Subgroups — C₂×#27161633 #38756000

Order 25216.

1168 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27161633	C₂	1	16	yes	maximal, direct product factor
2	C₂×#20263325	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#20263324	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#20263316	C₂	1	2	yes	maximal, complemented
5	C₂×#20263306	C₂	1	1	yes	maximal, characteristic
6	C₂×#20263304	C₂	1	2	yes	maximal
7	C₂×#20263302	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₃₉₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×Dic₃₉₄	C₂	1	2	yes	maximal, complemented
10	C₂×#20263283	C₂	1	2	yes	maximal
11	C₂×C₄²×D₁₉₇	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Q₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₃₉₄	C₂²	1	1	yes	characteristic
14	C₂²×Dic₃₉₄	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₉₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₁₉₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₉₇	C₂²	1	1	yes	characteristic
18	#20263325	C₂²	1	8	yes	complemented
19	#20263324	C₂²	1	8	yes	complemented
20	#20263316	C₂²	1	16	yes	complemented
21	#20263306	C₂²	1	8	yes	—
22	#20263304	C₂²	1	16	yes	—
23	#20263302	C₂²	1	8	yes	—
24	C₄×Dic₃₉₄	C₂²	1	8	yes	complemented
25	C₄×Dic₃₉₄	C₂²	1	16	yes	complemented
26	#20263283	C₂²	1	16	yes	—
27	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
29	C₂×#15691337	C₂²	1	2	yes	—
30	C₂×#15691337	C₂²	1	2	yes	—
31	C₂×#15691337	C₂²	1	2	yes	—
32	C₂×#15691337	C₂²	1	2	yes	—
33	C₂×#15691337	C₂²	1	2	yes	—
34	C₂×#15691335	C₂²	1	2	yes	—
35	C₂×#15691335	C₂²	1	2	yes	—
36	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
37	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
39	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
40	C₄²×D₁₉₇	C₂²	1	8	yes	—
41	C₂²×Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
42	C₄×Q₈×C₁₉₇	C₂²	1	8	yes	complemented
43	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	2	yes	—
45	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
46	C₂×C₄²×C₁₉₇	C₂²	1	2	yes	—
47	C₂³×D₁₉₇	C₂³	1	1	yes	characteristic
48	C₂×Dic₃₉₄	—	2	8	no	—
49	C₂×Dic₃₉₄	—	2	4	no	—
50	C₂×Dic₃₉₄	C₂³	1	8	yes	—
51	C₂×Dic₃₉₄	C₂³	1	4	yes	—
52	C₂²×Dic₁₉₇	C₂³	1	1	yes	characteristic
53	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
54	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
55	C₂²×Dic₁₉₇	C₂³	1	1	yes	characteristic
56	C₂²×Dic₁₉₇	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₁₉₇	—	2	4	no	—
58	C₂×C₄×D₁₉₇	—	2	4	no	—
59	C₂×C₄×D₁₉₇	Q₈	1	8	yes	complemented
60	C₂×C₄×D₁₉₇	C₂³	1	4	yes	—
61	C₂×C₄×D₁₉₇	C₂³	1	4	yes	—
62	C₂×C₄×D₁₉₇	C₂³	1	4	yes	—
63	C₄⋊Dic₁₉₇	C₂³	1	4	yes	—
64	C₄⋊Dic₁₉₇	C₂³	1	8	yes	—
65	#15691337	C₂³	1	8	yes	—
66	#15691337	C₂³	1	8	yes	—
67	#15691337	C₂³	1	8	yes	—
68	#15691337	C₂³	1	8	yes	—
69	#15691337	C₂³	1	8	yes	—
70	#15691335	C₂³	1	8	yes	—
71	#15691335	C₂³	1	8	yes	—
72	C₄×Dic₁₉₇	C₂³	1	4	yes	—
73	C₄×Dic₁₉₇	C₂³	1	4	yes	—
74	C₄×Dic₁₉₇	C₂³	1	4	yes	—
75	C₄×Dic₁₉₇	C₂³	1	8	yes	—
76	C₂×Q₈×C₁₉₇	—	2	4	no	—
77	C₂×Q₈×C₁₉₇	C₂³	1	4	yes	—
78	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
80	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
81	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
82	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₁₉₇	C₂³	1	8	yes	—
84	C₄⋊C₄×C₁₉₇	C₂³	1	4	yes	—
85	C₄²×C₁₉₇	C₂³	1	4	yes	—
86	C₄²×C₁₉₇	C₂³	1	8	yes	—
87	C₂²×D₁₉₇	—	2	2	no	—
88	C₂²×D₁₉₇	—	2	2	no	—
89	C₂²×D₁₉₇	C₂×Q₈	1	4	yes	—
90	C₂²×D₁₉₇	C₂⁴	1	2	yes	—
91	Dic₃₉₄	—	2	8	no	—
92	Dic₃₉₄	—	2	16	no	—
93	C₂×Dic₁₉₇	—	2	2	no	—
94	C₂×Dic₁₉₇	—	2	4	no	—
95	C₂×Dic₁₉₇	—	2	4	no	—
96	C₂×Dic₁₉₇	C₂×Q₈	1	4	yes	—
97	C₂×Dic₁₉₇	#41	1	4	yes	—
98	C₂×Dic₁₉₇	C₂⁴	1	2	yes	—
99	C₂×Dic₁₉₇	C₂⁴	1	4	yes	—
100	C₂×Dic₁₉₇	C₂⁴	1	4	yes	—
101	C₂×Dic₁₉₇	C₂⁴	1	2	yes	—
102	C₂×Dic₁₉₇	C₂⁴	1	2	yes	—
103	C₄×D₁₉₇	—	2	8	no	—
104	C₄×D₁₉₇	—	2	8	no	—
105	C₄×D₁₉₇	C₂×Q₈	1	16	yes	complemented
106	C₂³×C₁₉₇	C₂⁴	1	1	yes	characteristic
107	Q₈×C₁₉₇	—	2	8	no	—
108	C₂×C₄×C₁₉₇	—	2	4	no	—
109	C₂×C₄×C₁₉₇	—	2	2	no	—
110	C₂×C₄×C₁₉₇	—	2	4	no	—
111	C₂×C₄×C₁₉₇	C₂×Q₈	1	4	yes	—
112	C₂×C₄×C₁₉₇	#41	1	4	yes	—
113	C₂×C₄×C₁₉₇	C₂⁴	1	2	yes	—
114	C₂×C₄×C₁₉₇	C₂⁴	1	4	yes	—
115	C₂×C₄×C₁₉₇	C₂⁴	1	4	yes	—
116	C₂×C₄×C₁₉₇	C₂⁴	1	2	yes	—
117	C₂×C₄×C₁₉₇	C₂⁴	1	2	yes	—
118	C₂²×C₁₉₇	C₂²×Q₈	1	2	yes	—
119	C₂²×C₁₉₇	C₂×#41	1	2	yes	—
120	C₂²×C₁₉₇	C₂×#41	1	2	yes	—
121	C₂²×C₁₉₇	C₂⁵	1	1	yes	characteristic
122	C₂×D₁₉₇	—	2	8	no	—
123	C₂×D₁₉₇	—	2	2	no	—
124	C₂×D₁₉₇	—	2	2	no	—
125	C₂×D₁₉₇	C₂²×Q₈	1	4	yes	—
126	C₄×C₁₉₇	—	2	4	no	—
127	C₄×C₁₉₇	—	2	2	no	—
128	C₄×C₁₉₇	—	2	4	no	—
129	C₄×C₁₉₇	C₂²×Q₈	1	4	yes	—
130	C₄×C₁₉₇	C₂×#41	1	4	yes	—
131	Dic₁₉₇	—	2	2	no	—
132	Dic₁₉₇	—	2	4	no	—
133	Dic₁₉₇	—	2	4	no	—
134	Dic₁₉₇	C₂²×Q₈	1	4	yes	—
135	Dic₁₉₇	C₂×#41	1	4	yes	—
136	C₂×C₁₉₇	#533	1	4	yes	complemented
137	C₂×C₁₉₇	C₂³×Q₈	1	1	yes	characteristic
138	C₂×C₁₉₇	C₂²×#41	1	1	yes	characteristic
139	C₂×C₁₉₇	C₂²×#41	1	1	yes	characteristic
140	D₁₉₇	—	2	4	no	—
141	C₁₉₇	C₂×#533	1	1	yes	characteristic, complemented, Sylow
142	C₂×#533	—	197	1	no	maximal, Sylow
143	#533	—	197	16	no	—
144	C₂×C₄⋊Q₈	—	197	1	no	—
145	C₂×C₄⋊Q₈	—	197	1	no	—
146	C₂×#125	—	197	2	no	—
147	C₂×C₂²⋊Q₈	—	197	1	no	—
148	C₂×C₂²⋊Q₈	—	197	2	no	—
149	C₂×C₂²⋊Q₈	—	197	1	no	—
150	C₂×C₄×Q₈	—	197	1	no	—
151	C₂×C₄×Q₈	—	197	2	no	—
152	C₂×C₄×Q₈	D₁₉₇	1	1	yes	characteristic, complemented
153	C₂×#117	—	197	2	no	—
154	C₂²×C₄²	—	197	1	no	—
155	C₂²×Q₈	—	197	1	no	—
156	C₂²×Q₈	—	197	2	no	—
157	C₂²×Q₈	C₂×D₁₉₇	1	1	yes	characteristic
158	C₂³×C₄	—	197	1	no	—
159	C₂³×C₄	—	197	1	no	—
160	C₂³×C₄	—	197	1	no	—
161	C₄⋊Q₈	—	197	8	no	—
162	C₄⋊Q₈	—	197	8	no	—
163	#125	—	197	16	no	—
164	C₂²⋊Q₈	—	197	8	no	—
165	C₂²⋊Q₈	—	197	16	no	—
166	C₂²⋊Q₈	—	197	8	no	—
167	C₄×Q₈	—	197	16	no	—
168	C₄×Q₈	—	197	8	no	—
169	C₄×Q₈	C₂×D₁₉₇	1	8	yes	complemented
170	#117	—	197	16	no	—
171	C₂×C₄⋊C₄	—	197	2	no	—
172	C₂×C₄⋊C₄	—	197	2	no	—
173	C₂×C₄⋊C₄	—	197	2	no	—
174	C₂×C₄⋊C₄	—	197	2	no	—
175	C₂×C₄⋊C₄	—	197	2	no	—
176	C₂×C₄⋊C₄	—	197	2	no	—
177	C₂×C₄⋊C₄	—	197	1	no	—
178	C₂×C₄⋊C₄	C₂×D₁₉₇	1	2	yes	—
179	C₂×C₄⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
180	C₂×C₂²⋊C₄	—	197	2	no	—
181	C₂×C₂²⋊C₄	—	197	2	no	—
182	C₂×C₄²	—	197	1	no	—
183	C₂×C₄²	—	197	1	no	—
184	C₂×C₄²	—	197	2	no	—
185	C₂×C₄²	—	197	1	no	—
186	C₂×C₄²	—	197	8	no	—
187	C₂×C₄²	C₂×D₁₉₇	1	2	yes	—
188	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
189	C₂⁴	—	197	1	no	—
190	C₂×Q₈	—	2	4	no	—
191	C₂×Q₈	—	394	8	no	—
192	C₂×Q₈	—	394	4	no	—
193	C₂×Q₈	—	197	4	no	—
194	C₂×Q₈	—	197	8	no	—
195	C₂×Q₈	C₂²×D₁₉₇	1	4	yes	—
196	C₂²×C₄	—	394	4	no	—
197	C₂²×C₄	—	394	4	no	—
198	C₂²×C₄	—	197	1	no	—
199	C₂²×C₄	—	197	2	no	—
200	C₂²×C₄	—	197	2	no	—
201	C₂²×C₄	—	197	4	no	—
202	C₂²×C₄	—	197	4	no	—
203	C₂²×C₄	—	197	1	no	—
204	C₂²×C₄	—	197	1	no	—
205	C₂²×C₄	—	197	8	no	—
206	C₂²×C₄	—	197	4	no	—
207	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
208	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
209	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
210	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
211	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
212	C₄⋊C₄	—	197	8	no	—
213	C₄⋊C₄	—	197	8	no	—
214	C₄⋊C₄	—	197	8	no	—
215	C₄⋊C₄	—	197	8	no	—
216	C₄⋊C₄	—	197	8	no	—
217	C₄⋊C₄	—	197	8	no	—
218	C₄⋊C₄	—	197	4	no	—
219	C₄⋊C₄	C₂²×D₁₉₇	1	8	yes	—
220	C₄⋊C₄	C₂²×D₁₉₇	1	4	yes	—
221	C₂²⋊C₄	—	197	8	no	—
222	C₂²⋊C₄	—	197	8	no	—
223	C₄²	—	197	4	no	—
224	C₄²	—	197	4	no	—
225	C₄²	—	197	4	no	—
226	C₄²	—	197	8	no	—
227	C₄²	C₂²×D₁₉₇	1	4	yes	—
228	C₄²	C₂²×D₁₉₇	1	8	yes	—
229	C₂³	—	394	2	no	—
230	C₂³	—	394	2	no	—
231	C₂³	—	197	4	no	—
232	C₂³	—	197	2	no	—
233	C₂³	C₂³×D₁₉₇	1	1	yes	characteristic
234	Q₈	—	2	8	no	—
235	Q₈	—	394	16	no	—
236	Q₈	—	394	8	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	394	8	no	—
241	C₂×C₄	—	394	4	no	—
242	C₂×C₄	—	394	2	no	—
243	C₂×C₄	—	394	4	no	—
244	C₂×C₄	—	394	8	no	—
245	C₂×C₄	—	197	2	no	—
246	C₂×C₄	—	197	4	no	—
247	C₂×C₄	—	197	4	no	—
248	C₂×C₄	—	197	4	no	—
249	C₂×C₄	—	197	4	no	—
250	C₂×C₄	—	197	2	no	—
251	C₂×C₄	—	197	2	no	—
252	C₂×C₄	—	197	16	no	—
253	C₂×C₄	Q₈×D₁₉₇	1	4	yes	—
254	C₂×C₄	#15691358	1	4	yes	—
255	C₂×C₄	C₂³×D₁₉₇	1	2	yes	—
256	C₂×C₄	C₂³×D₁₉₇	1	4	yes	—
257	C₂×C₄	C₂³×D₁₉₇	1	4	yes	—
258	C₂×C₄	C₂³×D₁₉₇	1	2	yes	—
259	C₂×C₄	C₂³×D₁₉₇	1	2	yes	—
260	C₂²	—	394	8	no	—
261	C₂²	—	394	2	no	—
262	C₂²	—	394	2	no	—
263	C₂²	—	197	4	no	—
264	C₂²	C₂×Q₈×D₁₉₇	1	2	yes	—
265	C₂²	#41×D₁₉₇	1	2	yes	—
266	C₂²	C₂×#15691358	1	2	yes	—
267	C₂²	C₂⁴×D₁₉₇	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	4	no	—
270	C₄	—	2	4	no	—
271	C₄	—	394	4	no	—
272	C₄	—	394	2	no	—
273	C₄	—	394	4	no	—
274	C₄	—	197	4	no	—
275	C₄	—	197	4	no	—
276	C₄	C₂×Q₈×D₁₉₇	1	4	yes	—
277	C₄	C₂×#15691358	1	4	yes	—
278	C₂	—	394	4	no	—
279	C₂	#27161633	1	4	yes	direct product factor
280	C₂	C₂²×Q₈×D₁₉₇	1	1	yes	characteristic
281	C₂	C₂×#41×D₁₉₇	1	1	yes	characteristic
282	C₂	C₂²×#15691358	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₉₇
commutator, cosocle	C₂²×C₁₉₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₉₇
Fitting	C₂×C₄×Q₈×C₁₉₇	C₂
socle	C₂³×C₁₉₇	C₂⁴
prime radical	C₂³×D₁₉₇	C₂³
hypercenter	C₂×C₄×Q₈	D₁₉₇
hypocenter	C₁₉₇	C₂×#533
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #27161633	C₂	direct (N × H)
2. C₂×#20263325	C₂	semidirect
3. C₂×#20263324	C₂	semidirect
4. C₂×#20263316	C₂	semidirect
8. C₂×C₄×Dic₃₉₄	C₂	semidirect
9. C₂×C₄×Dic₃₉₄	C₂	semidirect
12. C₂×C₄×Q₈×C₁₉₇	C₂	semidirect
18. #20263325	C₂²	semidirect
19. #20263324	C₂²	semidirect
20. #20263316	C₂²	semidirect
24. C₄×Dic₃₉₄	C₂²	semidirect
25. C₄×Dic₃₉₄	C₂²	semidirect
42. C₄×Q₈×C₁₉₇	C₂²	semidirect
59. C₂×C₄×D₁₉₇	Q₈	semidirect
105. C₄×D₁₉₇	C₂×Q₈	semidirect
136. C₂×C₁₉₇	#533	semidirect
141. C₁₉₇	C₂×#533	semidirect
152. C₂×C₄×Q₈	D₁₉₇	semidirect
169. C₄×Q₈	C₂×D₁₉₇	semidirect
279. C₂	#27161633	direct (N × H)

Maximal subgroups

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

subgroup	index
#27161633	2
C₂×#20263325	2
C₂×#20263324	2
C₂×#20263316	2
C₂×#20263306	2
C₂×#20263304	2
C₂×#20263302	2
C₂×C₄×Dic₃₉₄	2
C₂×#20263283	2
C₂×C₄²×D₁₉₇	2
C₂×C₄×Q₈×C₁₉₇	2
C₂×#533	197

Subgroups — C2×#27161633 #38756000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27161633 #38756000