Subgroups — #38751682

Order 25216.

380 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27161609	C₂	1	1	yes	maximal, characteristic
2	#27161207	C₂	1	1	yes	maximal, characteristic, complemented
3	#27161203	C₂	1	1	yes	maximal, characteristic, complemented
4	#27161155	C₂	1	1	yes	maximal, characteristic, complemented
5	#27161150	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×#13096802	C₂	1	1	yes	maximal, characteristic, complemented
7	#27161142	C₂	1	1	yes	maximal, characteristic, complemented
8	#27161142	C₂	1	1	yes	maximal, characteristic, complemented
9	#27161139	C₂	1	1	yes	maximal, characteristic, complemented
10	#27161137	C₂	1	1	yes	maximal, characteristic, complemented
11	#27161133	C₂	1	1	yes	maximal, characteristic, complemented
12	#27161037	C₂	1	1	yes	maximal, characteristic, complemented
13	#27161036	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×#15691341	C₂²	1	1	yes	characteristic
17	C₂²×#13096802	C₂²	1	1	yes	characteristic
18	C₂²×#13096802	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₉₇	C₂²	1	1	yes	characteristic
20	#20263321	C₄	1	1	yes	characteristic
21	#20263319	C₄	1	1	yes	characteristic
22	#20263316	C₄	1	1	yes	characteristic
23	#20263312	C₄	1	1	yes	characteristic
24	#20263310	C₄	1	1	yes	characteristic
25	#20263307	C₄	1	1	yes	characteristic
26	#20263303	C₄	1	1	yes	characteristic
27	#20263298	C₄	1	1	yes	characteristic
28	C₄×Dic₃₉₄	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊D₁₉₇	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊D₁₉₇	C₂²	1	1	yes	characteristic
31	#20263285	C₂²	1	1	yes	characteristic
32	#20263283	C₂²	1	1	yes	characteristic
33	C₂×C₄×Dic₁₉₇	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	#20263236	C₂²	1	1	yes	characteristic
44	#20263236	C₂²	1	1	yes	characteristic
45	#20263236	C₂²	1	1	yes	characteristic
46	C₄×#13096802	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	C₂×Dic₃₉₄	C₂×C₄	1	1	yes	characteristic
60	C₂×C₂²⋊D₁₉₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₉₇	C₂×C₄	1	1	yes	characteristic
62	C₂²×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
63	C₂²×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₁₉₇	C₂³	1	1	yes	characteristic
65	#15691341	—	2	2	no	—
66	C₂×#13096802	—	2	2	no	—
67	C₂×#13096802	—	2	2	no	—
68	C₂×#13096802	C₂³	1	1	yes	characteristic
69	C₂×#13096802	C₂³	1	1	yes	characteristic
70	C₂×#13096802	C₂³	1	1	yes	characteristic
71	C₂×#13096802	C₂³	1	1	yes	characteristic
72	C₈×D₁₉₇	—	2	2	no	—
73	C₄⋊Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
74	C₄⋊Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
75	C₄⋊Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
76	#15691337	C₂×C₄	1	1	yes	characteristic
77	#15691337	C₂×C₄	1	1	yes	characteristic
78	#15691337	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	C₂²⋊Dic₁₉₇	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	C₂²⋊Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
86	#15691335	C₂×C₄	1	1	yes	characteristic
87	#15691335	C₂×C₄	1	1	yes	characteristic
88	#15691335	C₂×C₄	1	1	yes	characteristic
89	#15691335	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	C₂³	1	1	yes	characteristic
93	C₄×Dic₁₉₇	C₂³	1	1	yes	characteristic
94	C₄×Dic₁₉₇	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
98	M₁₆×C₁₉₇	—	2	2	no	—
99	C₂×C₈×C₁₉₇	—	2	2	no	—
100	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
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	C₄⋊C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
105	C₂²⋊C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
106	C₂²⋊C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
107	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
108	C₂²×D₁₉₇	C₂²×C₄	1	1	yes	characteristic
109	Dic₃₉₄	—	2	2	no	—
110	C₂²⋊D₁₉₇	—	2	2	no	—
111	C₂²⋊D₁₉₇	—	2	2	no	—
112	C₂×Dic₁₉₇	—	2	2	no	—
113	C₂×Dic₁₉₇	—	2	2	no	—
114	C₂×Dic₁₉₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×Dic₁₉₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×Dic₁₉₇	C₂²×C₄	1	1	yes	characteristic
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	C₄×D₁₉₇	—	2	1	no	—
122	C₄×D₁₉₇	—	2	1	no	—
123	#13096802	—	2	1	no	—
124	#13096802	—	2	1	no	—
125	#13096802	—	2	1	no	—
126	#13096802	#41	1	2	yes	—
127	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
128	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
129	D₄×C₁₉₇	—	2	2	no	—
130	C₂×C₄×C₁₉₇	—	2	1	no	—
131	C₂×C₄×C₁₉₇	—	2	1	no	—
132	C₂×C₄×C₁₉₇	—	2	1	no	—
133	C₂×C₄×C₁₉₇	—	2	1	no	—
134	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
139	C₈×C₁₉₇	—	2	1	no	—
140	C₈×C₁₉₇	—	2	1	no	—
141	C₈×C₁₉₇	—	2	1	no	—
142	C₈×C₁₉₇	#41	1	2	yes	—
143	C₂²×C₁₉₇	—	2	1	no	—
144	C₂²×C₁₉₇	—	2	1	no	—
145	C₂²×C₁₉₇	—	2	1	no	—
146	C₂²×C₁₉₇	—	2	1	no	—
147	C₂²×C₁₉₇	—	2	1	no	—
148	C₂²×C₁₉₇	#131	1	2	yes	—
149	C₂²×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
150	C₂×D₁₉₇	—	2	1	no	—
151	C₂×D₁₉₇	—	2	1	no	—
152	C₂×D₁₉₇	—	2	1	no	—
153	C₄×C₁₉₇	—	2	1	no	—
154	C₄×C₁₉₇	—	2	1	no	—
155	C₄×C₁₉₇	—	2	1	no	—
156	C₄×C₁₉₇	—	2	1	no	—
157	C₄×C₁₉₇	C₂×#41	1	1	yes	characteristic
158	C₄×C₁₉₇	C₂×#41	1	1	yes	characteristic
159	Dic₁₉₇	—	2	1	no	—
160	Dic₁₉₇	—	2	1	no	—
161	Dic₁₉₇	—	2	1	no	—
162	Dic₁₉₇	—	2	1	no	—
163	Dic₁₉₇	—	2	1	no	—
164	Dic₁₉₇	—	2	1	no	—
165	Dic₁₉₇	#131	1	2	yes	—
166	C₂×C₁₉₇	—	2	2	no	—
167	C₂×C₁₉₇	—	4	1	no	—
168	C₂×C₁₉₇	C₄×#41	1	1	yes	characteristic
169	C₂×C₁₉₇	C₂×#131	1	1	yes	characteristic
170	C₂×C₁₉₇	C₂×#131	1	1	yes	characteristic
171	D₁₉₇	—	4	1	no	—
172	C₁₉₇	#2966	1	1	yes	characteristic, complemented, Sylow
173	#2966	—	197	1	no	maximal, Sylow
174	#529	—	197	1	no	—
175	#446	—	197	1	no	—
176	C₈×Q₈	—	197	1	no	—
177	#436	—	197	1	no	—
178	#435	—	197	1	no	—
179	#435	D₁₉₇	1	1	yes	characteristic, complemented
180	C₈×D₄	—	197	1	no	—
181	C₈×D₄	—	197	1	no	—
182	C₈×D₄	—	197	1	no	—
183	#433	—	197	1	no	—
184	#433	—	197	1	no	—
185	#432	—	197	1	no	—
186	#405	—	197	1	no	—
187	#405	—	197	1	no	—
188	C₂×C₄×C₈	—	197	1	no	—
189	C₂×M₁₆	—	197	1	no	—
190	C₂×M₁₆	C₂×D₁₉₇	1	1	yes	characteristic
191	C₂²×C₈	—	197	1	no	—
192	C₂²×C₈	—	197	1	no	—
193	C₂²×C₈	—	197	1	no	—
194	C₂²×C₈	C₂×D₁₉₇	1	1	yes	characteristic
195	#126	—	197	1	no	—
196	#126	—	197	1	no	—
197	#125	—	197	1	no	—
198	#124	—	197	1	no	—
199	#123	—	197	1	no	—
200	#123	—	197	1	no	—
201	C₂²⋊Q₈	—	197	1	no	—
202	#121	—	197	1	no	—
203	C₄×Q₈	—	197	1	no	—
204	C₄×D₄	—	197	1	no	—
205	C₄×D₄	—	197	1	no	—
206	C₄×D₄	C₂×D₁₉₇	1	1	yes	characteristic
207	#117	—	197	1	no	—
208	#117	—	197	1	no	—
209	C₂×C₄²	—	197	1	no	—
210	C₄⋊C₈	—	197	1	no	—
211	C₄⋊C₈	—	197	1	no	—
212	C₄⋊C₈	—	197	1	no	—
213	C₄⋊C₈	—	197	1	no	—
214	C₄⋊C₈	—	197	1	no	—
215	C₄⋊C₈	C₂×D₁₉₇	1	1	yes	characteristic
216	C₂²⋊C₈	—	197	1	no	—
217	C₂²⋊C₈	—	197	1	no	—
218	C₂²⋊C₈	—	197	1	no	—
219	C₂²⋊C₈	—	197	1	no	—
220	C₂²⋊C₈	C₂×D₁₉₇	1	1	yes	characteristic
221	C₂²⋊C₈	C₂×D₁₉₇	1	1	yes	characteristic
222	#97	—	197	1	no	—
223	#97	—	197	1	no	—
224	#97	—	197	1	no	—
225	#97	C₂×D₁₉₇	1	1	yes	characteristic
226	C₄×C₈	—	197	1	no	—
227	C₄×C₈	—	197	1	no	—
228	C₄×C₈	—	197	1	no	—
229	C₄×C₈	—	197	1	no	—
230	C₄×C₈	—	197	1	no	—
231	C₄×C₈	—	197	1	no	—
232	C₂×Q₈	—	197	1	no	—
233	C₂×D₄	—	197	1	no	—
234	C₂×D₄	—	197	1	no	—
235	C₂×D₄	C₄×D₁₉₇	1	1	yes	characteristic
236	C₂²×C₄	—	197	1	no	—
237	C₂²×C₄	—	197	1	no	—
238	C₂²×C₄	—	197	1	no	—
239	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
240	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
241	M₁₆	—	2	2	no	—
242	M₁₆	—	394	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	394	2	no	—
245	C₂×C₈	—	394	2	no	—
246	C₂×C₈	—	394	2	no	—
247	C₂×C₈	—	197	1	no	—
248	C₂×C₈	—	197	1	no	—
249	C₂×C₈	—	197	1	no	—
250	C₂×C₈	—	197	1	no	—
251	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
253	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
254	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
255	C₄⋊C₄	—	197	1	no	—
256	C₄⋊C₄	—	197	1	no	—
257	C₄⋊C₄	—	197	1	no	—
258	C₄⋊C₄	—	197	1	no	—
259	C₄⋊C₄	—	197	1	no	—
260	C₄⋊C₄	—	197	1	no	—
261	C₄⋊C₄	—	197	1	no	—
262	C₄⋊C₄	—	197	1	no	—
263	C₄⋊C₄	—	197	1	no	—
264	C₄⋊C₄	C₄×D₁₉₇	1	1	yes	characteristic
265	C₂²⋊C₄	—	197	1	no	—
266	C₂²⋊C₄	—	197	1	no	—
267	C₂²⋊C₄	—	197	1	no	—
268	C₂²⋊C₄	—	197	1	no	—
269	C₂²⋊C₄	—	197	1	no	—
270	C₂²⋊C₄	—	197	1	no	—
271	C₂²⋊C₄	—	197	1	no	—
272	C₂²⋊C₄	—	197	1	no	—
273	C₂²⋊C₄	C₄×D₁₉₇	1	1	yes	characteristic
274	C₂²⋊C₄	C₄×D₁₉₇	1	1	yes	characteristic
275	C₄²	—	197	1	no	—
276	C₄²	—	197	1	no	—
277	C₄²	—	197	1	no	—
278	C₄²	—	197	1	no	—
279	C₄²	—	197	1	no	—
280	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
281	C₂³	—	197	1	no	—
282	C₂³	C₂×C₄×D₁₉₇	1	1	yes	characteristic
283	C₂³	C₂×C₄×D₁₉₇	1	1	yes	characteristic
284	Q₈	—	394	2	no	—
285	D₄	—	2	2	no	—
286	D₄	—	394	2	no	—
287	D₄	—	394	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	394	2	no	—
293	C₂×C₄	—	394	2	no	—
294	C₂×C₄	—	394	1	no	—
295	C₂×C₄	—	394	1	no	—
296	C₂×C₄	—	197	1	no	—
297	C₂×C₄	—	197	1	no	—
298	C₂×C₄	—	197	1	no	—
299	C₂×C₄	—	197	1	no	—
300	C₂×C₄	—	197	1	no	—
301	C₂×C₄	—	197	1	no	—
302	C₂×C₄	—	197	1	no	—
303	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
304	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
305	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
306	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	394	1	no	—
312	C₈	—	394	1	no	—
313	C₈	—	394	1	no	—
314	C₈	—	197	2	no	—
315	C₈	#15691357	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	394	1	no	—
322	C₂²	—	394	1	no	—
323	C₂²	—	394	1	no	—
324	C₂²	#20263334	1	2	yes	—
325	C₂²	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	394	1	no	—
331	C₄	—	394	1	no	—
332	C₄	—	394	1	no	—
333	C₄	—	394	1	no	—
334	C₄	—	394	1	no	—
335	C₄	—	394	1	no	—
336	C₄	—	197	2	no	—
337	C₄	#41×D₁₉₇	1	1	yes	characteristic
338	C₄	C₂×#15691357	1	1	yes	characteristic
339	C₂	—	2	2	no	—
340	C₂	—	4	1	no	—
341	C₂	—	788	1	no	—
342	C₂	C₄×#15691357	1	1	yes	characteristic
343	C₂	#131×D₁₉₇	1	1	yes	characteristic
344	C₂	C₂×#20263334	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	#20263298	C₄
cosocle	C₂×C₄×C₁₉₇	C₂⁴
hypercenter	#435	D₁₉₇
hypocenter	C₁₉₇	#2966
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. #27161207	C₂	semidirect
3. #27161203	C₂	semidirect
4. #27161155	C₂	semidirect
5. #27161150	C₂	semidirect
6. D₄×#13096802	C₂	semidirect
7. #27161142	C₂	semidirect
8. #27161142	C₂	semidirect
9. #27161139	C₂	semidirect
10. #27161137	C₂	semidirect
11. #27161133	C₂	semidirect
12. #27161037	C₂	semidirect
13. #27161036	C₂	semidirect
14. C₂×C₈×Dic₁₉₇	C₂	semidirect
15. #435×C₁₉₇	C₂	semidirect
172. C₁₉₇	#2966	semidirect
179. #435	D₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
#27161609	2
#27161207	2
#27161203	2
#27161155	2
#27161150	2
D₄×#13096802	2
#27161142	2
#27161139	2
#27161137	2
#27161133	2
#27161037	2
#27161036	2
C₂×C₈×Dic₁₉₇	2
#435×C₁₉₇	2
#2966	197