Subgroups — #38751930

Order 25216.

764 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20263346	C₂	1	2	yes	maximal, complemented
2	#27161590	C₂	1	1	yes	maximal, characteristic, complemented
3	#27161241	C₂	1	2	yes	maximal, complemented
4	#27161235	C₂	1	1	yes	maximal, characteristic, complemented
5	#27161230	C₂	1	1	yes	maximal, characteristic, complemented
6	#27161224	C₂	1	2	yes	maximal, complemented
7	#27161218	C₂	1	1	yes	maximal, characteristic, complemented
8	#27161213	C₂	1	1	yes	maximal, characteristic, complemented
9	#27161075	C₂	1	2	yes	maximal, complemented
10	C₂×C₂²⋊#13096802	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15691358	C₂²	1	1	yes	characteristic, complemented
13	#20263346	—	2	8	no	—
14	C₂×#15691348	C₂²	1	2	yes	complemented
15	C₂×#15691346	C₂²	1	2	yes	complemented
16	C₂×#15691345	C₂²	1	2	yes	complemented
17	C₂×#15691343	C₂²	1	2	yes	complemented
18	C₂²×#13096802	C₂²	1	2	yes	complemented
19	#20263309	—	2	1	no	—
20	#20263305	C₂²	1	1	yes	characteristic
21	#20263305	C₂²	1	1	yes	characteristic, complemented
22	#20263300	C₂²	1	1	yes	characteristic
23	#20263300	C₂²	1	1	yes	characteristic, complemented
24	#20263295	—	2	1	no	—
25	C₄×C₂²⋊D₁₉₇	—	2	1	no	—
26	C₄×C₂²⋊D₁₉₇	—	2	1	no	—
27	#20263285	C₂²	1	1	yes	characteristic, complemented
28	#20263250	C₂²	1	2	yes	complemented
29	#20263250	C₂²	1	2	yes	complemented
30	#20263247	C₂²	1	2	yes	complemented
31	#20263247	C₂²	1	2	yes	complemented
32	C₂²⋊#13096802	C₂²	1	1	yes	characteristic
33	C₂²⋊#13096802	C₂²	1	1	yes	characteristic
34	C₂²⋊#13096802	C₂²	1	2	yes	complemented
35	C₂×#41×C₁₉₇	—	2	8	no	—
36	C₂×#41×C₁₉₇	—	2	1	no	—
37	C₂×#41×C₁₉₇	—	2	1	no	—
38	C₂×#41×C₁₉₇	C₂²	1	2	yes	—
39	C₂×#41×C₁₉₇	C₂²	1	2	yes	—
40	C₂²×Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
41	C₂²×D₄×C₁₉₇	—	2	1	no	—
42	C₂²×D₄×C₁₉₇	C₂²	1	1	yes	characteristic
43	C₂³×C₄×C₁₉₇	—	2	1	no	—
44	C₂³×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
45	#15691358	—	2	4	no	—
46	C₂×Dic₃₉₄	C₂³	1	1	yes	characteristic
47	C₂×D₇₈₈	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₁₉₇	—	2	1	no	—
49	C₂×C₂²⋊D₁₉₇	—	2	1	no	—
50	C₂×C₂²⋊D₁₉₇	—	2	1	no	—
51	C₂×C₄×D₁₉₇	—	2	1	no	—
52	#15691348	—	2	4	no	—
53	#15691346	—	2	4	no	—
54	#15691345	—	2	4	no	—
55	#15691343	—	2	4	no	—
56	C₂×#13096802	—	2	4	no	—
57	C₂×#13096802	C₂³	1	2	yes	—
58	C₂×#13096802	C₂³	1	2	yes	—
59	C₄⋊Dic₁₉₇	C₂³	1	1	yes	characteristic
60	C₄⋊Dic₁₉₇	C₂³	1	1	yes	characteristic
61	#15691337	—	2	1	no	—
62	#15691337	—	2	1	no	—
63	C₂²⋊Dic₁₉₇	—	2	1	no	—
64	C₂²⋊Dic₁₉₇	—	2	1	no	—
65	C₂²⋊Dic₁₉₇	—	2	1	no	—
66	#15691335	—	2	1	no	—
67	#15691335	—	2	1	no	—
68	C₄×Dic₁₉₇	—	2	1	no	—
69	C₂⁴×C₁₉₇	—	2	1	no	—
70	C₂⁴×C₁₉₇	D₄	1	1	yes	characteristic
71	#41×C₁₉₇	—	2	8	no	—
72	#41×C₁₉₇	—	2	8	no	—
73	#41×C₁₉₇	—	2	4	no	—
74	#41×C₁₉₇	—	2	4	no	—
75	#41×C₁₉₇	—	2	4	no	—
76	#41×C₁₉₇	D₄	1	8	yes	complemented
77	C₂×Q₈×C₁₉₇	—	2	4	no	—
78	C₂×Q₈×C₁₉₇	—	2	1	no	—
79	C₂×Q₈×C₁₉₇	C₂³	1	2	yes	—
80	C₂×D₄×C₁₉₇	—	2	8	no	—
81	C₂×D₄×C₁₉₇	—	2	2	no	—
82	C₂×D₄×C₁₉₇	—	2	2	no	—
83	C₂×D₄×C₁₉₇	—	2	4	no	—
84	C₂×D₄×C₁₉₇	—	2	1	no	—
85	C₂×D₄×C₁₉₇	C₂³	1	2	yes	—
86	C₂²×C₄×C₁₉₇	—	2	8	no	—
87	C₂²×C₄×C₁₉₇	—	2	1	no	—
88	C₂²×C₄×C₁₉₇	—	2	2	no	—
89	C₂²×C₄×C₁₉₇	—	2	2	no	—
90	C₂²×C₄×C₁₉₇	—	2	4	no	—
91	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
94	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
97	C₂²×D₁₉₇	—	2	1	no	—
98	Dic₃₉₄	—	2	2	no	—
99	D₇₈₈	—	2	2	no	—
100	C₂²⋊D₁₉₇	—	4	2	no	—
101	C₂²⋊D₁₉₇	—	4	2	no	—
102	C₂²⋊D₁₉₇	—	4	2	no	—
103	C₂×Dic₁₉₇	—	2	1	no	—
104	C₂×Dic₁₉₇	—	2	1	no	—
105	C₂×Dic₁₉₇	—	2	1	no	—
106	C₄×D₁₉₇	—	4	2	no	—
107	#13096802	—	2	2	no	—
108	#13096802	—	2	2	no	—
109	C₂³×C₁₉₇	—	2	4	no	—
110	C₂³×C₁₉₇	—	2	2	no	—
111	C₂³×C₁₉₇	—	2	2	no	—
112	C₂³×C₁₉₇	—	2	2	no	—
113	C₂³×C₁₉₇	—	4	1	no	—
114	C₂³×C₁₉₇	—	4	2	no	—
115	C₂³×C₁₉₇	—	4	2	no	—
116	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
117	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
118	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
119	Q₈×C₁₉₇	—	2	4	no	—
120	Q₈×C₁₉₇	—	2	2	no	—
121	Q₈×C₁₉₇	C₂×D₄	1	4	yes	—
122	D₄×C₁₉₇	—	2	8	no	—
123	D₄×C₁₉₇	—	2	4	no	—
124	D₄×C₁₉₇	—	2	8	no	—
125	D₄×C₁₉₇	—	2	2	no	—
126	D₄×C₁₉₇	C₂×D₄	1	4	yes	—
127	C₂×C₄×C₁₉₇	—	2	4	no	—
128	C₂×C₄×C₁₉₇	—	2	8	no	—
129	C₂×C₄×C₁₉₇	—	2	2	no	—
130	C₂×C₄×C₁₉₇	—	2	2	no	—
131	C₂×C₄×C₁₉₇	—	2	2	no	—
132	C₂×C₄×C₁₉₇	—	2	8	no	—
133	C₂×C₄×C₁₉₇	—	2	2	no	—
134	C₂×C₄×C₁₉₇	—	2	2	no	—
135	C₂×C₄×C₁₉₇	—	2	2	no	—
136	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₉₇	C₂×D₄	1	4	yes	—
139	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
141	C₂²×C₁₉₇	—	2	1	no	—
142	C₂²×C₁₉₇	—	2	1	no	—
143	C₂²×C₁₉₇	—	2	1	no	—
144	C₂²×C₁₉₇	—	2	1	no	—
145	C₂²×C₁₉₇	—	2	1	no	—
146	C₂²×C₁₉₇	—	2	4	no	—
147	C₂²×C₁₉₇	—	2	1	no	—
148	C₂²×C₁₉₇	—	2	1	no	—
149	C₂²×C₁₉₇	—	4	4	no	—
150	C₂²×C₁₉₇	—	4	8	no	—
151	C₂²×C₁₉₇	—	4	2	no	—
152	C₂²×C₁₉₇	—	4	2	no	—
153	C₂²×C₁₉₇	#120	1	2	yes	—
154	C₂²×C₁₉₇	#135	1	2	yes	complemented
155	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₁₉₇	—	4	1	no	—
157	C₂×D₁₉₇	—	4	1	no	—
158	C₂×D₁₉₇	—	4	1	no	—
159	C₄×C₁₉₇	—	2	1	no	—
160	C₄×C₁₉₇	—	2	4	no	—
161	C₄×C₁₉₇	—	2	1	no	—
162	C₄×C₁₉₇	#120	1	2	yes	—
163	C₄×C₁₉₇	C₂²×D₄	1	2	yes	—
164	Dic₁₉₇	—	4	1	no	—
165	Dic₁₉₇	—	4	1	no	—
166	Dic₁₉₇	—	4	1	no	—
167	C₂×C₁₉₇	—	2	2	no	—
168	C₂×C₁₉₇	—	2	2	no	—
169	C₂×C₁₉₇	—	4	1	no	—
170	C₂×C₁₉₇	—	4	4	no	—
171	C₂×C₁₉₇	C₂×#120	1	1	yes	characteristic
172	C₂×C₁₉₇	C₂×#135	1	1	yes	characteristic
173	C₂×C₁₉₇	#575	1	1	yes	characteristic
174	D₁₉₇	—	8	1	no	—
175	C₁₉₇	#3002	1	1	yes	characteristic, complemented, Sylow
176	#3002	—	197	1	no	maximal, Sylow
177	C₂²×#41	D₁₉₇	1	1	yes	characteristic, complemented
178	C₂×#135	—	197	2	no	—
179	#525	—	197	1	no	—
180	#452	—	197	2	no	—
181	#451	—	197	1	no	—
182	#450	—	197	1	no	—
183	#449	—	197	2	no	—
184	#448	—	197	1	no	—
185	#447	—	197	1	no	—
186	#416	—	197	2	no	—
187	C₂×C₂²⋊C₈	—	197	1	no	—
188	C₂×#41	—	2	8	no	—
189	C₂×#41	—	2	1	no	—
190	C₂×#41	—	2	1	no	—
191	C₂×#41	—	197	1	no	—
192	C₂×#41	C₂×D₁₉₇	1	2	yes	—
193	C₂×#41	C₂×D₁₉₇	1	2	yes	—
194	C₂²×Q₈	C₂×D₁₉₇	1	1	yes	characteristic
195	C₂²×D₄	—	2	1	no	—
196	C₂²×D₄	C₂×D₁₉₇	1	1	yes	characteristic
197	C₂³×C₄	—	2	1	no	—
198	C₂³×C₄	C₂×D₁₉₇	1	1	yes	characteristic
199	#135	—	394	8	no	—
200	C₂×Q₁₆	—	197	2	no	—
201	C₂×SD₁₆	—	197	2	no	—
202	C₂×SD₁₆	—	197	2	no	—
203	C₂×D₈	—	197	2	no	—
204	C₂²×C₈	—	197	2	no	—
205	#123	—	394	1	no	—
206	C₂²⋊Q₈	—	197	1	no	—
207	C₂²⋊Q₈	—	197	1	no	—
208	#121	—	197	1	no	—
209	#121	—	197	1	no	—
210	#120	—	394	1	no	—
211	C₄×D₄	—	394	1	no	—
212	C₄×D₄	—	394	1	no	—
213	#117	—	197	1	no	—
214	#103	—	197	2	no	—
215	#103	—	197	2	no	—
216	#102	—	197	2	no	—
217	#102	—	197	2	no	—
218	C₂²⋊C₈	—	197	1	no	—
219	C₂²⋊C₈	—	197	1	no	—
220	C₂²⋊C₈	—	197	2	no	—
221	C₂⁴	—	2	1	no	—
222	C₂⁴	C₂²⋊D₁₉₇	1	1	yes	characteristic
223	#41	—	2	8	no	—
224	#41	—	2	8	no	—
225	#41	—	2	4	no	—
226	#41	—	2	4	no	—
227	#41	—	2	4	no	—
228	#41	—	394	4	no	—
229	#41	C₂²⋊D₁₉₇	1	8	yes	complemented
230	C₂×Q₈	—	2	4	no	—
231	C₂×Q₈	—	2	1	no	—
232	C₂×Q₈	—	197	1	no	—
233	C₂×Q₈	C₂²×D₁₉₇	1	2	yes	—
234	C₂×D₄	—	2	8	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	2	4	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	394	1	no	—
240	C₂×D₄	—	394	1	no	—
241	C₂×D₄	—	394	1	no	—
242	C₂×D₄	—	197	1	no	—
243	C₂×D₄	C₂²×D₁₉₇	1	2	yes	—
244	C₂²×C₄	—	2	8	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	394	1	no	—
250	C₂²×C₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
251	C₂²×C₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
252	C₂²×C₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
253	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
254	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
255	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
256	Q₁₆	—	394	4	no	—
257	SD₁₆	—	394	4	no	—
258	SD₁₆	—	394	4	no	—
259	D₈	—	394	4	no	—
260	C₂×C₈	—	394	4	no	—
261	C₂×C₈	—	197	2	no	—
262	C₂×C₈	—	197	2	no	—
263	C₄⋊C₄	—	394	1	no	—
264	C₄⋊C₄	—	394	1	no	—
265	C₄⋊C₄	—	197	1	no	—
266	C₄⋊C₄	—	197	1	no	—
267	C₂²⋊C₄	—	394	1	no	—
268	C₂²⋊C₄	—	394	1	no	—
269	C₂²⋊C₄	—	394	1	no	—
270	C₂²⋊C₄	—	394	1	no	—
271	C₂²⋊C₄	—	394	1	no	—
272	C₄²	—	394	1	no	—
273	C₂³	—	2	4	no	—
274	C₂³	—	2	2	no	—
275	C₂³	—	2	2	no	—
276	C₂³	—	2	2	no	—
277	C₂³	—	4	1	no	—
278	C₂³	—	4	2	no	—
279	C₂³	—	4	2	no	—
280	C₂³	—	394	1	no	—
281	C₂³	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
282	C₂³	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
283	C₂³	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
284	Q₈	—	2	4	no	—
285	Q₈	—	2	2	no	—
286	Q₈	—	394	2	no	—
287	Q₈	C₂×C₂²⋊D₁₉₇	1	4	yes	—
288	D₄	—	2	8	no	—
289	D₄	—	2	4	no	—
290	D₄	—	2	8	no	—
291	D₄	—	2	2	no	—
292	D₄	—	788	2	no	—
293	D₄	—	788	2	no	—
294	D₄	—	788	2	no	—
295	D₄	—	394	2	no	—
296	D₄	C₂×C₂²⋊D₁₉₇	1	4	yes	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	8	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	8	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	788	2	no	—
307	C₂×C₄	—	394	1	no	—
308	C₂×C₄	—	394	1	no	—
309	C₂×C₄	—	394	1	no	—
310	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
311	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
312	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	4	yes	—
313	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
314	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
315	C₈	—	394	2	no	—
316	C₈	—	394	2	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	4	no	—
325	C₂²	—	4	4	no	—
326	C₂²	—	4	8	no	—
327	C₂²	—	4	2	no	—
328	C₂²	—	4	2	no	—
329	C₂²	—	788	1	no	—
330	C₂²	—	788	1	no	—
331	C₂²	—	788	1	no	—
332	C₂²	#20263295	1	2	yes	—
333	C₂²	#20263346	1	2	yes	complemented
334	C₂²	C₂²×C₂²⋊D₁₉₇	1	1	yes	characteristic
335	C₄	—	2	1	no	—
336	C₄	—	2	4	no	—
337	C₄	—	2	1	no	—
338	C₄	—	788	1	no	—
339	C₄	—	788	1	no	—
340	C₄	—	788	1	no	—
341	C₄	#20263295	1	2	yes	—
342	C₄	C₂²×C₂²⋊D₁₉₇	1	2	yes	—
343	C₂	—	2	2	no	—
344	C₂	—	2	2	no	—
345	C₂	—	4	1	no	—
346	C₂	—	4	4	no	—
347	C₂	—	1576	1	no	—
348	C₂	C₂×#20263295	1	1	yes	characteristic
349	C₂	C₂×#20263346	1	1	yes	characteristic
350	C₂	#27161877	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₉₇
commutator, cosocle	C₂×C₄×C₁₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting	C₂²×#41×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂²×D₄
hypercenter	C₂²×#41	D₁₉₇
hypocenter	C₁₉₇	#3002
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#20263346	C₂	semidirect
2. #27161590	C₂	semidirect
3. #27161241	C₂	semidirect
4. #27161235	C₂	semidirect
5. #27161230	C₂	semidirect
6. #27161224	C₂	semidirect
7. #27161218	C₂	semidirect
8. #27161213	C₂	semidirect
9. #27161075	C₂	semidirect
10. C₂×C₂²⋊#13096802	C₂	semidirect
11. C₂²×#41×C₁₉₇	C₂	semidirect
12. C₂×#15691358	C₂²	semidirect
14. C₂×#15691348	C₂²	semidirect
15. C₂×#15691346	C₂²	semidirect
16. C₂×#15691345	C₂²	semidirect
17. C₂×#15691343	C₂²	semidirect
18. C₂²×#13096802	C₂²	semidirect
21. #20263305	C₂²	semidirect
23. #20263300	C₂²	semidirect
27. #20263285	C₂²	semidirect
28. #20263250	C₂²	semidirect
29. #20263250	C₂²	semidirect
30. #20263247	C₂²	semidirect
31. #20263247	C₂²	semidirect
34. C₂²⋊#13096802	C₂²	semidirect
76. #41×C₁₉₇	D₄	semidirect
154. C₂²×C₁₉₇	#135	semidirect
175. C₁₉₇	#3002	semidirect
177. C₂²×#41	D₁₉₇	semidirect
229. #41	C₂²⋊D₁₉₇	semidirect
333. C₂²	#20263346	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20263346	2
#27161590	2
#27161241	2
#27161235	2
#27161230	2
#27161224	2
#27161218	2
#27161213	2
#27161075	2
C₂×C₂²⋊#13096802	2
C₂²×#41×C₁₉₇	2
#3002	197