Subgroups — C₂×#27161590 #38755954

Order 25216.

1880 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15691358	C₂	1	1	yes	maximal, characteristic, complemented
2	#27161590	C₂	1	16	yes	maximal, direct product factor
3	C₂×#20263309	C₂	1	2	yes	maximal, complemented
4	C₂×#20263305	C₂	1	2	yes	maximal, complemented
5	C₂×#20263300	C₂	1	2	yes	maximal, complemented
6	C₂×#20263295	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₂²⋊D₁₉₇	C₂	1	4	yes	maximal, complemented
8	C₂×#20263285	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×C₄×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15691358	—	2	8	no	—
11	C₂×#15691358	C₂²	1	8	yes	complemented
12	C₂²×Dic₃₉₄	C₂²	1	1	yes	characteristic, complemented
13	C₂²×D₇₈₈	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₂²⋊D₁₉₇	C₂²	1	2	yes	—
15	C₂²×C₂²⋊D₁₉₇	C₂²	1	4	yes	—
16	C₂²×C₄×D₁₉₇	C₂²	1	2	yes	complemented
17	#20263309	C₂²	1	16	yes	complemented
18	#20263305	C₂²	1	16	yes	complemented
19	#20263300	C₂²	1	16	yes	complemented
20	#20263295	C₂²	1	16	yes	complemented
21	C₄×C₂²⋊D₁₉₇	C₂²	1	32	yes	complemented
22	#20263285	C₂²	1	8	yes	complemented
23	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	complemented
24	C₂×#15691337	C₂²	1	4	yes	complemented
25	C₂×C₂²⋊Dic₁₉₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊Dic₁₉₇	C₂²	1	4	yes	—
27	C₂×#15691335	C₂²	1	4	yes	complemented
28	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	complemented
29	C₂⁵×C₁₉₇	C₂²	1	1	yes	characteristic
30	C₂³×C₄×C₁₉₇	—	2	8	no	—
31	C₂³×C₄×C₁₉₇	C₂²	1	2	yes	—
32	C₂³×C₄×C₁₉₇	C₂²	1	2	yes	—
33	C₂³×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
34	C₂³×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
35	C₂³×C₄×C₁₉₇	C₂²	1	8	yes	complemented
36	C₂³×D₁₉₇	C₂³	1	2	yes	—
37	#15691358	—	2	32	no	—
38	C₂×Dic₃₉₄	—	2	4	no	—
39	C₂×Dic₃₉₄	C₂³	1	4	yes	complemented
40	C₂×D₇₈₈	—	2	4	no	—
41	C₂×D₇₈₈	C₂³	1	4	yes	complemented
42	C₂×C₂²⋊D₁₉₇	—	2	8	no	—
43	C₂×C₂²⋊D₁₉₇	—	2	16	no	—
44	C₂×C₂²⋊D₁₉₇	C₂³	1	16	yes	—
45	C₂×C₂²⋊D₁₉₇	C₂³	1	8	yes	—
46	C₂²×Dic₁₉₇	C₂³	1	4	yes	—
47	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
48	C₂×C₄×D₁₉₇	—	2	8	no	—
49	C₂×C₄×D₁₉₇	C₂³	1	8	yes	complemented
50	C₄⋊Dic₁₉₇	C₂³	1	8	yes	complemented
51	#15691337	C₂³	1	16	yes	complemented
52	C₂²⋊Dic₁₉₇	C₂³	1	8	yes	—
53	C₂²⋊Dic₁₉₇	C₂³	1	16	yes	—
54	#15691335	C₂³	1	16	yes	complemented
55	C₄×Dic₁₉₇	C₂³	1	8	yes	complemented
56	C₂⁴×C₁₉₇	—	2	4	no	—
57	C₂⁴×C₁₉₇	—	2	8	no	—
58	C₂⁴×C₁₉₇	C₂³	1	4	yes	—
59	C₂⁴×C₁₉₇	C₂³	1	1	yes	characteristic
60	C₂⁴×C₁₉₇	C₂³	1	2	yes	—
61	C₂²×C₄×C₁₉₇	—	2	4	no	—
62	C₂²×C₄×C₁₉₇	—	2	8	no	—
63	C₂²×C₄×C₁₉₇	—	2	8	no	—
64	C₂²×C₄×C₁₉₇	—	2	32	no	—
65	C₂²×C₄×C₁₉₇	D₄	1	8	yes	complemented
66	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
67	C₂²×C₄×C₁₉₇	C₂³	1	8	yes	—
68	C₂²×C₄×C₁₉₇	C₂³	1	8	yes	—
69	C₂²×C₄×C₁₉₇	C₂³	1	4	yes	—
70	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₁₉₇	C₂³	1	4	yes	—
72	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
73	C₂²×D₁₉₇	—	2	4	no	—
74	C₂²×D₁₉₇	—	2	8	no	—
75	C₂²×D₁₉₇	C₂⁴	1	4	yes	—
76	Dic₃₉₄	—	2	8	no	—
77	D₇₈₈	—	2	8	no	—
78	C₂²⋊D₁₉₇	—	2	16	no	—
79	C₂²⋊D₁₉₇	—	2	32	no	—
80	C₂×Dic₁₉₇	—	2	4	no	—
81	C₂×Dic₁₉₇	—	2	8	no	—
82	C₂×Dic₁₉₇	C₂⁴	1	8	yes	—
83	C₂×Dic₁₉₇	C₂⁴	1	4	yes	—
84	C₄×D₁₉₇	—	2	16	no	—
85	C₂³×C₁₉₇	—	2	32	no	—
86	C₂³×C₁₉₇	—	2	8	no	—
87	C₂³×C₁₉₇	—	2	4	no	—
88	C₂³×C₁₉₇	—	2	4	no	—
89	C₂³×C₁₉₇	—	2	4	no	—
90	C₂³×C₁₉₇	—	2	16	no	—
91	C₂³×C₁₉₇	C₂×D₄	1	4	yes	—
92	C₂³×C₁₉₇	#41	1	8	yes	complemented
93	C₂³×C₁₉₇	C₂⁴	1	2	yes	—
94	C₂³×C₁₉₇	C₂⁴	1	4	yes	—
95	C₂³×C₁₉₇	C₂⁴	1	1	yes	characteristic
96	C₂×C₄×C₁₉₇	—	2	8	no	—
97	C₂×C₄×C₁₉₇	—	2	4	no	—
98	C₂×C₄×C₁₉₇	—	2	16	no	—
99	C₂×C₄×C₁₉₇	—	2	16	no	—
100	C₂×C₄×C₁₉₇	C₂×D₄	1	4	yes	—
101	C₂×C₄×C₁₉₇	C₂×D₄	1	4	yes	—
102	C₂×C₄×C₁₉₇	C₂×D₄	1	16	yes	complemented
103	C₂×C₄×C₁₉₇	C₂⁴	1	4	yes	—
104	C₂×C₄×C₁₉₇	C₂⁴	1	2	yes	—
105	C₂×C₄×C₁₉₇	C₂⁴	1	2	yes	—
106	C₂²×C₁₉₇	—	2	32	no	—
107	C₂²×C₁₉₇	—	2	8	no	—
108	C₂²×C₁₉₇	—	2	16	no	—
109	C₂²×C₁₉₇	—	2	4	no	—
110	C₂²×C₁₉₇	—	2	4	no	—
111	C₂²×C₁₉₇	—	2	4	no	—
112	C₂²×C₁₉₇	C₂²×D₄	1	4	yes	—
113	C₂²×C₁₉₇	C₂²×D₄	1	2	yes	—
114	C₂²×C₁₉₇	C₂×#41	1	4	yes	—
115	C₂²×C₁₉₇	C₂×#41	1	8	yes	complemented
116	C₂²×C₁₉₇	C₂⁵	1	1	yes	characteristic
117	C₂×D₁₉₇	—	2	4	no	—
118	C₂×D₁₉₇	—	2	8	no	—
119	C₂×D₁₉₇	—	4	8	no	—
120	C₄×C₁₉₇	—	2	4	no	—
121	C₄×C₁₉₇	C₂²×D₄	1	4	yes	—
122	C₄×C₁₉₇	C₂²×D₄	1	4	yes	—
123	Dic₁₉₇	—	2	4	no	—
124	Dic₁₉₇	—	2	8	no	—
125	C₂×C₁₉₇	—	2	4	no	—
126	C₂×C₁₉₇	—	2	8	no	—
127	C₂×C₁₉₇	#525	1	4	yes	complemented
128	C₂×C₁₉₇	C₂³×D₄	1	1	yes	characteristic
129	C₂×C₁₉₇	C₂²×#41	1	2	yes	—
130	D₁₉₇	—	4	4	no	—
131	C₁₉₇	C₂×#525	1	1	yes	characteristic, complemented, Sylow
132	C₂×#525	—	197	1	no	maximal, Sylow
133	C₂²×#41	—	197	1	no	—
134	C₂⁴×C₄	D₁₉₇	1	1	yes	characteristic, complemented
135	#525	—	197	16	no	—
136	C₂×#123	—	197	2	no	—
137	C₂×C₂²⋊Q₈	—	197	2	no	—
138	C₂×#121	—	197	2	no	—
139	C₂×#120	—	197	2	no	—
140	C₂×C₄×D₄	—	197	4	no	—
141	C₂×#117	—	197	1	no	—
142	C₂⁵	C₂×D₁₉₇	1	1	yes	characteristic
143	C₂×#41	—	394	8	no	—
144	C₂×#41	—	197	8	no	—
145	C₂²×Q₈	—	197	1	no	—
146	C₂²×D₄	—	197	4	no	—
147	C₂²×D₄	—	197	2	no	—
148	C₂²×D₄	—	197	1	no	—
149	C₂³×C₄	—	2	8	no	—
150	C₂³×C₄	—	197	2	no	—
151	C₂³×C₄	C₂×D₁₉₇	1	2	yes	—
152	C₂³×C₄	C₂×D₁₉₇	1	1	yes	characteristic
153	C₂³×C₄	C₂×D₁₉₇	1	2	yes	—
154	C₂³×C₄	C₂×D₁₉₇	1	1	yes	characteristic
155	C₂³×C₄	C₂×D₁₉₇	1	8	yes	complemented
156	#123	—	197	16	no	—
157	C₂²⋊Q₈	—	197	16	no	—
158	#121	—	197	16	no	—
159	#120	—	197	16	no	—
160	C₄×D₄	—	197	32	no	—
161	#117	—	197	8	no	—
162	C₂×C₄⋊C₄	—	197	2	no	—
163	C₂×C₄⋊C₄	—	197	4	no	—
164	C₂×C₂²⋊C₄	—	197	2	no	—
165	C₂×C₂²⋊C₄	—	197	4	no	—
166	C₂×C₂²⋊C₄	—	197	4	no	—
167	C₂×C₄²	—	197	2	no	—
168	C₂⁴	—	2	4	no	—
169	C₂⁴	—	2	8	no	—
170	C₂⁴	—	197	2	no	—
171	C₂⁴	C₂²×D₁₉₇	1	4	yes	—
172	C₂⁴	C₂²×D₁₉₇	1	1	yes	characteristic
173	C₂⁴	C₂²×D₁₉₇	1	2	yes	—
174	#41	—	394	32	no	—
175	C₂×Q₈	—	394	4	no	—
176	C₂×Q₈	—	197	4	no	—
177	C₂×D₄	—	394	8	no	—
178	C₂×D₄	—	394	16	no	—
179	C₂×D₄	—	394	4	no	—
180	C₂×D₄	—	197	16	no	—
181	C₂×D₄	—	197	8	no	—
182	C₂×D₄	—	197	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	—	2	8	no	—
186	C₂²×C₄	—	2	32	no	—
187	C₂²×C₄	—	394	8	no	—
188	C₂²×C₄	—	197	4	no	—
189	C₂²×C₄	—	197	2	no	—
190	C₂²×C₄	—	197	8	no	—
191	C₂²×C₄	C₂²⋊D₁₉₇	1	8	yes	complemented
192	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
193	C₂²×C₄	C₂²×D₁₉₇	1	8	yes	—
194	C₂²×C₄	C₂²×D₁₉₇	1	8	yes	—
195	C₂²×C₄	C₂²×D₁₉₇	1	4	yes	—
196	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
197	C₂²×C₄	C₂²×D₁₉₇	1	4	yes	—
198	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
199	C₄⋊C₄	—	197	16	no	—
200	C₄⋊C₄	—	197	8	no	—
201	C₂²⋊C₄	—	197	8	no	—
202	C₂²⋊C₄	—	197	16	no	—
203	C₂²⋊C₄	—	197	16	no	—
204	C₄²	—	197	8	no	—
205	C₂³	—	2	32	no	—
206	C₂³	—	2	8	no	—
207	C₂³	—	2	4	no	—
208	C₂³	—	2	4	no	—
209	C₂³	—	2	4	no	—
210	C₂³	—	2	16	no	—
211	C₂³	—	394	4	no	—
212	C₂³	—	394	8	no	—
213	C₂³	—	197	4	no	—
214	C₂³	C₂×C₂²⋊D₁₉₇	1	4	yes	—
215	C₂³	#15691358	1	8	yes	complemented
216	C₂³	C₂³×D₁₉₇	1	2	yes	—
217	C₂³	C₂³×D₁₉₇	1	4	yes	—
218	C₂³	C₂³×D₁₉₇	1	1	yes	characteristic
219	Q₈	—	394	8	no	—
220	D₄	—	394	16	no	—
221	D₄	—	394	32	no	—
222	D₄	—	394	8	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	16	no	—
226	C₂×C₄	—	2	16	no	—
227	C₂×C₄	—	394	4	no	—
228	C₂×C₄	—	394	8	no	—
229	C₂×C₄	—	394	16	no	—
230	C₂×C₄	—	197	8	no	—
231	C₂×C₄	—	197	4	no	—
232	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	4	yes	—
233	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	4	yes	—
234	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	16	yes	complemented
235	C₂×C₄	C₂³×D₁₉₇	1	4	yes	—
236	C₂×C₄	C₂³×D₁₉₇	1	2	yes	—
237	C₂×C₄	C₂³×D₁₉₇	1	2	yes	—
238	C₂²	—	2	32	no	—
239	C₂²	—	2	4	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	8	no	—
243	C₂²	—	2	16	no	—
244	C₂²	—	788	8	no	—
245	C₂²	—	394	4	no	—
246	C₂²	—	394	8	no	—
247	C₂²	C₂²×C₂²⋊D₁₉₇	1	4	yes	—
248	C₂²	C₂²×C₂²⋊D₁₉₇	1	2	yes	—
249	C₂²	C₂×#15691358	1	4	yes	—
250	C₂²	C₂×#15691358	1	8	yes	complemented
251	C₂²	C₂⁴×D₁₉₇	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	394	4	no	—
254	C₄	—	394	8	no	—
255	C₄	C₂²×C₂²⋊D₁₉₇	1	4	yes	—
256	C₄	C₂²×C₂²⋊D₁₉₇	1	4	yes	—
257	C₂	—	2	4	no	—
258	C₂	—	2	8	no	—
259	C₂	—	788	4	no	—
260	C₂	#27161590	1	4	yes	direct product factor
261	C₂	C₂³×C₂²⋊D₁₉₇	1	1	yes	characteristic
262	C₂	C₂²×#15691358	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₉₇
commutator, cosocle	C₂²×C₁₉₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₉₇
Fitting	C₂⁴×C₄×C₁₉₇	C₂
socle	C₂³×C₁₉₇	C₂⁴
hypercenter	C₂⁴×C₄	D₁₉₇
hypocenter	C₁₉₇	C₂×#525
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#15691358	C₂	semidirect
2. #27161590	C₂	direct (N × H)
3. C₂×#20263309	C₂	semidirect
4. C₂×#20263305	C₂	semidirect
5. C₂×#20263300	C₂	semidirect
6. C₂×#20263295	C₂	semidirect
7. C₂×C₄×C₂²⋊D₁₉₇	C₂	semidirect
8. C₂×#20263285	C₂	semidirect
9. C₂⁴×C₄×C₁₉₇	C₂	semidirect
11. C₂×#15691358	C₂²	semidirect
12. C₂²×Dic₃₉₄	C₂²	semidirect
13. C₂²×D₇₈₈	C₂²	semidirect
16. C₂²×C₄×D₁₉₇	C₂²	semidirect
17. #20263309	C₂²	semidirect
18. #20263305	C₂²	semidirect
19. #20263300	C₂²	semidirect
20. #20263295	C₂²	semidirect
21. C₄×C₂²⋊D₁₉₇	C₂²	semidirect
22. #20263285	C₂²	semidirect
23. C₂×C₄⋊Dic₁₉₇	C₂²	semidirect
24. C₂×#15691337	C₂²	semidirect
27. C₂×#15691335	C₂²	semidirect
28. C₂×C₄×Dic₁₉₇	C₂²	semidirect
35. C₂³×C₄×C₁₉₇	C₂²	semidirect
39. C₂×Dic₃₉₄	C₂³	semidirect
41. C₂×D₇₈₈	C₂³	semidirect
49. C₂×C₄×D₁₉₇	C₂³	semidirect
50. C₄⋊Dic₁₉₇	C₂³	semidirect
51. #15691337	C₂³	semidirect
54. #15691335	C₂³	semidirect
55. C₄×Dic₁₉₇	C₂³	semidirect
65. C₂²×C₄×C₁₉₇	D₄	semidirect
92. C₂³×C₁₉₇	#41	semidirect
102. C₂×C₄×C₁₉₇	C₂×D₄	semidirect
115. C₂²×C₁₉₇	C₂×#41	semidirect
127. C₂×C₁₉₇	#525	semidirect
131. C₁₉₇	C₂×#525	semidirect
134. C₂⁴×C₄	D₁₉₇	semidirect
155. C₂³×C₄	C₂×D₁₉₇	semidirect
191. C₂²×C₄	C₂²⋊D₁₉₇	semidirect
215. C₂³	#15691358	semidirect
234. C₂×C₄	C₂×C₂²⋊D₁₉₇	semidirect
250. C₂²	C₂×#15691358	semidirect
260. C₂	#27161590	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#15691358	2
#27161590	2
C₂×#20263309	2
C₂×#20263305	2
C₂×#20263300	2
C₂×#20263295	2
C₂×C₄×C₂²⋊D₁₉₇	2
C₂×#20263285	2
C₂⁴×C₄×C₁₉₇	2
C₂×#525	197

Subgroups — C2×#27161590 #38755954

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27161590 #38755954