Subgroups — C₂×#27161689 #38756078

Order 25216.

1004 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27161689	C₂	1	16	yes	maximal, direct product factor
2	C₂×#20263324	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#20263319	C₂	1	2	yes	maximal
4	C₂×#20263319	C₂	1	2	yes	maximal
5	C₂×#20263317	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#20263317	C₂	1	2	yes	maximal, complemented
7	C₂×#20263316	C₂	1	2	yes	maximal, complemented
8	C₂×#20263305	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₈×Dic₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#20263291	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#20263285	C₂	1	1	yes	maximal, characteristic
12	C₂×C₂²⋊Q₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic
13	C₂²×Dic₃₉₄	C₂²	1	1	yes	characteristic
14	#20263324	C₂²	1	8	yes	complemented
15	#20263319	C₂²	1	16	yes	—
16	#20263319	C₂²	1	16	yes	—
17	#20263317	C₂²	1	8	yes	complemented
18	#20263317	C₂²	1	16	yes	complemented
19	#20263316	C₂²	1	16	yes	complemented
20	#20263305	C₂²	1	8	yes	—
21	Q₈×Dic₁₉₇	C₂²	1	8	yes	complemented
22	#20263291	C₂²	1	8	yes	complemented
23	#20263285	C₂²	1	8	yes	—
24	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
25	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
27	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
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×C₂²⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
36	C₂×C₂²⋊Dic₁₉₇	C₂²	1	2	yes	—
37	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
38	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
39	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
40	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
42	C₂³×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
43	C₂²⋊Q₈×C₁₉₇	C₂²	1	8	yes	—
44	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	2	yes	—
46	C₂×C₂²⋊C₄×C₁₉₇	C₂²	1	2	yes	—
47	C₂×Dic₃₉₄	—	2	4	no	—
48	C₂×Dic₃₉₄	C₂³	1	4	yes	—
49	C₂²×Dic₁₉₇	C₂³	1	1	yes	characteristic
50	C₂²×Dic₁₉₇	C₂³	1	1	yes	characteristic
51	C₂²×Dic₁₉₇	C₂³	1	1	yes	characteristic
52	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
53	C₂²×Dic₁₉₇	C₂³	1	1	yes	characteristic
54	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
55	C₄⋊Dic₁₉₇	C₂³	1	4	yes	—
56	C₄⋊Dic₁₉₇	C₂³	1	4	yes	—
57	C₄⋊Dic₁₉₇	C₂³	1	4	yes	—
58	C₄⋊Dic₁₉₇	C₂³	1	8	yes	—
59	C₄⋊Dic₁₉₇	C₂³	1	8	yes	—
60	#15691337	C₂³	1	8	yes	—
61	#15691337	C₂³	1	8	yes	—
62	#15691337	C₂³	1	8	yes	—
63	#15691337	C₂³	1	8	yes	—
64	#15691337	C₂³	1	8	yes	—
65	C₂²⋊Dic₁₉₇	C₂³	1	4	yes	—
66	C₂²⋊Dic₁₉₇	C₂³	1	4	yes	—
67	C₂²⋊Dic₁₉₇	C₂³	1	8	yes	—
68	C₄×Dic₁₉₇	C₂³	1	4	yes	—
69	C₄×Dic₁₉₇	C₂³	1	4	yes	—
70	C₄×Dic₁₉₇	C₂³	1	8	yes	—
71	C₄×Dic₁₉₇	C₂³	1	8	yes	—
72	C₂⁴×C₁₉₇	C₂³	1	1	yes	characteristic
73	C₂×Q₈×C₁₉₇	—	2	4	no	—
74	C₂×Q₈×C₁₉₇	C₂³	1	4	yes	—
75	C₂²×C₄×C₁₉₇	—	2	4	no	—
76	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
77	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
78	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₁₉₇	C₂³	1	4	yes	—
80	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₁₉₇	C₂³	1	4	yes	—
82	C₄⋊C₄×C₁₉₇	C₂³	1	8	yes	—
83	C₂²⋊C₄×C₁₉₇	C₂³	1	8	yes	—
84	Dic₃₉₄	—	2	8	no	—
85	C₂×Dic₁₉₇	—	2	4	no	—
86	C₂×Dic₁₉₇	—	2	2	no	—
87	C₂×Dic₁₉₇	—	2	2	no	—
88	C₂×Dic₁₉₇	—	2	2	no	—
89	C₂×Dic₁₉₇	—	2	2	no	—
90	C₂×Dic₁₉₇	—	2	4	no	—
91	C₂×Dic₁₉₇	C₂⁴	1	2	yes	—
92	C₂×Dic₁₉₇	C₂⁴	1	2	yes	—
93	C₂×Dic₁₉₇	C₂⁴	1	2	yes	—
94	C₂×Dic₁₉₇	C₂⁴	1	4	yes	—
95	C₂×Dic₁₉₇	C₂⁴	1	2	yes	—
96	C₂×Dic₁₉₇	C₂⁴	1	4	yes	—
97	C₂³×C₁₉₇	—	2	2	no	—
98	C₂³×C₁₉₇	—	2	2	no	—
99	C₂³×C₁₉₇	—	2	2	no	—
100	C₂³×C₁₉₇	C₂⁴	1	2	yes	—
101	C₂³×C₁₉₇	C₂⁴	1	1	yes	characteristic
102	Q₈×C₁₉₇	—	2	8	no	—
103	C₂×C₄×C₁₉₇	—	2	2	no	—
104	C₂×C₄×C₁₉₇	—	2	4	no	—
105	C₂×C₄×C₁₉₇	—	2	4	no	—
106	C₂×C₄×C₁₉₇	—	2	8	no	—
107	C₂×C₄×C₁₉₇	#41	1	4	yes	—
108	C₂×C₄×C₁₉₇	C₂⁴	1	4	yes	—
109	C₂×C₄×C₁₉₇	C₂⁴	1	4	yes	—
110	C₂×C₄×C₁₉₇	C₂⁴	1	2	yes	—
111	C₂×C₄×C₁₉₇	C₂⁴	1	2	yes	—
112	C₂²×C₁₉₇	—	2	2	no	—
113	C₂²×C₁₉₇	—	2	2	no	—
114	C₂²×C₁₉₇	—	2	2	no	—
115	C₂²×C₁₉₇	—	4	4	no	—
116	C₂²×C₁₉₇	C₂×#41	1	2	yes	—
117	C₂²×C₁₉₇	2₋¹⁺⁴	1	2	yes	—
118	C₂²×C₁₉₇	2₋¹⁺⁴	1	2	yes	—
119	C₂²×C₁₉₇	C₂⁵	1	1	yes	characteristic
120	C₄×C₁₉₇	—	2	2	no	—
121	C₄×C₁₉₇	—	2	4	no	—
122	C₄×C₁₉₇	—	2	4	no	—
123	C₄×C₁₉₇	C₂×#41	1	4	yes	—
124	Dic₁₉₇	—	2	4	no	—
125	Dic₁₉₇	—	2	2	no	—
126	Dic₁₉₇	—	2	2	no	—
127	Dic₁₉₇	—	2	2	no	—
128	Dic₁₉₇	—	2	2	no	—
129	Dic₁₉₇	—	2	4	no	—
130	C₂×C₁₉₇	—	4	2	no	—
131	C₂×C₁₉₇	#541	1	4	yes	complemented
132	C₂×C₁₉₇	C₂²×#41	1	1	yes	characteristic
133	C₂×C₁₉₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
134	C₂×C₁₉₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
135	C₁₉₇	C₂×#541	1	1	yes	characteristic, complemented, Sylow
136	C₂×#541	—	197	1	no	maximal, Sylow
137	#541	—	197	16	no	—
138	C₂×C₄⋊Q₈	—	197	1	no	—
139	C₂×#126	—	197	2	no	—
140	C₂×#126	—	197	2	no	—
141	C₂×#125	—	197	1	no	—
142	C₂×#125	—	197	2	no	—
143	C₂×#125	—	197	2	no	—
144	C₂×C₂²⋊Q₈	—	197	1	no	—
145	C₂×C₂²⋊Q₈	D₁₉₇	1	1	yes	characteristic
146	C₂×C₄×Q₈	—	197	1	no	—
147	C₂×C₄×Q₈	—	197	1	no	—
148	C₂×#117	—	197	1	no	—
149	C₂²×Q₈	—	197	1	no	—
150	C₂²×Q₈	C₂×D₁₉₇	1	1	yes	characteristic
151	C₂³×C₄	C₂×D₁₉₇	1	1	yes	characteristic
152	C₄⋊Q₈	—	197	8	no	—
153	#126	—	197	16	no	—
154	#126	—	197	16	no	—
155	#125	—	197	16	no	—
156	#125	—	197	8	no	—
157	#125	—	197	16	no	—
158	C₂²⋊Q₈	—	197	8	no	—
159	C₂²⋊Q₈	C₂×D₁₉₇	1	8	yes	—
160	C₄×Q₈	—	197	8	no	—
161	C₄×Q₈	—	197	8	no	—
162	#117	—	197	8	no	—
163	C₂×C₄⋊C₄	—	197	1	no	—
164	C₂×C₄⋊C₄	—	197	1	no	—
165	C₂×C₄⋊C₄	—	197	1	no	—
166	C₂×C₄⋊C₄	—	197	2	no	—
167	C₂×C₄⋊C₄	—	197	2	no	—
168	C₂×C₄⋊C₄	—	197	2	no	—
169	C₂×C₄⋊C₄	—	197	2	no	—
170	C₂×C₄⋊C₄	—	197	2	no	—
171	C₂×C₄⋊C₄	—	197	2	no	—
172	C₂×C₄⋊C₄	—	197	2	no	—
173	C₂×C₄⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
174	C₂×C₄⋊C₄	C₂×D₁₉₇	1	2	yes	—
175	C₂×C₂²⋊C₄	—	197	1	no	—
176	C₂×C₂²⋊C₄	—	197	1	no	—
177	C₂×C₂²⋊C₄	—	197	2	no	—
178	C₂×C₂²⋊C₄	C₂×D₁₉₇	1	2	yes	—
179	C₂×C₄²	—	197	1	no	—
180	C₂×C₄²	—	197	1	no	—
181	C₂×C₄²	—	197	2	no	—
182	C₂×C₄²	—	197	2	no	—
183	C₂⁴	C₂²×D₁₉₇	1	1	yes	characteristic
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	—	394	4	no	—
186	C₂×Q₈	—	197	4	no	—
187	C₂×Q₈	C₂²×D₁₉₇	1	4	yes	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	197	1	no	—
190	C₂²×C₄	—	197	1	no	—
191	C₂²×C₄	—	197	1	no	—
192	C₂²×C₄	—	197	1	no	—
193	C₂²×C₄	—	197	2	no	—
194	C₂²×C₄	—	197	2	no	—
195	C₂²×C₄	C₂²×D₁₉₇	1	4	yes	—
196	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
197	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
198	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
199	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
200	C₄⋊C₄	—	197	8	no	—
201	C₄⋊C₄	—	197	8	no	—
202	C₄⋊C₄	—	197	8	no	—
203	C₄⋊C₄	—	197	8	no	—
204	C₄⋊C₄	—	197	8	no	—
205	C₄⋊C₄	—	197	4	no	—
206	C₄⋊C₄	—	197	4	no	—
207	C₄⋊C₄	—	197	4	no	—
208	C₄⋊C₄	—	197	8	no	—
209	C₄⋊C₄	—	197	8	no	—
210	C₄⋊C₄	C₂²×D₁₉₇	1	4	yes	—
211	C₄⋊C₄	C₂²×D₁₉₇	1	8	yes	—
212	C₂²⋊C₄	—	197	4	no	—
213	C₂²⋊C₄	—	197	4	no	—
214	C₂²⋊C₄	—	197	8	no	—
215	C₂²⋊C₄	C₂²×D₁₉₇	1	8	yes	—
216	C₄²	—	197	4	no	—
217	C₄²	—	197	4	no	—
218	C₄²	—	197	8	no	—
219	C₄²	—	197	8	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	C₂³×D₁₉₇	1	2	yes	—
224	C₂³	C₂³×D₁₉₇	1	1	yes	characteristic
225	Q₈	—	2	8	no	—
226	Q₈	—	394	8	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	394	4	no	—
232	C₂×C₄	—	394	2	no	—
233	C₂×C₄	—	394	2	no	—
234	C₂×C₄	—	394	2	no	—
235	C₂×C₄	—	394	2	no	—
236	C₂×C₄	—	394	4	no	—
237	C₂×C₄	—	197	2	no	—
238	C₂×C₄	—	197	2	no	—
239	C₂×C₄	—	197	2	no	—
240	C₂×C₄	—	197	2	no	—
241	C₂×C₄	—	197	4	no	—
242	C₂×C₄	—	197	4	no	—
243	C₂×C₄	#15691357	1	4	yes	—
244	C₂×C₄	C₂³×D₁₉₇	1	4	yes	—
245	C₂×C₄	C₂³×D₁₉₇	1	4	yes	—
246	C₂×C₄	C₂³×D₁₉₇	1	2	yes	—
247	C₂×C₄	C₂³×D₁₉₇	1	2	yes	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	4	4	no	—
252	C₂²	C₂×#15691357	1	2	yes	—
253	C₂²	#20263373	1	2	yes	—
254	C₂²	#20263374	1	2	yes	—
255	C₂²	C₂⁴×D₁₉₇	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	2	4	no	—
258	C₄	—	2	4	no	—
259	C₄	—	394	4	no	—
260	C₄	—	394	2	no	—
261	C₄	—	394	2	no	—
262	C₄	—	394	2	no	—
263	C₄	—	394	2	no	—
264	C₄	—	394	4	no	—
265	C₄	C₂×#15691357	1	4	yes	—
266	C₂	—	4	2	no	—
267	C₂	#27161689	1	4	yes	direct product factor
268	C₂	C₂²×#15691357	1	1	yes	characteristic
269	C₂	C₂×#20263373	1	1	yes	characteristic
270	C₂	C₂×#20263374	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₉₇
commutator, cosocle	C₂²×C₁₉₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₉₇
Fitting	C₂×C₂²⋊Q₈×C₁₉₇	C₂
socle	C₂³×C₁₉₇	C₂⁴
prime radical	C₂⁴×C₁₉₇	C₂³
hypercenter	C₂×C₂²⋊Q₈	D₁₉₇
hypocenter	C₁₉₇	C₂×#541
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #27161689	C₂	direct (N × H)
2. C₂×#20263324	C₂	semidirect
5. C₂×#20263317	C₂	semidirect
6. C₂×#20263317	C₂	semidirect
7. C₂×#20263316	C₂	semidirect
9. C₂×Q₈×Dic₁₉₇	C₂	semidirect
10. C₂×#20263291	C₂	semidirect
14. #20263324	C₂²	semidirect
17. #20263317	C₂²	semidirect
18. #20263317	C₂²	semidirect
19. #20263316	C₂²	semidirect
21. Q₈×Dic₁₉₇	C₂²	semidirect
22. #20263291	C₂²	semidirect
131. C₂×C₁₉₇	#541	semidirect
135. C₁₉₇	C₂×#541	semidirect
267. C₂	#27161689	direct (N × H)

Maximal subgroups

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

subgroup	index
#27161689	2
C₂×#20263324	2
C₂×#20263319	2
C₂×#20263317	2
C₂×#20263316	2
C₂×#20263305	2
C₂×Q₈×Dic₁₉₇	2
C₂×#20263291	2
C₂×#20263285	2
C₂×C₂²⋊Q₈×C₁₉₇	2
C₂×#541	197

Subgroups — C2×#27161689 #38756078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27161689 #38756078