Subgroups — C₄⋊C₄×#11719917 #27382086

Order 13568.

1242 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810647	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11719917	C₂	1	2	yes	maximal
3	C₂×C₄×#11719917	C₂	1	1	yes	maximal, characteristic
4	#20274590	C₂	1	4	yes	maximal, complemented
5	#20274584	C₂	1	2	yes	maximal, complemented
6	C₄⋊C₄×#88322	C₂	1	2	yes	maximal, complemented
7	#20274409	C₂	1	2	yes	maximal
8	C₂×C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11719917	C₂²	1	2	yes	—
10	C₂²×#11719917	C₂²	1	1	yes	characteristic
11	C₂²×#11719917	C₂²	1	1	yes	characteristic
12	#16810647	C₄	1	8	yes	complemented
13	C₂×C₄⋊#88322	C₂²	1	2	yes	—
14	C₂×C₄⋊#88322	C₂²	1	2	yes	—
15	C₄×#11719917	—	2	8	no	—
16	C₄×#11719917	C₄	1	8	yes	complemented
17	C₂×#13133128	C₂²	1	4	yes	—
18	C₂×#13133128	C₂²	1	2	yes	—
19	C₂×C₄×#88322	C₂²	1	4	yes	—
20	C₂×C₄×#88322	C₂²	1	2	yes	—
21	#16810378	C₂²	1	4	yes	—
22	#16810378	C₂²	1	4	yes	—
23	C₂²×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
24	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
25	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
27	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
28	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
30	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
31	C₂×#95×C₅₃	C₄	1	2	yes	—
32	C₂×#11719917	—	2	4	no	—
33	C₂×#11719917	—	2	2	no	—
34	C₂×#11719917	—	2	4	no	—
35	C₂×#11719917	—	2	2	no	—
36	C₂×#11719917	C₂×C₄	1	4	yes	—
37	C₂×#11719917	C₂×C₄	1	8	yes	—
38	C₂×#11719917	C₂×C₄	1	4	yes	—
39	C₂×#11719917	D₄	1	4	yes	—
40	C₂×#11719917	Q₈	1	4	yes	—
41	C₂²×#88322	C₂³	1	2	yes	—
42	C₂²×#88322	C₂³	1	4	yes	—
43	C₂²×#88322	C₂³	1	2	yes	—
44	C₄⋊#88322	C₂×C₄	1	8	yes	complemented
45	C₄⋊#88322	C₂×C₄	1	8	yes	complemented
46	#13133128	—	2	8	no	—
47	#13133128	C₂×C₄	1	8	yes	complemented
48	C₄×#88322	—	2	8	no	—
49	C₄×#88322	C₂×C₄	1	8	yes	complemented
50	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
51	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
55	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
57	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₅₃	—	2	4	no	—
59	C₂×C₄²×C₅₃	—	2	4	no	—
60	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
61	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
62	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
63	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
66	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
67	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	#95×C₅₃	C₂×C₄	1	2	yes	—
70	#95×C₅₃	C₂×C₄	1	4	yes	—
71	#95×C₅₃	C₂×C₄	1	2	yes	—
72	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	—	2	4	no	—
74	C₂²×C₄×C₅₃	—	2	8	no	—
75	C₂²×C₄×C₅₃	—	2	8	no	—
76	C₂²×C₄×C₅₃	—	4	2	no	—
77	C₂²×C₄×C₅₃	—	4	4	no	—
78	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
79	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
80	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
81	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
85	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
91	C₂²×C₄×C₅₃	#41	1	2	yes	—
92	C₂²×C₄×C₅₃	#41	1	2	yes	—
93	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₅₃	—	2	8	no	—
95	C₄⋊C₄×C₅₃	—	2	4	no	—
96	C₄⋊C₄×C₅₃	M₁₆	1	8	yes	complemented
97	C₄²×C₅₃	—	2	8	no	—
98	C₄²×C₅₃	—	2	2	no	—
99	C₄²×C₅₃	—	2	2	no	—
100	C₄²×C₅₃	—	2	2	no	—
101	C₄²×C₅₃	—	2	2	no	—
102	C₄²×C₅₃	C₄²	1	4	yes	—
103	C₄²×C₅₃	C₄²	1	4	yes	—
104	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
105	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
106	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
107	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
108	#11719917	—	2	8	no	—
109	#11719917	—	2	16	no	—
110	#11719917	C₄⋊C₄	1	16	yes	direct product factor
111	C₂×#88322	—	2	4	no	—
112	C₂×#88322	—	2	2	no	—
113	C₂×#88322	—	2	2	no	—
114	C₂×#88322	—	2	4	no	—
115	C₂×#88322	C₂²×C₄	1	4	yes	—
116	C₂×#88322	C₂²×C₄	1	8	yes	—
117	C₂×#88322	C₂²×C₄	1	4	yes	—
118	C₂×#88322	C₂×D₄	1	4	yes	—
119	C₂×#88322	C₂×Q₈	1	4	yes	—
120	C₂³×C₅₃	—	2	2	no	—
121	C₂³×C₅₃	—	2	2	no	—
122	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
123	C₂³×C₅₃	C₄×D₄	1	2	yes	—
124	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
125	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	—	2	4	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	4	no	—
131	C₂×C₄×C₅₃	—	2	8	no	—
132	C₂×C₄×C₅₃	—	2	4	no	—
133	C₂×C₄×C₅₃	—	2	4	no	—
134	C₂×C₄×C₅₃	—	2	4	no	—
135	C₂×C₄×C₅₃	—	4	4	no	—
136	C₂×C₄×C₅₃	—	4	16	no	—
137	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
138	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
139	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
140	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
141	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
142	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
143	C₂×C₄×C₅₃	#117	1	4	yes	—
144	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
145	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
146	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
147	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
148	C₂×C₄×C₅₃	C₂×M₁₆	1	8	yes	—
149	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
150	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
156	#88322	—	2	4	no	—
157	#88322	—	2	8	no	—
158	#88322	C₂×C₄⋊C₄	1	8	yes	complemented
159	C₄⋊C₄×M₁₆	—	53	1	no	maximal, Sylow
160	C₂²×C₅₃	—	2	2	no	—
161	C₂²×C₅₃	—	2	2	no	—
162	C₂²×C₅₃	—	2	4	no	—
163	C₂²×C₅₃	—	2	4	no	—
164	C₂²×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
165	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
166	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
168	C₂²×C₅₃	C₂²×M₁₆	1	2	yes	—
169	C₂²×C₅₃	#568	1	2	yes	—
170	C₄×C₅₃	—	2	2	no	—
171	C₄×C₅₃	—	2	2	no	—
172	C₄×C₅₃	—	2	2	no	—
173	C₄×C₅₃	—	2	8	no	—
174	C₄×C₅₃	—	4	4	no	—
175	C₄×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
176	C₄×C₅₃	C₄×M₁₆	1	4	yes	complemented
177	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
178	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
179	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
180	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
181	C₂×#423	—	53	1	no	—
182	C₂×C₄×M₁₆	—	53	1	no	—
183	C₂×C₄×M₁₆	—	53	2	no	—
184	C₂×C₄×C₄⋊C₄	D₅₃	1	1	yes	characteristic
185	#1771	—	53	4	no	—
186	#1770	—	53	2	no	—
187	C₈×C₄⋊C₄	—	53	2	no	—
188	#1728	—	53	2	no	—
189	C₂×C₅₃	—	2	4	no	—
190	C₂×C₅₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
191	C₂×C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic
192	C₂×C₅₃	#2873	1	1	yes	characteristic
193	C₂×C₅₃	D₄×M₁₆	1	2	yes	—
194	C₂×C₅₃	Q₈×M₁₆	1	2	yes	—
195	C₂²×M₁₆	—	53	1	no	—
196	C₂²×M₁₆	—	53	2	no	—
197	C₂²×M₁₆	—	53	1	no	—
198	C₂²×C₄⋊C₄	Dic₅₃	1	2	yes	—
199	C₂²×C₄²	C₂×D₅₃	1	2	yes	—
200	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
201	#423	—	53	8	no	—
202	C₂×C₄⋊C₈	—	53	2	no	—
203	C₂×C₄⋊C₈	—	53	2	no	—
204	C₄×M₁₆	—	53	8	no	—
205	C₄×M₁₆	—	106	8	no	—
206	C₂×#97	—	53	2	no	—
207	C₂×#97	—	53	4	no	—
208	C₂×C₄×C₈	—	53	2	no	—
209	C₂×C₄×C₈	—	53	4	no	—
210	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	—
211	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	—
212	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
213	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
214	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
215	C₂×#95	Dic₅₃	1	2	yes	—
216	#336	—	53	4	no	—
217	#336	—	53	4	no	—
218	C₅₃	C₄⋊C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
219	C₂³×C₄	C₂×Dic₅₃	1	4	yes	—
220	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
221	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
222	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
223	C₂×M₁₆	—	53	4	no	—
224	C₂×M₁₆	—	53	4	no	—
225	C₂×M₁₆	—	53	4	no	—
226	C₂×M₁₆	—	53	4	no	—
227	C₂×M₁₆	—	53	8	no	—
228	C₂×M₁₆	—	106	4	no	—
229	C₂×M₁₆	—	106	2	no	—
230	C₂×M₁₆	—	106	2	no	—
231	C₂×M₁₆	—	106	4	no	—
232	C₂²×C₈	—	53	2	no	—
233	C₂²×C₈	—	53	2	no	—
234	C₂²×C₈	—	53	4	no	—
235	C₂×C₄⋊C₄	—	2	8	no	—
236	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
237	C₂×C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
238	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
239	C₂×C₄²	—	2	4	no	—
240	C₂×C₄²	—	2	4	no	—
241	C₂×C₄²	C₄×D₅₃	1	4	yes	—
242	C₂×C₄²	C₄×D₅₃	1	4	yes	—
243	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
244	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
245	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
246	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
247	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
248	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
249	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
250	C₄⋊C₈	—	53	8	no	—
251	C₄⋊C₈	—	53	8	no	—
252	#97	—	53	8	no	—
253	#97	—	106	8	no	—
254	C₄×C₈	—	53	8	no	—
255	C₄×C₈	—	106	8	no	—
256	#95	C₂×Dic₅₃	1	2	yes	—
257	#95	C₂×Dic₅₃	1	4	yes	—
258	#95	C₂×Dic₅₃	1	2	yes	—
259	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
260	C₂²×C₄	—	2	4	no	—
261	C₂²×C₄	—	2	8	no	—
262	C₂²×C₄	—	2	8	no	—
263	C₂²×C₄	—	4	2	no	—
264	C₂²×C₄	—	4	4	no	—
265	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
266	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
267	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
268	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
269	C₂²×C₄	D₄×D₅₃	1	2	yes	—
270	C₂²×C₄	#11719947	1	2	yes	—
271	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
272	C₂²×C₄	#11719949	1	2	yes	—
273	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
274	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
275	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
276	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
277	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
278	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
279	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
280	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
281	M₁₆	—	53	16	no	—
282	M₁₆	—	106	8	no	—
283	M₁₆	—	106	16	no	—
284	C₂×C₈	—	53	4	no	—
285	C₂×C₈	—	53	4	no	—
286	C₂×C₈	—	53	4	no	—
287	C₂×C₈	—	53	4	no	—
288	C₂×C₈	—	53	8	no	—
289	C₂×C₈	—	106	4	no	—
290	C₂×C₈	—	106	2	no	—
291	C₂×C₈	—	106	2	no	—
292	C₂×C₈	—	106	4	no	—
293	C₄⋊C₄	—	2	8	no	—
294	C₄⋊C₄	—	2	4	no	—
295	C₄⋊C₄	#11719917	1	8	yes	direct product factor
296	C₄²	—	2	8	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	2	2	no	—
299	C₄²	—	2	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	C₄×Dic₅₃	1	4	yes	—
302	C₄²	C₄×Dic₅₃	1	4	yes	—
303	C₄²	C₂×C₄×D₅₃	1	2	yes	—
304	C₄²	C₂×C₄×D₅₃	1	2	yes	—
305	C₄²	C₂×C₄×D₅₃	1	2	yes	—
306	C₄²	C₂×C₄×D₅₃	1	2	yes	—
307	C₂³	—	2	2	no	—
308	C₂³	—	2	2	no	—
309	C₂³	C₂×C₄×Dic₅₃	1	2	yes	—
310	C₂³	D₄×Dic₅₃	1	2	yes	—
311	C₂³	Q₈×Dic₅₃	1	2	yes	—
312	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	8	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	4	4	no	—
323	C₂×C₄	—	4	16	no	—
324	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
325	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
326	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
327	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
328	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
329	C₂×C₄	C₄⋊C₄×D₅₃	1	4	yes	—
330	C₂×C₄	#13133175	1	4	yes	—
331	C₂×C₄	#13133179	1	4	yes	—
332	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
333	C₂×C₄	#13133182	1	4	yes	—
334	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
335	C₂×C₄	C₂×#11719917	1	8	yes	—
336	C₂×C₄	C₂×#11719917	1	4	yes	—
337	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
338	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
339	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
340	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
341	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
342	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
343	C₈	—	53	8	no	—
344	C₈	—	106	4	no	—
345	C₈	—	106	8	no	—
346	C₂²	—	2	4	no	—
347	C₂²	—	2	4	no	—
348	C₂²	—	2	2	no	—
349	C₂²	—	2	2	no	—
350	C₂²	C₄⋊C₄×Dic₅₃	1	4	yes	—
351	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
352	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
353	C₂²	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
354	C₂²	C₂²×#11719917	1	2	yes	—
355	C₂²	#16811390	1	2	yes	—
356	C₄	—	2	8	no	—
357	C₄	—	2	2	no	—
358	C₄	—	2	2	no	—
359	C₄	—	2	2	no	—
360	C₄	—	4	4	no	—
361	C₄	C₄⋊C₄×Dic₅₃	1	4	yes	—
362	C₄	C₄×#11719917	1	4	yes	complemented
363	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
364	C₄	C₂×#13133175	1	1	yes	characteristic
365	C₄	C₂×#13133179	1	1	yes	characteristic
366	C₄	C₂×#13133182	1	1	yes	characteristic
367	C₂	—	2	4	no	—
368	C₂	C₂×C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
369	C₂	C₂×C₄×#11719917	1	1	yes	characteristic
370	C₂	#20283134	1	1	yes	characteristic
371	C₂	D₄×#11719917	1	2	yes	—
372	C₂	Q₈×#11719917	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂²×C₄²
Fitting	C₂×C₄×C₄⋊C₄×C₅₃	C₂
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×C₄×C₄⋊C₄	D₅₃
hypocenter	C₅₃	C₄⋊C₄×M₁₆
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. #20274590	C₂	semidirect
5. #20274584	C₂	semidirect
6. C₄⋊C₄×#88322	C₂	semidirect
12. #16810647	C₄	semidirect
16. C₄×#11719917	C₄	semidirect
44. C₄⋊#88322	C₂×C₄	semidirect
45. C₄⋊#88322	C₂×C₄	semidirect
47. #13133128	C₂×C₄	semidirect
49. C₄×#88322	C₂×C₄	semidirect
96. C₄⋊C₄×C₅₃	M₁₆	semidirect
110. #11719917	C₄⋊C₄	direct (N × H)
158. #88322	C₂×C₄⋊C₄	semidirect
176. C₄×C₅₃	C₄×M₁₆	semidirect
218. C₅₃	C₄⋊C₄×M₁₆	semidirect
295. C₄⋊C₄	#11719917	direct (N × H)
362. C₄	C₄×#11719917	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810647	2
C₂×C₄×#11719917	2
#20274590	2
#20274584	2
C₄⋊C₄×#88322	2
#20274409	2
C₂×C₄×C₄⋊C₄×C₅₃	2
C₄⋊C₄×M₁₆	53

Subgroups — C4⋊C4×#11719917 #27382086

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊C₄×#11719917 #27382086