₂³×D₅₃	C₂⁴	1	12	yes	—
97	C₂²×C₄×C₅₃	—	2	16	no	—
98	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
99	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
100	C₂²×C₄×C₅₃	C₂×D₄	1	96	yes	complemented
101	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
102	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
103	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
104	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
105	C₂×C₂²⋊D₅₃	—	2	96	no	—
106	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
107	C₂²×Dic₅₃	—	2	48	no	—
108	C₂²×Dic₅₃	—	2	8	no	—
109	C₂²×Dic₅₃	—	2	8	no	—
110	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
111	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
112	C₂²×Dic₅₃	C₂⁴	1	16	yes	—
113	#11719945	—	2	256	no	—
114	C₂×D₂₁₂	—	2	192	no	—
115	C₂×D₂₁₂	—	2	48	no	—
116	C₂×D₂₁₂	C₂⁴	1	64	yes	—
117	C₂×D₂₁₂	C₂⁴	1	16	yes	complemented
118	C₂×C₄×D₅₃	—	2	96	no	—
119	C₂×C₄×D₅₃	—	2	96	no	—
120	C₂×C₄×D₅₃	—	2	96	no	—
121	C₂×C₄×D₅₃	—	2	256	no	—
122	C₂×C₄×D₅₃	C₂⁴	1	32	yes	—
123	C₂×C₄×D₅₃	C₂⁴	1	32	yes	—
124	C₂×C₄×D₅₃	C₂⁴	1	32	yes	complemented
125	C₂×Dic₁₀₆	—	2	48	no	—
126	C₂×Dic₁₀₆	C₂⁴	1	16	yes	complemented
127	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
128	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	—
129	C₄²×C₅₃	C₂⁴	1	32	yes	complemented
130	#11719921	C₂⁴	1	64	yes	—
131	#11719921	C₂⁴	1	64	yes	complemented
132	C₄⋊Dic₅₃	C₂⁴	1	64	yes	complemented
133	C₂³×C₅₃	—	2	32	no	—
134	C₂³×C₅₃	—	2	24	no	—
135	C₂³×C₅₃	C₂²×D₄	1	24	yes	—
136	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
137	C₂³×C₅₃	C₂×#41	1	8	yes	—
138	C₂³×C₅₃	C₂⁵	1	4	yes	—
139	C₂³×C₅₃	C₂⁵	1	3	yes	—
140	C₂²×D₅₃	—	2	384	no	—
141	C₂²×D₅₃	—	2	64	no	—
142	C₂²×D₅₃	—	2	32	no	—
143	C₂²×D₅₃	—	2	24	no	—
144	C₂²×D₅₃	—	2	24	no	—
145	C₂²×D₅₃	—	2	64	no	—
146	C₂²×D₅₃	—	2	24	no	—
147	C₂²×D₅₃	—	2	48	no	—
148	C₂²×D₅₃	—	4	32	no	—
149	C₂²×D₅₃	C₂×#41	1	48	yes	complemented
150	C₂²×D₅₃	C₂⁵	1	8	yes	—
151	C₂²×D₅₃	C₂⁵	1	8	yes	—
152	C₂×C₄×C₅₃	—	2	48	no	—
153	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
154	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
155	C₂×C₄×C₅₃	C₂²×D₄	1	64	yes	complemented
156	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
157	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
158	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
159	C₂²⋊D₅₃	—	2	64	no	—
160	C₂×Dic₅₃	—	2	24	no	—
161	C₂×Dic₅₃	—	2	32	no	—
162	C₂×Dic₅₃	—	2	24	no	—
163	C₂×Dic₅₃	C₂⁵	1	8	yes	—
164	C₂×Dic₅₃	C₂⁵	1	8	yes	—
165	D₂₁₂	—	2	128	no	—
166	D₂₁₂	—	2	32	no	—
167	C₄×D₅₃	—	2	64	no	—
168	C₄×D₅₃	—	2	64	no	—
169	C₄×D₅₃	—	2	64	no	—
170	Dic₁₀₆	—	2	32	no	—
171	C₂²×#525	—	53	1	no	maximal, Sylow
172	C₂²×C₅₃	—	2	8	no	—
173	C₂²×C₅₃	—	2	48	no	—
174	C₂²×C₅₃	#525	1	16	yes	complemented
175	C₂²×C₅₃	C₂³×D₄	1	8	yes	—
176	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
177	C₂²×C₅₃	C₂²×#41	1	12	yes	—
178	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
179	C₂×D₅₃	—	2	128	no	—
180	C₂×D₅₃	—	2	8	no	—
181	C₂×D₅₃	—	2	8	no	—
182	C₂×D₅₃	—	2	96	no	—
183	C₂×D₅₃	—	2	8	no	—
184	C₂×D₅₃	—	2	16	no	—
185	C₂×D₅₃	—	4	48	no	—
186	C₂×D₅₃	C₂²×#41	1	16	yes	complemented
187	C₄×C₅₃	—	2	16	no	—
188	C₄×C₅₃	C₂³×D₄	1	8	yes	—
189	C₄×C₅₃	C₂³×D₄	1	8	yes	—
190	Dic₅₃	—	2	8	no	—
191	Dic₅₃	—	2	8	no	—
192	C₂³×#41	—	53	1	no	—
193	C₂⁵×C₄	—	53	1	no	—
194	C₂×#525	—	53	48	no	—
195	C₂²×#123	—	53	2	no	—
196	C₂²×C₂²⋊Q₈	—	53	2	no	—
197	C₂²×#121	—	53	2	no	—
198	C₂²×#120	—	53	2	no	—
199	C₂²×C₄×D₄	—	53	4	no	—
200	C₂²×#117	D₅₃	1	1	yes	characteristic, complemented
201	C₂×C₅₃	—	2	8	no	—
202	C₂×C₅₃	C₂×#525	1	12	yes	complemented
203	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
204	C₂×C₅₃	C₂³×#41	1	2	yes	—
205	D₅₃	—	2	16	no	—
206	D₅₃	—	4	8	no	—
207	C₂⁶	—	53	1	no	—
208	C₂²×#41	—	53	24	no	—
209	C₂²×#41	—	106	16	no	—
210	C₂³×Q₈	—	53	1	no	—
211	C₂³×D₄	—	53	2	no	—
212	C₂³×D₄	—	53	1	no	—
213	C₂³×D₄	—	53	4	no	—
214	C₂⁴×C₄	—	53	1	no	—
215	C₂⁴×C₄	—	53	2	no	—
216	C₂⁴×C₄	—	53	24	no	—
217	C₂⁴×C₄	—	53	2	no	—
218	C₂⁴×C₄	—	53	2	no	—
219	C₂⁴×C₄	—	106	16	no	—
220	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
221	#525	—	53	256	no	—
222	C₂×#123	—	53	48	no	—
223	C₂×C₂²⋊Q₈	—	53	48	no	—
224	C₂×#121	—	53	48	no	—
225	C₂×#120	—	53	48	no	—
226	C₂×C₄×D₄	—	53	96	no	—
227	C₂×#117	C₂×D₅₃	1	24	yes	complemented
228	C₂²×C₄⋊C₄	—	53	4	no	—
229	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	complemented
230	C₂²×C₂²⋊C₄	—	53	4	no	—
231	C₂²×C₂²⋊C₄	—	53	4	no	—
232	C₂²×C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
233	C₂²×C₄²	C₂×D₅₃	1	2	yes	complemented
234	C₅₃	C₂²×#525	1	1	yes	characteristic, complemented, Sylow
235	C₂⁵	—	53	2	no	—
236	C₂⁵	—	53	2	no	—
237	C₂⁵	—	53	12	no	—
238	C₂⁵	—	106	16	no	—
239	C₂⁵	—	106	8	no	—
240	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
241	C₂×#41	—	53	64	no	—
242	C₂×#41	—	106	192	no	—
243	C₂²×Q₈	—	53	12	no	—
244	C₂²×Q₈	—	106	8	no	—
245	C₂²×D₄	—	53	24	no	—
246	C₂²×D₄	—	53	48	no	—
247	C₂²×D₄	—	53	12	no	—
248	C₂²×D₄	—	106	32	no	—
249	C₂²×D₄	—	106	16	no	—
250	C₂²×D₄	—	106	8	no	—
251	C₂³×C₄	—	53	2	no	—
252	C₂³×C₄	—	53	2	no	—
253	C₂³×C₄	—	53	12	no	—
254	C₂³×C₄	—	53	24	no	—
255	C₂³×C₄	—	53	24	no	—
256	C₂³×C₄	—	53	24	no	—
257	C₂³×C₄	—	53	64	no	—
258	C₂³×C₄	—	106	8	no	—
259	C₂³×C₄	—	106	16	no	—
260	C₂³×C₄	—	106	16	no	—
261	C₂³×C₄	—	106	16	no	—
262	C₂³×C₄	—	106	192	no	—
263	C₂³×C₄	D₂₁₂	1	16	yes	complemented
264	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
265	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
266	C₂³×C₄	C₂²×D₅₃	1	12	yes	—
267	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
268	#123	—	53	128	no	—
269	C₂²⋊Q₈	—	53	128	no	—
270	#121	—	53	128	no	—
271	#120	—	53	128	no	—
272	C₄×D₄	—	53	256	no	—
273	#117	C₂²×D₅₃	1	64	yes	complemented
274	C₂×C₄⋊C₄	—	53	48	no	—
275	C₂×C₄⋊C₄	C₂²×D₅₃	1	24	yes	complemented
276	C₂×C₂²⋊C₄	—	53	48	no	—
277	C₂×C₂²⋊C₄	—	53	48	no	—
278	C₂×C₂²⋊C₄	C₂²×D₅₃	1	24	yes	—
279	C₂×C₄²	C₂²×D₅₃	1	24	yes	complemented
280	C₂⁴	—	2	8	no	—
281	C₂⁴	—	53	16	no	—
282	C₂⁴	—	53	12	no	—
283	C₂⁴	—	53	12	no	—
284	C₂⁴	—	53	16	no	—
285	C₂⁴	—	106	96	no	—
286	C₂⁴	—	106	128	no	—
287	C₂⁴	—	106	48	no	—
288	C₂⁴	—	106	8	no	—
289	C₂⁴	—	106	8	no	—
290	C₂⁴	—	106	8	no	—
291	C₂⁴	—	106	16	no	—
292	C₂⁴	C₂×D₂₁₂	1	8	yes	—
293	C₂⁴	C₂³×D₅₃	1	6	yes	—
294	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
295	#41	—	106	256	no	—
296	C₂×Q₈	—	53	16	no	—
297	C₂×Q₈	—	106	48	no	—
298	C₂×D₄	—	53	32	no	—
299	C₂×D₄	—	53	64	no	—
300	C₂×D₄	—	53	16	no	—
301	C₂×D₄	—	106	192	no	—
302	C₂×D₄	—	106	96	no	—
303	C₂×D₄	—	106	48	no	—
304	C₂²×C₄	—	2	16	no	—
305	C₂²×C₄	—	53	12	no	—
306	C₂²×C₄	—	53	12	no	—
307	C₂²×C₄	—	53	16	no	—
308	C₂²×C₄	—	53	32	no	—
309	C₂²×C₄	—	53	32	no	—
310	C₂²×C₄	—	53	32	no	—
311	C₂²×C₄	—	106	48	no	—
312	C₂²×C₄	—	106	8	no	—
313	C₂²×C₄	—	106	8	no	—
314	C₂²×C₄	—	106	96	no	—
315	C₂²×C₄	—	106	96	no	—
316	C₂²×C₄	—	106	96	no	—
317	C₂²×C₄	—	106	256	no	—
318	C₂²×C₄	C₂×D₂₁₂	1	8	yes	—
319	C₂²×C₄	C₂×D₂₁₂	1	8	yes	—
320	C₂²×C₄	C₂×D₂₁₂	1	96	yes	complemented
321	C₂²×C₄	C₂³×D₅₃	1	24	yes	—
322	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
323	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
324	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
325	C₄⋊C₄	—	53	64	no	—
326	C₄⋊C₄	C₂³×D₅₃	1	32	yes	complemented
327	C₂²⋊C₄	—	53	64	no	—
328	C₂²⋊C₄	—	53	64	no	—
329	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	—
330	C₄²	C₂³×D₅₃	1	32	yes	complemented
331	C₂³	—	2	32	no	—
332	C₂³	—	2	24	no	—
333	C₂³	—	53	48	no	—
334	C₂³	—	53	8	no	—
335	C₂³	—	53	8	no	—
336	C₂³	—	106	64	no	—
337	C₂³	—	106	384	no	—
338	C₂³	—	106	64	no	—
339	C₂³	—	106	32	no	—
340	C₂³	—	106	24	no	—
341	C₂³	—	106	24	no	—
342	C₂³	—	106	24	no	—
343	C₂³	—	106	48	no	—
344	C₂³	—	212	32	no	—
345	C₂³	C₂²×D₂₁₂	1	24	yes	—
346	C₂³	C₂²×D₂₁₂	1	4	yes	—
347	C₂³	#41×D₅₃	1	8	yes	—
348	C₂³	C₂⁴×D₅₃	1	4	yes	—
349	C₂³	C₂⁴×D₅₃	1	3	yes	—
350	Q₈	—	106	32	no	—
351	D₄	—	106	128	no	—
352	D₄	—	106	64	no	—
353	D₄	—	106	32	no	—
354	C₂×C₄	—	2	48	no	—
355	C₂×C₄	—	53	8	no	—
356	C₂×C₄	—	53	8	no	—
357	C₂×C₄	—	106	32	no	—
358	C₂×C₄	—	106	24	no	—
359	C₂×C₄	—	106	24	no	—
360	C₂×C₄	—	106	64	no	—
361	C₂×C₄	—	106	64	no	—
362	C₂×C₄	—	106	64	no	—
363	C₂×C₄	C₂²×D₂₁₂	1	24	yes	—
364	C₂×C₄	C₂²×D₂₁₂	1	24	yes	—
365	C₂×C₄	C₂²×D₂₁₂	1	64	yes	complemented
366	C₂×C₄	C₂⁴×D₅₃	1	16	yes	—
367	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
368	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
369	C₂²	—	2	48	no	—
370	C₂²	—	2	8	no	—
371	C₂²	—	53	16	no	—
372	C₂²	—	106	128	no	—
373	C₂²	—	106	96	no	—
374	C₂²	—	106	8	no	—
375	C₂²	—	106	8	no	—
376	C₂²	—	106	8	no	—
377	C₂²	—	106	16	no	—
378	C₂²	—	212	48	no	—
379	C₂²	#16811135	1	16	yes	direct product factor
380	C₂²	C₂³×D₂₁₂	1	8	yes	—
381	C₂²	C₂³×D₂₁₂	1	6	yes	—
382	C₂²	C₂×#41×D₅₃	1	12	yes	—
383	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
384	C₄	—	2	16	no	—
385	C₄	—	106	8	no	—
386	C₄	—	106	8	no	—
387	C₄	C₂³×D₂₁₂	1	8	yes	—
388	C₄	C₂³×D₂₁₂	1	8	yes	—
389	C₂	—	2	8	no	—
390	C₂	—	106	16	no	—
391	C₂	—	212	8	no	—
392	C₂	C₂×#16811135	1	12	yes	direct product factor
393	C₂	C₂⁴×D₂₁₂	1	1	yes	characteristic
394	C₂	C₂²×#41×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×#117×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂²×#117	D₅₃
hypocenter	C₅₃	C₂²×#525
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
1. C₂³×#11719945	C₂	semidirect
2. C₂⁴×C₄×D₅₃	C₂	semidirect
3. C₂×#16811135	C₂	direct (N × H)
4. C₂²×#13133201	C₂	semidirect
5. C₂²×#13133197	C₂	semidirect
6. C₂²×#13133192	C₂	semidirect
7. C₂²×#13133187	C₂	semidirect
8. C₂²×C₄×D₂₁₂	C₂	semidirect
9. C₂²×#117×C₅₃	C₂	semidirect
12. C₂²×#11719945	C₂²	semidirect
13. C₂³×Dic₁₀₆	C₂²	semidirect
15. C₂³×D₂₁₂	C₂²	semidirect
21. C₂³×C₄×D₅₃	C₂²	semidirect
22. C₂³×C₄×D₅₃	C₂²	semidirect
23. #16811135	C₂²	direct (N × H)
24. C₂×#13133201	C₂²	semidirect
25. C₂×#13133197	C₂²	semidirect
26. C₂×#13133192	C₂²	semidirect
27. C₂×#13133187	C₂²	semidirect
28. C₂×C₄×D₂₁₂	C₂²	semidirect
29. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
31. C₂²×#11719921	C₂²	semidirect
33. C₂×#117×C₅₃	C₂²	semidirect
34. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
36. C₂²×C₄²×C₅₃	C₂²	semidirect
43. C₂×#11719945	C₂³	semidirect
45. C₂²×Dic₁₀₆	C₂³	semidirect
49. C₂²×D₂₁₂	C₂³	semidirect
62. C₂²×C₄×D₅₃	C₂³	semidirect
63. C₂²×C₄×D₅₃	C₂³	semidirect
64. #13133201	C₂³	semidirect
65. #13133197	C₂³	semidirect
66. #13133192	C₂³	semidirect
67. #13133187	C₂³	semidirect
68. C₄×D₂₁₂	C₂³	semidirect
69. C₂×C₄⋊Dic₅₃	C₂³	semidirect
71. C₂×#11719921	C₂³	semidirect
73. C₂³×C₄×C₅₃	D₄	semidirect
78. #117×C₅₃	C₂³	semidirect
79. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
81. C₂×C₄²×C₅₃	C₂³	semidirect
93. C₂³×D₅₃	#41	semidirect
100. C₂²×C₄×C₅₃	C₂×D₄	semidirect
117. C₂×D₂₁₂	C₂⁴	semidirect
124. C₂×C₄×D₅₃	C₂⁴	semidirect
126. C₂×Dic₁₀₆	C₂⁴	semidirect
127. C₄⋊C₄×C₅₃	C₂⁴	semidirect
129. C₄²×C₅₃	C₂⁴	semidirect
131. #11719921	C₂⁴	semidirect
132. C₄⋊Dic₅₃	C₂⁴	semidirect
149. C₂²×D₅₃	C₂×#41	semidirect
155. C₂×C₄×C₅₃	C₂²×D₄	semidirect
174. C₂²×C₅₃	#525	semidirect
186. C₂×D₅₃	C₂²×#41	semidirect
200. C₂²×#117	D₅₃	semidirect
202. C₂×C₅₃	C₂×#525	semidirect
227. C₂×#117	C₂×D₅₃	semidirect
229. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
233. C₂²×C₄²	C₂×D₅₃	semidirect
234. C₅₃	C₂²×#525	semidirect
263. C₂³×C₄	D₂₁₂	semidirect
273. #117	C₂²×D₅₃	semidirect
275. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
279. C₂×C₄²	C₂²×D₅₃	semidirect
320. C₂²×C₄	C₂×D₂₁₂	semidirect
326. C₄⋊C₄	C₂³×D₅₃	semidirect
330. C₄²	C₂³×D₅₃	semidirect
365. C₂×C₄	C₂²×D₂₁₂	semidirect
379. C₂²	#16811135	direct (N × H)
392. C₂	C₂×#16811135	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11719945	2
C₂⁴×C₄×D₅₃	2
C₂×#16811135	2
C₂²×#13133201	2
C₂²×#13133197	2
C₂²×#13133192	2
C₂²×#13133187	2
C₂²×C₄×D₂₁₂	2
C₂²×#117×C₅₃	2
C₂²×#525	53

Subgroups — C22×#16811135 #28290750

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811135 #28290750