Subgroups — #38752027

Order 25216.

712 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20263357	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	#27161235	C₂	1	2	yes	maximal, complemented
6	#27161224	C₂	1	2	yes	maximal, complemented
7	#27161218	C₂	1	2	yes	maximal, complemented
8	#27161080	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#20263250	C₂	1	1	yes	maximal, characteristic, complemented
10	#27161046	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×2₋¹⁺⁴×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15691358	C₂²	1	1	yes	characteristic
13	#20263357	—	2	4	no	—
14	#20263346	—	2	4	no	—
15	C₂×#15691348	C₂²	1	1	yes	characteristic
16	C₂×#15691348	C₂²	1	2	yes	—
17	C₂×#15691346	C₂²	1	1	yes	characteristic
18	C₂×#15691345	C₂²	1	1	yes	characteristic
19	C₂×#15691345	C₂²	1	2	yes	—
20	C₂×#15691343	C₂²	1	1	yes	characteristic
21	C₂×#15691342	C₂²	1	1	yes	characteristic
22	C₂²×#13096802	C₂²	1	1	yes	characteristic
23	#20263305	C₂²	1	2	yes	—
24	#20263302	—	2	1	no	—
25	#20263300	C₂²	1	2	yes	—
26	#20263298	—	2	2	no	—
27	#20263297	—	2	1	no	—
28	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
29	#20263250	C₂²	1	2	yes	—
30	#20263250	C₂²	1	2	yes	—
31	#20263250	C₂²	1	2	yes	—
32	#20263247	C₂²	1	2	yes	—
33	C₂²⋊#13096802	C₂²	1	2	yes	—
34	C₂²⋊#13096802	C₂²	1	2	yes	—
35	2₋¹⁺⁴×C₁₉₇	—	2	4	no	—
36	2₋¹⁺⁴×C₁₉₇	—	2	4	no	—
37	C₂×#41×C₁₉₇	—	2	1	no	—
38	C₂×#41×C₁₉₇	—	2	2	no	—
39	C₂×#41×C₁₉₇	C₂²	1	2	yes	—
40	C₂×#41×C₁₉₇	C₂²	1	1	yes	characteristic
41	C₂×#41×C₁₉₇	C₂²	1	1	yes	characteristic
42	C₂²×Q₈×C₁₉₇	—	2	1	no	—
43	C₂²×Q₈×C₁₉₇	C₂²	1	2	yes	—
44	C₂²×Q₈×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	#15691348	—	2	4	no	—
55	#15691346	—	2	2	no	—
56	#15691345	—	2	2	no	—
57	#15691345	—	2	4	no	—
58	#15691343	—	2	2	no	—
59	#15691342	—	2	2	no	—
60	C₂×#13096802	—	2	2	no	—
61	C₂×#13096802	C₂³	1	2	yes	—
62	C₂×#13096802	C₂³	1	1	yes	characteristic
63	C₂×#13096802	C₂³	1	1	yes	characteristic
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	#41×C₁₉₇	—	2	16	no	—
70	#41×C₁₉₇	—	2	8	no	—
71	#41×C₁₉₇	—	2	4	no	—
72	#41×C₁₉₇	—	2	4	no	—
73	#41×C₁₉₇	—	2	2	no	—
74	#41×C₁₉₇	—	2	2	no	—
75	#41×C₁₉₇	—	2	2	no	—
76	#41×C₁₉₇	D₄	1	4	yes	complemented
77	C₂×Q₈×C₁₉₇	—	2	8	no	—
78	C₂×Q₈×C₁₉₇	—	2	4	no	—
79	C₂×Q₈×C₁₉₇	—	2	4	no	—
80	C₂×Q₈×C₁₉₇	—	2	2	no	—
81	C₂×Q₈×C₁₉₇	—	2	1	no	—
82	C₂×Q₈×C₁₉₇	—	2	2	no	—
83	C₂×Q₈×C₁₉₇	D₄	1	1	yes	characteristic
84	C₂×Q₈×C₁₉₇	D₄	1	4	yes	complemented
85	C₂×Q₈×C₁₉₇	C₂³	1	1	yes	characteristic
86	C₂×Q₈×C₁₉₇	C₂³	1	2	yes	—
87	C₂×D₄×C₁₉₇	—	2	2	no	—
88	C₂×D₄×C₁₉₇	—	2	1	no	—
89	C₂×D₄×C₁₉₇	D₄	1	1	yes	characteristic
90	C₂×D₄×C₁₉₇	D₄	1	2	yes	—
91	C₂×D₄×C₁₉₇	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₁₉₇	—	2	2	no	—
93	C₂²×C₄×C₁₉₇	—	2	1	no	—
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	1	no	—
113	C₂³×C₁₉₇	C₂×D₄	1	2	yes	—
114	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₁₉₇	—	2	8	no	—
116	Q₈×C₁₉₇	—	2	2	no	—
117	Q₈×C₁₉₇	—	2	4	no	—
118	Q₈×C₁₉₇	—	2	1	no	—
119	Q₈×C₁₉₇	—	2	2	no	—
120	Q₈×C₁₉₇	C₂×D₄	1	2	yes	—
121	Q₈×C₁₉₇	C₂×D₄	1	4	yes	—
122	D₄×C₁₉₇	—	2	8	no	—
123	D₄×C₁₉₇	—	2	4	no	—
124	D₄×C₁₉₇	—	2	4	no	—
125	D₄×C₁₉₇	—	2	2	no	—
126	D₄×C₁₉₇	—	2	1	no	—
127	D₄×C₁₉₇	C₂×D₄	1	2	yes	—
128	C₂×C₄×C₁₉₇	—	2	8	no	—
129	C₂×C₄×C₁₉₇	—	2	8	no	—
130	C₂×C₄×C₁₉₇	—	2	4	no	—
131	C₂×C₄×C₁₉₇	—	2	2	no	—
132	C₂×C₄×C₁₉₇	—	2	1	no	—
133	C₂×C₄×C₁₉₇	—	2	4	no	—
134	C₂×C₄×C₁₉₇	—	2	2	no	—
135	C₂×C₄×C₁₉₇	—	2	2	no	—
136	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
137	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
139	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
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	2	no	—
144	C₂²×C₁₉₇	—	2	2	no	—
145	C₂²×C₁₉₇	—	2	2	no	—
146	C₂²×C₁₉₇	—	2	2	no	—
147	C₂²×C₁₉₇	—	4	1	no	—
148	C₂²×C₁₉₇	—	4	1	no	—
149	C₂²×C₁₉₇	#120	1	2	yes	—
150	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₁₉₇	—	4	1	no	—
152	C₂×D₁₉₇	—	4	1	no	—
153	C₂×D₁₉₇	—	4	1	no	—
154	C₄×C₁₉₇	—	2	4	no	—
155	C₄×C₁₉₇	—	2	2	no	—
156	C₄×C₁₉₇	—	2	2	no	—
157	C₄×C₁₉₇	#120	1	2	yes	—
158	C₄×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
159	C₄×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
160	Dic₁₉₇	—	4	1	no	—
161	Dic₁₉₇	—	4	2	no	—
162	C₂×C₁₉₇	—	2	2	no	—
163	C₂×C₁₉₇	—	4	2	no	—
164	C₂×C₁₉₇	—	4	2	no	—
165	C₂×C₁₉₇	C₂×#120	1	1	yes	characteristic
166	C₂×C₁₉₇	#577	1	1	yes	characteristic
167	C₂×C₁₉₇	#578	1	1	yes	characteristic
168	D₁₉₇	—	8	1	no	—
169	C₁₉₇	#3013	1	1	yes	characteristic, complemented, Sylow
170	#3013	—	197	1	no	maximal, Sylow
171	C₂×2₋¹⁺⁴	D₁₉₇	1	1	yes	characteristic, complemented
172	C₂×#137	—	197	1	no	—
173	C₂×#135	—	197	1	no	—
174	#537	—	197	1	no	—
175	#452	—	197	2	no	—
176	#451	—	197	2	no	—
177	#449	—	197	2	no	—
178	#448	—	197	2	no	—
179	#417	—	197	1	no	—
180	C₂×#103	—	197	1	no	—
181	#408	—	197	1	no	—
182	2₋¹⁺⁴	—	2	4	no	—
183	2₋¹⁺⁴	—	2	4	no	—
184	C₂×#41	—	2	1	no	—
185	C₂×#41	—	2	2	no	—
186	C₂×#41	—	197	1	no	—
187	C₂×#41	C₂×D₁₉₇	1	2	yes	—
188	C₂×#41	C₂×D₁₉₇	1	1	yes	characteristic
189	C₂×#41	C₂×D₁₉₇	1	1	yes	characteristic
190	C₂²×Q₈	—	2	1	no	—
191	C₂²×Q₈	C₂×D₁₉₇	1	2	yes	—
192	C₂²×Q₈	C₂×D₁₉₇	1	1	yes	characteristic
193	#137	—	394	4	no	—
194	#135	—	394	4	no	—
195	C₂×Q₁₆	—	197	1	no	—
196	C₂×Q₁₆	—	197	2	no	—
197	C₂×SD₁₆	—	197	1	no	—
198	C₂×SD₁₆	—	197	2	no	—
199	C₂×SD₁₆	—	197	1	no	—
200	C₂×D₈	—	197	1	no	—
201	C₂×M₁₆	—	197	1	no	—
202	C₂²×C₈	—	197	1	no	—
203	C₂²⋊Q₈	—	394	1	no	—
204	C₂²⋊Q₈	—	197	2	no	—
205	#121	—	394	1	no	—
206	#121	—	394	2	no	—
207	#121	—	197	2	no	—
208	C₂×C₄⋊C₄	—	197	1	no	—
209	#103	—	197	2	no	—
210	#103	—	197	2	no	—
211	#103	—	197	2	no	—
212	#102	—	197	2	no	—
213	C₂²⋊C₈	—	197	2	no	—
214	C₂²⋊C₈	—	197	2	no	—
215	#41	—	2	16	no	—
216	#41	—	2	8	no	—
217	#41	—	2	4	no	—
218	#41	—	2	4	no	—
219	#41	—	2	2	no	—
220	#41	—	2	2	no	—
221	#41	—	2	2	no	—
222	#41	—	394	2	no	—
223	#41	—	394	2	no	—
224	#41	C₂²⋊D₁₉₇	1	4	yes	complemented
225	C₂×Q₈	—	2	8	no	—
226	C₂×Q₈	—	2	4	no	—
227	C₂×Q₈	—	2	4	no	—
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	2	1	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	197	1	no	—
232	C₂×Q₈	C₂²⋊D₁₉₇	1	1	yes	characteristic
233	C₂×Q₈	C₂²⋊D₁₉₇	1	4	yes	complemented
234	C₂×Q₈	C₂²×D₁₉₇	1	1	yes	characteristic
235	C₂×Q₈	C₂²×D₁₉₇	1	2	yes	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	—	394	1	no	—
239	C₂×D₄	—	394	2	no	—
240	C₂×D₄	—	197	1	no	—
241	C₂×D₄	C₂²⋊D₁₉₇	1	2	yes	—
242	C₂×D₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
243	C₂×D₄	C₂²×D₁₉₇	1	1	yes	characteristic
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	394	1	no	—
249	C₂²×C₄	—	394	1	no	—
250	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
251	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
252	Q₁₆	—	394	2	no	—
253	Q₁₆	—	394	4	no	—
254	SD₁₆	—	394	2	no	—
255	SD₁₆	—	394	4	no	—
256	SD₁₆	—	394	2	no	—
257	D₈	—	394	2	no	—
258	M₁₆	—	394	2	no	—
259	C₂×C₈	—	394	2	no	—
260	C₂×C₈	—	197	2	no	—
261	C₂×C₈	—	197	1	no	—
262	C₂×C₈	—	197	1	no	—
263	C₄⋊C₄	—	394	2	no	—
264	C₄⋊C₄	—	394	1	no	—
265	C₄⋊C₄	—	197	2	no	—
266	C₂²⋊C₄	—	394	2	no	—
267	C₂²⋊C₄	—	394	2	no	—
268	C₂³	—	2	1	no	—
269	C₂³	—	394	1	no	—
270	C₂³	C₂×C₂²⋊D₁₉₇	1	2	yes	—
271	C₂³	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
272	Q₈	—	2	8	no	—
273	Q₈	—	2	2	no	—
274	Q₈	—	2	4	no	—
275	Q₈	—	2	1	no	—
276	Q₈	—	2	2	no	—
277	Q₈	—	394	1	no	—
278	Q₈	—	394	1	no	—
279	Q₈	C₂×C₂²⋊D₁₉₇	1	2	yes	—
280	Q₈	C₂×C₂²⋊D₁₉₇	1	4	yes	—
281	D₄	—	2	8	no	—
282	D₄	—	2	4	no	—
283	D₄	—	2	4	no	—
284	D₄	—	2	2	no	—
285	D₄	—	2	1	no	—
286	D₄	—	788	2	no	—
287	D₄	—	1576	2	no	—
288	D₄	—	394	1	no	—
289	D₄	—	394	1	no	—
290	D₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
291	C₂×C₄	—	2	8	no	—
292	C₂×C₄	—	2	8	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	788	2	no	—
300	C₂×C₄	—	1576	1	no	—
301	C₂×C₄	—	394	2	no	—
302	C₂×C₄	—	394	1	no	—
303	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
304	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
305	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
306	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
307	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
308	C₈	—	394	1	no	—
309	C₈	—	394	1	no	—
310	C₈	—	394	2	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	788	1	no	—
320	C₂²	—	788	1	no	—
321	C₂²	—	788	1	no	—
322	C₂²	#20263295	1	2	yes	—
323	C₂²	C₂²×C₂²⋊D₁₉₇	1	1	yes	characteristic
324	C₄	—	2	4	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	788	2	no	—
328	C₄	—	788	1	no	—
329	C₄	#20263295	1	2	yes	—
330	C₄	C₂²×C₂²⋊D₁₉₇	1	1	yes	characteristic
331	C₄	C₂²×C₂²⋊D₁₉₇	1	1	yes	characteristic
332	C₂	—	2	2	no	—
333	C₂	—	4	2	no	—
334	C₂	—	4	2	no	—
335	C₂	—	1576	1	no	—
336	C₂	C₂×#20263295	1	1	yes	characteristic
337	C₂	#27161887	1	1	yes	characteristic
338	C₂	#27161894	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₉₇
commutator, cosocle	C₂×C₄×C₁₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting	C₂×2₋¹⁺⁴×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂²×D₄
hypercenter	C₂×2₋¹⁺⁴	D₁₉₇
hypocenter	C₁₉₇	#3013
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₂×#20263357	C₂	semidirect
2. C₂×#20263346	C₂	semidirect
3. #27161657	C₂	semidirect
4. #27161241	C₂	semidirect
5. #27161235	C₂	semidirect
6. #27161224	C₂	semidirect
7. #27161218	C₂	semidirect
8. #27161080	C₂	semidirect
9. C₂×#20263250	C₂	semidirect
10. #27161046	C₂	semidirect
11. C₂×2₋¹⁺⁴×C₁₉₇	C₂	semidirect
76. #41×C₁₉₇	D₄	semidirect
84. C₂×Q₈×C₁₉₇	D₄	semidirect
169. C₁₉₇	#3013	semidirect
171. C₂×2₋¹⁺⁴	D₁₉₇	semidirect
224. #41	C₂²⋊D₁₉₇	semidirect
233. C₂×Q₈	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×#20263357	2
C₂×#20263346	2
#27161657	2
#27161241	2
#27161235	2
#27161224	2
#27161218	2
#27161080	2
C₂×#20263250	2
#27161046	2
C₂×2₋¹⁺⁴×C₁₉₇	2
#3013	197