Subgroups — #38751672

Order 25216.

412 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15691357	C₂	1	1	yes	maximal, characteristic
2	#27161208	C₂	1	1	yes	maximal, characteristic, complemented
3	#27161207	C₂	1	1	yes	maximal, characteristic, complemented
4	#27161155	C₂	1	1	yes	maximal, characteristic, complemented
5	#27161154	C₂	1	1	yes	maximal, characteristic, complemented
6	#27161151	C₂	1	1	yes	maximal, characteristic, complemented
7	#27161150	C₂	1	1	yes	maximal, characteristic, complemented
8	#27161147	C₂	1	1	yes	maximal, characteristic, complemented
9	#27161131	C₂	1	1	yes	maximal, characteristic, complemented
10	#27161129	C₂	1	1	yes	maximal, characteristic, complemented
11	#27161129	C₂	1	1	yes	maximal, characteristic, complemented
12	#27161031	C₂	1	1	yes	maximal, characteristic, complemented
13	M₁₆×Dic₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₈×Dic₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
15	#435×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#15691357	C₄	1	1	yes	characteristic
17	C₂×#15691342	C₂²	1	1	yes	characteristic
18	C₂×#15691341	C₂²	1	1	yes	characteristic
19	C₂×#15691341	C₂²	1	1	yes	characteristic
20	C₂²×#13096802	C₂²	1	1	yes	characteristic
21	C₄×Dic₃₉₄	C₂²	1	1	yes	characteristic
22	#20263291	C₄	1	1	yes	characteristic
23	D₄×Dic₁₉₇	C₄	1	1	yes	characteristic
24	C₄×C₂²⋊D₁₉₇	C₂²	1	1	yes	characteristic
25	C₄×C₂²⋊D₁₉₇	C₂²	1	1	yes	characteristic
26	#20263286	C₄	1	1	yes	characteristic
27	#20263286	C₄	1	1	yes	characteristic
28	#20263284	C₄	1	1	yes	characteristic
29	#20263282	C₄	1	1	yes	characteristic
30	#20263282	C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
33	C₄²×D₁₉₇	C₂²	1	1	yes	characteristic
34	C₄⋊#13096802	C₂²	1	1	yes	characteristic
35	#20263255	C₂²	1	1	yes	characteristic
36	#20263255	C₂²	1	1	yes	characteristic
37	#20263255	C₂²	1	1	yes	characteristic
38	#20263255	C₂²	1	1	yes	characteristic
39	C₂²⋊#13096802	C₂²	1	1	yes	characteristic
40	C₂²⋊#13096802	C₂²	1	1	yes	characteristic
41	#20263238	C₂²	1	1	yes	characteristic
42	#20263238	C₂²	1	1	yes	characteristic
43	#20263237	C₂²	1	1	yes	characteristic
44	#20263236	C₂²	1	1	yes	characteristic
45	#20263236	C₂²	1	1	yes	characteristic
46	C₈×Dic₁₉₇	C₂²	1	1	yes	characteristic
47	C₈×Dic₁₉₇	C₂²	1	1	yes	characteristic
48	C₈×Dic₁₉₇	C₂²	1	1	yes	characteristic
49	C₈×Dic₁₉₇	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₉₇	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₉₇	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₁₉₇	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₁₉₇	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₁₉₇	C₂²	1	1	yes	characteristic
57	C₂²⋊C₈×C₁₉₇	C₂²	1	1	yes	characteristic
58	#97×C₁₉₇	C₂²	1	1	yes	characteristic
59	#15691357	—	2	4	no	—
60	C₂×Dic₃₉₄	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₉₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₉₇	C₂×C₄	1	1	yes	characteristic
63	C₂²×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
64	C₂²×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
65	C₂²×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
66	C₂²×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×D₁₉₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×D₁₉₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×D₁₉₇	C₂³	1	1	yes	characteristic
70	#15691342	—	2	2	no	—
71	#15691341	—	2	2	no	—
72	#15691341	—	2	2	no	—
73	C₂×#13096802	—	2	2	no	—
74	C₂×#13096802	C₂³	1	1	yes	characteristic
75	C₂×#13096802	C₂³	1	1	yes	characteristic
76	C₂×#13096802	C₂³	1	1	yes	characteristic
77	C₂×#13096802	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
79	#15691337	C₂×C₄	1	1	yes	characteristic
80	#15691337	C₂×C₄	1	1	yes	characteristic
81	#15691337	C₂×C₄	1	1	yes	characteristic
82	#15691337	C₂×C₄	1	1	yes	characteristic
83	C₂²⋊Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
85	#15691335	C₂×C₄	1	1	yes	characteristic
86	#15691335	C₂×C₄	1	1	yes	characteristic
87	C₄×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
88	C₄×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
89	C₄×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
90	C₄×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
91	C₄×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
92	C₄×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₁₉₇	C₂³	1	1	yes	characteristic
94	C₄×Dic₁₉₇	C₂³	1	1	yes	characteristic
95	C₄×Dic₁₉₇	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
99	M₁₆×C₁₉₇	—	2	2	no	—
100	C₂×C₈×C₁₉₇	—	2	2	no	—
101	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
106	C₂²⋊C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
107	C₂²⋊C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
108	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
109	C₂²×D₁₉₇	C₂²×C₄	1	1	yes	characteristic
110	Dic₃₉₄	—	2	2	no	—
111	C₂²⋊D₁₉₇	—	2	2	no	—
112	C₂²⋊D₁₉₇	—	2	2	no	—
113	C₂×Dic₁₉₇	—	2	2	no	—
114	C₂×Dic₁₉₇	—	2	2	no	—
115	C₂×Dic₁₉₇	—	2	2	no	—
116	C₂×Dic₁₉₇	M₁₆	1	4	yes	complemented
117	C₂×Dic₁₉₇	C₂²×C₄	1	1	yes	characteristic
118	C₂×Dic₁₉₇	C₂²×C₄	1	1	yes	characteristic
119	C₂×Dic₁₉₇	C₂²×C₄	1	1	yes	characteristic
120	C₂×Dic₁₉₇	C₂²×C₄	1	1	yes	characteristic
121	C₂×Dic₁₉₇	C₂²×C₄	1	1	yes	characteristic
122	C₂×Dic₁₉₇	C₂²×C₄	1	1	yes	characteristic
123	C₂×Dic₁₉₇	C₂²×C₄	1	1	yes	characteristic
124	C₄×D₁₉₇	—	2	1	no	—
125	C₄×D₁₉₇	—	2	1	no	—
126	C₄×D₁₉₇	—	2	2	no	—
127	C₄×D₁₉₇	—	2	2	no	—
128	#13096802	—	2	1	no	—
129	#13096802	—	2	1	no	—
130	#13096802	—	2	1	no	—
131	#13096802	#41	1	2	yes	complemented
132	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
133	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
134	D₄×C₁₉₇	—	2	2	no	—
135	C₂×C₄×C₁₉₇	—	2	1	no	—
136	C₂×C₄×C₁₉₇	—	2	1	no	—
137	C₂×C₄×C₁₉₇	—	2	1	no	—
138	C₂×C₄×C₁₉₇	—	2	1	no	—
139	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
142	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
143	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
144	C₈×C₁₉₇	—	2	1	no	—
145	C₈×C₁₉₇	—	2	1	no	—
146	C₈×C₁₉₇	—	2	1	no	—
147	C₈×C₁₉₇	#41	1	2	yes	complemented
148	C₂²×C₁₉₇	—	2	1	no	—
149	C₂²×C₁₉₇	—	2	1	no	—
150	C₂²×C₁₉₇	—	2	1	no	—
151	C₂²×C₁₉₇	—	2	1	no	—
152	C₂²×C₁₉₇	—	2	1	no	—
153	C₂²×C₁₉₇	C₂×M₁₆	1	2	yes	—
154	C₂²×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
155	C₂×D₁₉₇	—	2	1	no	—
156	C₂×D₁₉₇	—	2	1	no	—
157	C₂×D₁₉₇	—	2	1	no	—
158	C₄×C₁₉₇	—	2	1	no	—
159	C₄×C₁₉₇	—	2	1	no	—
160	C₄×C₁₉₇	—	2	1	no	—
161	C₄×C₁₉₇	—	2	1	no	—
162	C₄×C₁₉₇	C₂×#41	1	1	yes	characteristic
163	C₄×C₁₉₇	C₂×#41	1	1	yes	characteristic
164	Dic₁₉₇	—	2	1	no	—
165	Dic₁₉₇	—	2	1	no	—
166	Dic₁₉₇	—	2	1	no	—
167	Dic₁₉₇	—	2	1	no	—
168	Dic₁₉₇	C₂×M₁₆	1	2	yes	—
169	Dic₁₉₇	C₂×M₁₆	1	2	yes	—
170	Dic₁₉₇	#131	1	2	yes	—
171	C₂×C₁₉₇	—	2	2	no	—
172	C₂×C₁₉₇	—	4	1	no	—
173	C₂×C₁₉₇	C₄×#41	1	1	yes	characteristic
174	C₂×C₁₉₇	C₂²×M₁₆	1	1	yes	characteristic
175	C₂×C₁₉₇	C₂×#131	1	1	yes	characteristic
176	D₁₉₇	—	4	1	no	—
177	C₁₉₇	#2965	1	1	yes	characteristic, complemented, Sylow
178	#2965	—	197	1	no	maximal, Sylow
179	C₄×#41	—	197	1	no	—
180	#446	—	197	1	no	—
181	#446	—	197	1	no	—
182	#436	—	197	1	no	—
183	#436	—	197	1	no	—
184	#435	—	197	1	no	—
185	#435	—	197	1	no	—
186	#435	—	197	1	no	—
187	#435	D₁₉₇	1	1	yes	characteristic, complemented
188	#431	—	197	1	no	—
189	#431	—	197	1	no	—
190	#431	—	197	1	no	—
191	C₄×M₁₆	—	197	1	no	—
192	C₄×M₁₆	—	197	1	no	—
193	C₂×C₄×C₈	—	197	1	no	—
194	C₂×#41	—	197	1	no	—
195	C₂×M₁₆	—	197	1	no	—
196	C₂×M₁₆	—	197	1	no	—
197	C₂×M₁₆	—	197	1	no	—
198	C₂×M₁₆	C₂×D₁₉₇	1	1	yes	characteristic
199	C₂²×C₈	—	197	1	no	—
200	C₂²×C₈	C₂×D₁₉₇	1	1	yes	characteristic
201	C₄×Q₈	—	197	1	no	—
202	C₄×Q₈	—	197	1	no	—
203	C₄×D₄	—	197	1	no	—
204	C₄×D₄	—	197	1	no	—
205	C₄×D₄	—	197	1	no	—
206	C₄×D₄	—	197	1	no	—
207	C₄×D₄	—	197	1	no	—
208	C₄×D₄	C₂×D₁₉₇	1	1	yes	characteristic
209	#117	—	197	1	no	—
210	#117	—	197	1	no	—
211	#117	—	197	1	no	—
212	C₂×C₄²	—	197	1	no	—
213	C₂×C₄²	—	197	1	no	—
214	C₂×C₄²	—	197	1	no	—
215	C₄⋊C₈	—	197	1	no	—
216	C₄⋊C₈	—	197	1	no	—
217	C₄⋊C₈	—	197	1	no	—
218	C₄⋊C₈	—	197	1	no	—
219	C₄⋊C₈	—	197	1	no	—
220	C₄⋊C₈	C₂×D₁₉₇	1	1	yes	characteristic
221	C₂²⋊C₈	—	197	1	no	—
222	C₂²⋊C₈	—	197	1	no	—
223	C₂²⋊C₈	—	197	1	no	—
224	C₂²⋊C₈	—	197	1	no	—
225	C₂²⋊C₈	C₂×D₁₉₇	1	1	yes	characteristic
226	C₂²⋊C₈	C₂×D₁₉₇	1	1	yes	characteristic
227	#97	—	197	1	no	—
228	#97	—	197	1	no	—
229	#97	—	197	1	no	—
230	#97	C₂×D₁₉₇	1	1	yes	characteristic
231	C₄×C₈	—	197	1	no	—
232	C₄×C₈	—	197	1	no	—
233	C₄×C₈	—	197	1	no	—
234	C₄×C₈	—	197	1	no	—
235	C₄×C₈	—	197	1	no	—
236	C₄×C₈	—	197	1	no	—
237	#41	—	394	4	no	—
238	C₂×Q₈	—	197	1	no	—
239	C₂×D₄	—	197	1	no	—
240	C₂×D₄	—	197	1	no	—
241	C₂×D₄	C₄×D₁₉₇	1	1	yes	characteristic
242	C₂²×C₄	—	197	1	no	—
243	C₂²×C₄	—	197	1	no	—
244	C₂²×C₄	—	197	1	no	—
245	C₂²×C₄	—	197	1	no	—
246	C₂²×C₄	—	197	1	no	—
247	C₂²×C₄	—	197	1	no	—
248	C₂²×C₄	—	197	1	no	—
249	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
250	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
251	M₁₆	—	2	2	no	—
252	M₁₆	—	394	2	no	—
253	M₁₆	—	394	2	no	—
254	M₁₆	—	394	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	394	2	no	—
257	C₂×C₈	—	197	1	no	—
258	C₂×C₈	—	197	1	no	—
259	C₂×C₈	—	197	1	no	—
260	C₂×C₈	—	197	1	no	—
261	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
265	C₄⋊C₄	—	197	1	no	—
266	C₄⋊C₄	—	197	1	no	—
267	C₄⋊C₄	—	197	1	no	—
268	C₄⋊C₄	—	197	1	no	—
269	C₄⋊C₄	—	197	1	no	—
270	C₄⋊C₄	C₄×D₁₉₇	1	1	yes	characteristic
271	C₂²⋊C₄	—	197	1	no	—
272	C₂²⋊C₄	—	197	1	no	—
273	C₂²⋊C₄	—	197	1	no	—
274	C₂²⋊C₄	—	197	1	no	—
275	C₂²⋊C₄	C₄×D₁₉₇	1	1	yes	characteristic
276	C₂²⋊C₄	C₄×D₁₉₇	1	1	yes	characteristic
277	C₄²	—	197	1	no	—
278	C₄²	—	197	1	no	—
279	C₄²	—	197	1	no	—
280	C₄²	—	197	1	no	—
281	C₄²	—	197	1	no	—
282	C₄²	—	197	1	no	—
283	C₄²	—	197	1	no	—
284	C₄²	—	197	1	no	—
285	C₄²	—	197	1	no	—
286	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
287	C₂³	—	197	1	no	—
288	C₂³	C₂×C₄×D₁₉₇	1	1	yes	characteristic
289	C₂³	C₂×C₄×D₁₉₇	1	1	yes	characteristic
290	Q₈	—	394	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	394	2	no	—
293	D₄	—	394	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	394	2	no	—
299	C₂×C₄	—	394	2	no	—
300	C₂×C₄	—	394	2	no	—
301	C₂×C₄	—	394	2	no	—
302	C₂×C₄	—	394	2	no	—
303	C₂×C₄	—	394	1	no	—
304	C₂×C₄	—	394	1	no	—
305	C₂×C₄	—	197	1	no	—
306	C₂×C₄	—	197	1	no	—
307	C₂×C₄	—	197	1	no	—
308	C₂×C₄	—	197	1	no	—
309	C₂×C₄	—	197	4	no	—
310	C₂×C₄	—	197	1	no	—
311	C₂×C₄	—	197	1	no	—
312	C₂×C₄	—	197	1	no	—
313	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
316	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
317	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	394	1	no	—
322	C₈	—	394	1	no	—
323	C₈	—	394	1	no	—
324	C₈	—	197	2	no	—
325	C₈	#15691357	1	2	yes	complemented
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	394	1	no	—
332	C₂²	—	394	1	no	—
333	C₂²	—	394	1	no	—
334	C₂²	M₁₆×D₁₉₇	1	2	yes	—
335	C₂²	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	394	1	no	—
341	C₄	—	394	1	no	—
342	C₄	—	394	1	no	—
343	C₄	—	394	1	no	—
344	C₄	—	197	2	no	—
345	C₄	—	197	2	no	—
346	C₄	—	197	2	no	—
347	C₄	#41×D₁₉₇	1	1	yes	characteristic
348	C₄	C₂×#15691357	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	4	1	no	—
351	C₂	—	788	1	no	—
352	C₂	C₄×#15691357	1	1	yes	characteristic
353	C₂	C₂×M₁₆×D₁₉₇	1	1	yes	characteristic
354	C₂	#131×D₁₉₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉₇
commutator, socle	C₂²×C₁₉₇	C₂³×C₄
Fitting	#435×C₁₉₇	C₂
prime radical	C₂×#15691357	C₄
cosocle	C₂×C₄×C₁₉₇	C₂⁴
hypercenter	#435	D₁₉₇
hypocenter	C₁₉₇	#2965
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #435

Complements

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

N	H ≅ G/N	type
2. #27161208	C₂	semidirect
3. #27161207	C₂	semidirect
4. #27161155	C₂	semidirect
5. #27161154	C₂	semidirect
6. #27161151	C₂	semidirect
7. #27161150	C₂	semidirect
8. #27161147	C₂	semidirect
9. #27161131	C₂	semidirect
10. #27161129	C₂	semidirect
11. #27161129	C₂	semidirect
12. #27161031	C₂	semidirect
13. M₁₆×Dic₁₉₇	C₂	semidirect
14. C₂×C₈×Dic₁₉₇	C₂	semidirect
15. #435×C₁₉₇	C₂	semidirect
116. C₂×Dic₁₉₇	M₁₆	semidirect
131. #13096802	#41	semidirect
147. C₈×C₁₉₇	#41	semidirect
177. C₁₉₇	#2965	semidirect
187. #435	D₁₉₇	semidirect
325. C₈	#15691357	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15691357	2
#27161208	2
#27161207	2
#27161155	2
#27161154	2
#27161151	2
#27161150	2
#27161147	2
#27161131	2
#27161129	2
#27161031	2
M₁₆×Dic₁₉₇	2
C₂×C₈×Dic₁₉₇	2
#435×C₁₉₇	2
#2965	197