Subgroups — #38752016

Order 25216.

764 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20263352	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#20263346	C₂	1	1	yes	maximal, characteristic, complemented
3	#27161657	C₂	1	1	yes	maximal, characteristic, complemented
4	#27161241	C₂	1	2	yes	maximal, complemented
5	#27161230	C₂	1	2	yes	maximal, complemented
6	#27161224	C₂	1	2	yes	maximal, complemented
7	#27161213	C₂	1	2	yes	maximal, complemented
8	#27161080	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#20263247	C₂	1	1	yes	maximal, characteristic, complemented
10	#27161046	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×H(5,𝔽₂)×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15691358	C₂²	1	1	yes	characteristic, complemented
13	#20263352	—	2	4	no	—
14	#20263346	—	2	4	no	—
15	C₂×#15691348	C₂²	1	1	yes	characteristic, complemented
16	C₂×#15691346	C₂²	1	1	yes	characteristic, complemented
17	C₂×#15691346	C₂²	1	2	yes	complemented
18	C₂×#15691345	C₂²	1	1	yes	characteristic, complemented
19	C₂×#15691343	C₂²	1	1	yes	characteristic, complemented
20	C₂×#15691343	C₂²	1	2	yes	complemented
21	C₂×#15691342	C₂²	1	1	yes	characteristic, complemented
22	C₂²×#13096802	C₂²	1	1	yes	characteristic, complemented
23	#20263305	C₂²	1	2	yes	complemented
24	#20263302	—	2	1	no	—
25	#20263300	C₂²	1	2	yes	complemented
26	#20263298	—	2	2	no	—
27	#20263297	—	2	1	no	—
28	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic, complemented
29	#20263250	C₂²	1	2	yes	complemented
30	#20263247	C₂²	1	2	yes	complemented
31	#20263247	C₂²	1	2	yes	complemented
32	#20263247	C₂²	1	2	yes	complemented
33	C₂²⋊#13096802	C₂²	1	2	yes	complemented
34	C₂²⋊#13096802	C₂²	1	2	yes	complemented
35	H(5,𝔽₂)×C₁₉₇	—	2	4	no	—
36	H(5,𝔽₂)×C₁₉₇	—	2	4	no	—
37	C₂×#41×C₁₉₇	—	2	1	no	—
38	C₂×#41×C₁₉₇	C₂²	1	2	yes	—
39	C₂×#41×C₁₉₇	C₂²	1	1	yes	characteristic
40	C₂×#41×C₁₉₇	C₂²	1	1	yes	characteristic
41	C₂²×D₄×C₁₉₇	—	2	1	no	—
42	C₂²×D₄×C₁₉₇	—	2	2	no	—
43	C₂²×D₄×C₁₉₇	C₂²	1	2	yes	—
44	C₂²×D₄×C₁₉₇	C₂²	1	1	yes	characteristic
45	#15691358	—	2	2	no	—
46	#15691358	—	2	2	no	—
47	C₂×Dic₃₉₄	C₂³	1	1	yes	characteristic
48	C₂×D₇₈₈	C₂³	1	1	yes	characteristic
49	C₂×C₂²⋊D₁₉₇	—	2	1	no	—
50	C₂×C₂²⋊D₁₉₇	—	2	2	no	—
51	C₂²×Dic₁₉₇	—	2	1	no	—
52	C₂×C₄×D₁₉₇	—	2	1	no	—
53	#15691348	—	2	2	no	—
54	#15691346	—	2	2	no	—
55	#15691346	—	2	4	no	—
56	#15691345	—	2	2	no	—
57	#15691343	—	2	2	no	—
58	#15691343	—	2	4	no	—
59	#15691342	—	2	2	no	—
60	C₂×#13096802	—	2	2	no	—
61	C₂×#13096802	C₂³	1	1	yes	characteristic
62	C₂×#13096802	C₂³	1	1	yes	characteristic
63	C₂×#13096802	C₂³	1	2	yes	—
64	C₄⋊Dic₁₉₇	—	2	1	no	—
65	C₄⋊Dic₁₉₇	C₂³	1	2	yes	—
66	#15691337	—	2	2	no	—
67	C₂²⋊Dic₁₉₇	—	2	2	no	—
68	#15691335	—	2	2	no	—
69	C₂⁴×C₁₉₇	—	2	2	no	—
70	C₂⁴×C₁₉₇	—	2	1	no	—
71	#41×C₁₉₇	—	2	8	no	—
72	#41×C₁₉₇	—	2	4	no	—
73	#41×C₁₉₇	—	2	4	no	—
74	#41×C₁₉₇	—	2	2	no	—
75	#41×C₁₉₇	—	2	2	no	—
76	#41×C₁₉₇	—	2	2	no	—
77	#41×C₁₉₇	D₄	1	4	yes	complemented
78	C₂×Q₈×C₁₉₇	D₄	1	1	yes	characteristic
79	C₂×Q₈×C₁₉₇	C₂³	1	1	yes	characteristic
80	C₂×D₄×C₁₉₇	—	2	16	no	—
81	C₂×D₄×C₁₉₇	—	2	8	no	—
82	C₂×D₄×C₁₉₇	—	2	2	no	—
83	C₂×D₄×C₁₉₇	—	2	1	no	—
84	C₂×D₄×C₁₉₇	—	2	4	no	—
85	C₂×D₄×C₁₉₇	—	2	4	no	—
86	C₂×D₄×C₁₉₇	—	2	2	no	—
87	C₂×D₄×C₁₉₇	—	2	1	no	—
88	C₂×D₄×C₁₉₇	—	2	2	no	—
89	C₂×D₄×C₁₉₇	D₄	1	1	yes	characteristic
90	C₂×D₄×C₁₉₇	D₄	1	2	yes	—
91	C₂×D₄×C₁₉₇	D₄	1	4	yes	complemented
92	C₂×D₄×C₁₉₇	C₂³	1	1	yes	characteristic
93	C₂×D₄×C₁₉₇	C₂³	1	2	yes	—
94	C₂²×C₄×C₁₉₇	—	2	2	no	—
95	C₂²×C₄×C₁₉₇	—	2	1	no	—
96	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
97	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
98	C₂²×D₁₉₇	—	2	1	no	—
99	Dic₃₉₄	—	2	1	no	—
100	Dic₃₉₄	—	2	1	no	—
101	D₇₈₈	—	2	1	no	—
102	D₇₈₈	—	2	1	no	—
103	C₂²⋊D₁₉₇	—	4	2	no	—
104	C₂²⋊D₁₉₇	—	8	2	no	—
105	C₂×Dic₁₉₇	—	2	2	no	—
106	C₂×Dic₁₉₇	—	2	1	no	—
107	C₂×Dic₁₉₇	—	8	1	no	—
108	C₄×D₁₉₇	—	4	2	no	—
109	#13096802	—	2	1	no	—
110	#13096802	—	2	1	no	—
111	#13096802	—	2	2	no	—
112	C₂³×C₁₉₇	—	2	8	no	—
113	C₂³×C₁₉₇	—	2	2	no	—
114	C₂³×C₁₉₇	—	2	1	no	—
115	C₂³×C₁₉₇	—	2	4	no	—
116	C₂³×C₁₉₇	—	8	1	no	—
117	C₂³×C₁₉₇	—	8	1	no	—
118	C₂³×C₁₉₇	—	8	2	no	—
119	C₂³×C₁₉₇	—	8	2	no	—
120	C₂³×C₁₉₇	C₂×D₄	1	2	yes	—
121	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
122	Q₈×C₁₉₇	—	2	2	no	—
123	Q₈×C₁₉₇	—	2	1	no	—
124	Q₈×C₁₉₇	C₂×D₄	1	2	yes	—
125	D₄×C₁₉₇	—	2	8	no	—
126	D₄×C₁₉₇	—	2	8	no	—
127	D₄×C₁₉₇	—	2	4	no	—
128	D₄×C₁₉₇	—	2	4	no	—
129	D₄×C₁₉₇	—	2	2	no	—
130	D₄×C₁₉₇	—	2	4	no	—
131	D₄×C₁₉₇	—	2	2	no	—
132	D₄×C₁₉₇	—	2	1	no	—
133	D₄×C₁₉₇	C₂×D₄	1	2	yes	—
134	D₄×C₁₉₇	C₂×D₄	1	4	yes	—
135	C₂×C₄×C₁₉₇	—	2	8	no	—
136	C₂×C₄×C₁₉₇	—	2	1	no	—
137	C₂×C₄×C₁₉₇	—	2	4	no	—
138	C₂×C₄×C₁₉₇	—	2	2	no	—
139	C₂×C₄×C₁₉₇	—	2	2	no	—
140	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
141	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
143	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
144	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
145	C₂²×C₁₉₇	—	2	1	no	—
146	C₂²×C₁₉₇	—	2	1	no	—
147	C₂²×C₁₉₇	—	2	2	no	—
148	C₂²×C₁₉₇	—	2	4	no	—
149	C₂²×C₁₉₇	—	2	2	no	—
150	C₂²×C₁₉₇	—	2	2	no	—
151	C₂²×C₁₉₇	—	2	2	no	—
152	C₂²×C₁₉₇	—	4	2	no	—
153	C₂²×C₁₉₇	—	4	1	no	—
154	C₂²×C₁₉₇	—	4	2	no	—
155	C₂²×C₁₉₇	—	4	1	no	—
156	C₂²×C₁₉₇	—	8	4	no	—
157	C₂²×C₁₉₇	—	8	8	no	—
158	C₂²×C₁₉₇	#120	1	2	yes	—
159	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
160	C₂×D₁₉₇	—	4	1	no	—
161	C₂×D₁₉₇	—	4	1	no	—
162	C₂×D₁₉₇	—	4	1	no	—
163	C₄×C₁₉₇	—	2	2	no	—
164	C₄×C₁₉₇	—	2	2	no	—
165	C₄×C₁₉₇	#120	1	2	yes	—
166	C₄×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
167	C₄×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
168	Dic₁₉₇	—	4	1	no	—
169	Dic₁₉₇	—	4	2	no	—
170	C₂×C₁₉₇	—	2	2	no	—
171	C₂×C₁₉₇	—	4	2	no	—
172	C₂×C₁₉₇	—	4	4	no	—
173	C₂×C₁₉₇	—	4	2	no	—
174	C₂×C₁₉₇	C₂×#120	1	1	yes	characteristic
175	C₂×C₁₉₇	#576	1	1	yes	characteristic
176	C₂×C₁₉₇	#577	1	1	yes	characteristic
177	D₁₉₇	—	8	1	no	—
178	C₁₉₇	#3012	1	1	yes	characteristic, complemented, Sylow
179	#3012	—	197	1	no	maximal, Sylow
180	C₂×H(5,𝔽₂)	D₁₉₇	1	1	yes	characteristic, complemented
181	C₂×AGL(1,ℤ/8)	—	197	1	no	—
182	C₂×#135	—	197	1	no	—
183	#537	—	197	1	no	—
184	#452	—	197	2	no	—
185	#450	—	197	2	no	—
186	#449	—	197	2	no	—
187	#447	—	197	2	no	—
188	#417	—	197	1	no	—
189	C₂×#102	—	197	1	no	—
190	#408	—	197	1	no	—
191	H(5,𝔽₂)	—	2	4	no	—
192	H(5,𝔽₂)	—	2	4	no	—
193	C₂×#41	—	2	1	no	—
194	C₂×#41	—	197	1	no	—
195	C₂×#41	C₂×D₁₉₇	1	2	yes	—
196	C₂×#41	C₂×D₁₉₇	1	1	yes	characteristic
197	C₂×#41	C₂×D₁₉₇	1	1	yes	characteristic
198	C₂²×D₄	—	2	2	no	—
199	C₂²×D₄	—	2	1	no	—
200	C₂²×D₄	C₂×D₁₉₇	1	2	yes	—
201	C₂²×D₄	C₂×D₁₉₇	1	1	yes	characteristic
202	AGL(1,ℤ/8)	—	394	4	no	—
203	#135	—	394	4	no	—
204	C₂×Q₁₆	—	197	1	no	—
205	C₂×SD₁₆	—	197	1	no	—
206	C₂×SD₁₆	—	197	1	no	—
207	C₂×SD₁₆	—	197	2	no	—
208	C₂×D₈	—	197	1	no	—
209	C₂×D₈	—	197	2	no	—
210	C₂×M₁₆	—	197	1	no	—
211	C₂²×C₈	—	197	1	no	—
212	C₂²⋊Q₈	—	394	1	no	—
213	C₂²⋊Q₈	—	197	2	no	—
214	#121	—	394	1	no	—
215	#121	—	394	2	no	—
216	#121	—	197	2	no	—
217	C₂×C₄⋊C₄	—	197	1	no	—
218	#103	—	197	2	no	—
219	#102	—	197	2	no	—
220	#102	—	197	2	no	—
221	#102	—	197	2	no	—
222	C₂²⋊C₈	—	197	2	no	—
223	C₂²⋊C₈	—	197	2	no	—
224	C₂⁴	—	2	2	no	—
225	C₂⁴	—	2	1	no	—
226	#41	—	2	8	no	—
227	#41	—	2	4	no	—
228	#41	—	2	4	no	—
229	#41	—	2	2	no	—
230	#41	—	2	2	no	—
231	#41	—	2	2	no	—
232	#41	—	394	2	no	—
233	#41	—	394	2	no	—
234	#41	C₂²⋊D₁₉₇	1	4	yes	complemented
235	C₂×Q₈	—	197	1	no	—
236	C₂×Q₈	C₂²⋊D₁₉₇	1	1	yes	characteristic
237	C₂×Q₈	C₂²×D₁₉₇	1	1	yes	characteristic
238	C₂×D₄	—	2	16	no	—
239	C₂×D₄	—	2	8	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	—	2	4	no	—
243	C₂×D₄	—	2	4	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	394	1	no	—
248	C₂×D₄	—	394	2	no	—
249	C₂×D₄	—	197	1	no	—
250	C₂×D₄	C₂²⋊D₁₉₇	1	2	yes	—
251	C₂×D₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
252	C₂×D₄	C₂²⋊D₁₉₇	1	4	yes	complemented
253	C₂×D₄	C₂²×D₁₉₇	1	1	yes	characteristic
254	C₂×D₄	C₂²×D₁₉₇	1	2	yes	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	394	1	no	—
258	C₂²×C₄	—	394	1	no	—
259	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
260	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
261	Q₁₆	—	394	2	no	—
262	SD₁₆	—	394	2	no	—
263	SD₁₆	—	394	2	no	—
264	SD₁₆	—	394	4	no	—
265	D₈	—	394	2	no	—
266	D₈	—	394	4	no	—
267	M₁₆	—	394	2	no	—
268	C₂×C₈	—	394	2	no	—
269	C₂×C₈	—	197	1	no	—
270	C₂×C₈	—	197	1	no	—
271	C₂×C₈	—	197	2	no	—
272	C₄⋊C₄	—	394	2	no	—
273	C₄⋊C₄	—	394	1	no	—
274	C₄⋊C₄	—	197	2	no	—
275	C₂²⋊C₄	—	394	2	no	—
276	C₂²⋊C₄	—	394	2	no	—
277	C₂³	—	2	8	no	—
278	C₂³	—	2	2	no	—
279	C₂³	—	2	1	no	—
280	C₂³	—	2	4	no	—
281	C₂³	—	8	1	no	—
282	C₂³	—	8	1	no	—
283	C₂³	—	8	2	no	—
284	C₂³	—	8	2	no	—
285	C₂³	—	394	1	no	—
286	C₂³	C₂×C₂²⋊D₁₉₇	1	2	yes	—
287	C₂³	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
288	Q₈	—	2	2	no	—
289	Q₈	—	2	1	no	—
290	Q₈	—	394	1	no	—
291	Q₈	—	394	1	no	—
292	Q₈	C₂×C₂²⋊D₁₉₇	1	2	yes	—
293	D₄	—	2	8	no	—
294	D₄	—	2	8	no	—
295	D₄	—	2	4	no	—
296	D₄	—	2	4	no	—
297	D₄	—	2	2	no	—
298	D₄	—	2	4	no	—
299	D₄	—	2	2	no	—
300	D₄	—	2	1	no	—
301	D₄	—	788	2	no	—
302	D₄	—	1576	2	no	—
303	D₄	—	394	1	no	—
304	D₄	—	394	1	no	—
305	D₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
306	D₄	C₂×C₂²⋊D₁₉₇	1	4	yes	—
307	C₂×C₄	—	2	8	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	788	2	no	—
313	C₂×C₄	—	1576	1	no	—
314	C₂×C₄	—	394	2	no	—
315	C₂×C₄	—	394	1	no	—
316	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
317	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
318	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
319	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
320	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
321	C₈	—	394	1	no	—
322	C₈	—	394	1	no	—
323	C₈	—	394	2	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	2	4	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	4	2	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	2	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	8	4	no	—
336	C₂²	—	8	8	no	—
337	C₂²	—	788	1	no	—
338	C₂²	—	788	1	no	—
339	C₂²	—	788	1	no	—
340	C₂²	#20263295	1	2	yes	—
341	C₂²	C₂²×C₂²⋊D₁₉₇	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	2	2	no	—
344	C₄	—	788	2	no	—
345	C₄	—	788	1	no	—
346	C₄	#20263295	1	2	yes	—
347	C₄	C₂²×C₂²⋊D₁₉₇	1	1	yes	characteristic
348	C₄	C₂²×C₂²⋊D₁₉₇	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	4	2	no	—
351	C₂	—	4	4	no	—
352	C₂	—	4	2	no	—
353	C₂	—	1576	1	no	—
354	C₂	C₂×#20263295	1	1	yes	characteristic
355	C₂	#27161882	1	1	yes	characteristic
356	C₂	#27161890	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₉₇
commutator, cosocle	C₂×C₄×C₁₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting	C₂×H(5,𝔽₂)×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂²×D₄
hypercenter	C₂×H(5,𝔽₂)	D₁₉₇
hypocenter	C₁₉₇	#3012
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂×#20263352	C₂	semidirect
2. C₂×#20263346	C₂	semidirect
3. #27161657	C₂	semidirect
4. #27161241	C₂	semidirect
5. #27161230	C₂	semidirect
6. #27161224	C₂	semidirect
7. #27161213	C₂	semidirect
8. #27161080	C₂	semidirect
9. C₂×#20263247	C₂	semidirect
10. #27161046	C₂	semidirect
11. C₂×H(5,𝔽₂)×C₁₉₇	C₂	semidirect
12. C₂×#15691358	C₂²	semidirect
15. C₂×#15691348	C₂²	semidirect
16. C₂×#15691346	C₂²	semidirect
17. C₂×#15691346	C₂²	semidirect
18. C₂×#15691345	C₂²	semidirect
19. C₂×#15691343	C₂²	semidirect
20. C₂×#15691343	C₂²	semidirect
21. C₂×#15691342	C₂²	semidirect
22. C₂²×#13096802	C₂²	semidirect
23. #20263305	C₂²	semidirect
25. #20263300	C₂²	semidirect
28. C₂×C₄⋊Dic₁₉₇	C₂²	semidirect
29. #20263250	C₂²	semidirect
30. #20263247	C₂²	semidirect
31. #20263247	C₂²	semidirect
32. #20263247	C₂²	semidirect
33. C₂²⋊#13096802	C₂²	semidirect
34. C₂²⋊#13096802	C₂²	semidirect
77. #41×C₁₉₇	D₄	semidirect
91. C₂×D₄×C₁₉₇	D₄	semidirect
178. C₁₉₇	#3012	semidirect
180. C₂×H(5,𝔽₂)	D₁₉₇	semidirect
234. #41	C₂²⋊D₁₉₇	semidirect
252. C₂×D₄	C₂²⋊D₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20263352	2
C₂×#20263346	2
#27161657	2
#27161241	2
#27161230	2
#27161224	2
#27161213	2
#27161080	2
C₂×#20263247	2
#27161046	2
C₂×H(5,𝔽₂)×C₁₉₇	2
#3012	197