Subgroups — C₂×#27160975 #38745161

Order 25216.

800 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₁₉₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#15691337	C₂	1	2	yes	maximal
3	C₂²×C₄×Dic₁₉₇	C₂	1	1	yes	maximal, characteristic
4	#27160975	C₂	1	8	yes	maximal, direct product factor
5	C₂×#20263233	C₂	1	2	yes	maximal
6	C₂²×C₄⋊C₄×C₁₉₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₉₇	C₂²	1	2	yes	—
8	C₂³×Dic₁₉₇	C₂²	1	1	yes	characteristic
9	C₂³×Dic₁₉₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₉₇	C₄	1	8	yes	complemented
11	C₂×C₄⋊Dic₁₉₇	C₂²	1	4	yes	—
12	C₂×#15691337	—	2	8	no	—
13	C₂×#15691337	C₂²	1	8	yes	—
14	C₂×C₄×Dic₁₉₇	—	2	4	no	—
15	C₂×C₄×Dic₁₉₇	C₂²	1	4	yes	—
16	#20263233	C₂²	1	8	yes	—
17	C₂³×C₄×C₁₉₇	C₂²	1	2	yes	—
18	C₂³×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₉₇	—	2	4	no	—
20	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	4	yes	—
21	C₂²×Dic₁₉₇	—	2	2	no	—
22	C₂²×Dic₁₉₇	—	2	2	no	—
23	C₂²×Dic₁₉₇	—	2	4	no	—
24	C₂²×Dic₁₉₇	—	2	2	no	—
25	C₂²×Dic₁₉₇	—	2	4	no	—
26	C₂²×Dic₁₉₇	—	2	2	no	—
27	C₂²×Dic₁₉₇	C₂×C₄	1	8	yes	—
28	C₂²×Dic₁₉₇	D₄	1	4	yes	—
29	C₂²×Dic₁₉₇	Q₈	1	4	yes	—
30	C₂²×Dic₁₉₇	C₂³	1	4	yes	—
31	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
32	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
33	C₄⋊Dic₁₉₇	C₂×C₄	1	16	yes	complemented
34	#15691337	—	2	16	no	—
35	C₄×Dic₁₉₇	—	2	8	no	—
36	C₂⁴×C₁₉₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₉₇	—	2	4	no	—
38	C₂²×C₄×C₁₉₇	—	2	4	no	—
39	C₂²×C₄×C₁₉₇	—	2	4	no	—
40	C₂²×C₄×C₁₉₇	C₂×C₄	1	4	yes	—
41	C₂²×C₄×C₁₉₇	D₄	1	4	yes	—
42	C₂²×C₄×C₁₉₇	Q₈	1	4	yes	—
43	C₂²×C₄×C₁₉₇	C₂³	1	4	yes	—
44	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
45	C₄⋊C₄×C₁₉₇	—	2	8	no	—
46	C₂×Dic₁₉₇	—	2	16	no	—
47	C₂×Dic₁₉₇	—	2	8	no	—
48	C₂×Dic₁₉₇	—	2	8	no	—
49	C₂×Dic₁₉₇	—	2	2	no	—
50	C₂×Dic₁₉₇	—	2	2	no	—
51	C₂×Dic₁₉₇	—	2	4	no	—
52	C₂×Dic₁₉₇	—	2	2	no	—
53	C₂×Dic₁₉₇	—	2	4	no	—
54	C₂×Dic₁₉₇	—	2	2	no	—
55	C₂×Dic₁₉₇	C₂²×C₄	1	8	yes	—
56	C₂×Dic₁₉₇	C₂×D₄	1	4	yes	—
57	C₂×Dic₁₉₇	C₂×Q₈	1	4	yes	—
58	C₂³×C₁₉₇	C₂²×C₄	1	2	yes	—
59	C₂³×C₁₉₇	C₂×D₄	1	2	yes	—
60	C₂³×C₁₉₇	C₂×D₄	1	2	yes	—
61	C₂³×C₁₉₇	C₂×Q₈	1	2	yes	—
62	C₂³×C₁₉₇	C₂×Q₈	1	2	yes	—
63	C₂³×C₁₉₇	#41	1	2	yes	—
64	C₂³×C₁₉₇	#41	1	2	yes	—
65	C₂³×C₁₉₇	C₂⁴	1	1	yes	characteristic
66	C₂×C₄×C₁₉₇	—	2	4	no	—
67	C₂×C₄×C₁₉₇	—	2	4	no	—
68	C₂×C₄×C₁₉₇	—	2	4	no	—
69	C₂×C₄×C₁₉₇	—	4	8	no	—
70	C₂×C₄×C₁₉₇	C₄⋊C₄	1	16	yes	complemented
71	C₂×C₄×C₁₉₇	C₂²×C₄	1	4	yes	—
72	C₂×C₄×C₁₉₇	C₂×D₄	1	4	yes	—
73	C₂×C₄×C₁₉₇	C₂×Q₈	1	4	yes	—
74	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	4	yes	—
75	C₂²×C₁₉₇	C₄×D₄	1	4	yes	—
76	C₂²×C₁₉₇	C₄×Q₈	1	4	yes	—
77	C₂²×C₁₉₇	#121	1	4	yes	—
78	C₂²×C₁₉₇	C₂²⋊Q₈	1	4	yes	—
79	C₂²×C₁₉₇	#125	1	4	yes	—
80	C₂²×C₁₉₇	C₄⋊Q₈	1	4	yes	—
81	C₂²×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
82	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
83	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
84	C₂²×C₁₉₇	C₂²×Q₈	1	1	yes	characteristic
85	C₂²×C₁₉₇	C₂²×Q₈	1	1	yes	characteristic
86	C₂²×C₁₉₇	C₂×#41	1	1	yes	characteristic
87	C₂²×C₁₉₇	C₂×#41	1	1	yes	characteristic
88	C₄×C₁₉₇	—	4	4	no	—
89	C₄×C₁₉₇	C₂×C₄⋊C₄	1	8	yes	complemented
90	Dic₁₉₇	—	2	8	no	—
91	Dic₁₉₇	—	2	4	no	—
92	Dic₁₉₇	—	2	4	no	—
93	C₂×C₁₉₇	#389	1	8	yes	complemented
94	C₂×C₁₉₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₁₉₇	C₂×C₄×D₄	1	1	yes	characteristic
96	C₂×C₁₉₇	C₂×C₄×Q₈	1	1	yes	characteristic
97	C₂×C₁₉₇	C₂×#121	1	1	yes	characteristic
98	C₂×C₁₉₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₁₉₇	C₂×#125	1	1	yes	characteristic
100	C₂×C₁₉₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₁₉₇	C₂×#389	1	1	yes	characteristic, complemented, Sylow
102	C₂×#389	—	197	1	no	maximal, Sylow
103	C₂²×C₄⋊C₄	—	197	2	no	—
104	C₂²×C₄⋊C₄	—	197	1	no	—
105	C₂²×C₄⋊C₄	D₁₉₇	1	1	yes	characteristic
106	C₂²×C₄²	—	197	1	no	—
107	#389	—	197	8	no	—
108	C₂×#95	—	197	2	no	—
109	C₂³×C₄	—	197	2	no	—
110	C₂³×C₄	—	197	1	no	—
111	C₂³×C₄	—	197	1	no	—
112	C₂³×C₄	C₂×D₁₉₇	1	2	yes	—
113	C₂³×C₄	C₂×D₁₉₇	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₄⋊C₄	—	394	8	no	—
116	C₂×C₄⋊C₄	—	197	8	no	—
117	C₂×C₄⋊C₄	—	197	8	no	—
118	C₂×C₄⋊C₄	—	197	4	no	—
119	C₂×C₄⋊C₄	C₂×D₁₉₇	1	4	yes	—
120	C₂×C₄²	—	394	4	no	—
121	C₂×C₄²	—	197	4	no	—
122	#95	—	197	8	no	—
123	C₂⁴	C₂²×D₁₉₇	1	1	yes	characteristic
124	C₂²×C₄	—	2	4	no	—
125	C₂²×C₄	—	2	4	no	—
126	C₂²×C₄	—	2	4	no	—
127	C₂²×C₄	—	394	4	no	—
128	C₂²×C₄	—	394	2	no	—
129	C₂²×C₄	—	394	2	no	—
130	C₂²×C₄	—	394	2	no	—
131	C₂²×C₄	—	394	4	no	—
132	C₂²×C₄	—	394	2	no	—
133	C₂²×C₄	—	197	8	no	—
134	C₂²×C₄	—	197	4	no	—
135	C₂²×C₄	—	197	4	no	—
136	C₂²×C₄	—	197	2	no	—
137	C₂²×C₄	—	197	4	no	—
138	C₂²×C₄	—	197	2	no	—
139	C₂²×C₄	C₄×D₁₉₇	1	4	yes	—
140	C₂²×C₄	C₂²⋊D₁₉₇	1	4	yes	—
141	C₂²×C₄	Dic₃₉₄	1	4	yes	—
142	C₂²×C₄	C₂²×D₁₉₇	1	4	yes	—
143	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
144	C₄⋊C₄	—	2	8	no	—
145	C₄⋊C₄	—	394	16	no	—
146	C₄⋊C₄	—	197	16	no	—
147	C₄²	—	394	8	no	—
148	C₂³	C₂×C₄×D₁₉₇	1	2	yes	—
149	C₂³	D₄×D₁₉₇	1	2	yes	—
150	C₂³	C₂×C₂²⋊D₁₉₇	1	2	yes	—
151	C₂³	Q₈×D₁₉₇	1	2	yes	—
152	C₂³	C₂×Dic₃₉₄	1	2	yes	—
153	C₂³	#15691357	1	2	yes	—
154	C₂³	#15691359	1	2	yes	—
155	C₂³	C₂³×D₁₉₇	1	1	yes	characteristic
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	4	8	no	—
160	C₂×C₄	—	394	16	no	—
161	C₂×C₄	—	394	8	no	—
162	C₂×C₄	—	394	8	no	—
163	C₂×C₄	—	394	4	no	—
164	C₂×C₄	—	394	2	no	—
165	C₂×C₄	—	394	2	no	—
166	C₂×C₄	—	394	2	no	—
167	C₂×C₄	—	394	4	no	—
168	C₂×C₄	—	394	2	no	—
169	C₂×C₄	—	197	8	no	—
170	C₂×C₄	—	197	4	no	—
171	C₂×C₄	—	197	4	no	—
172	C₂×C₄	#15691337	1	16	yes	complemented
173	C₂×C₄	C₂×C₄×D₁₉₇	1	4	yes	—
174	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	4	yes	—
175	C₂×C₄	C₂×Dic₃₉₄	1	4	yes	—
176	C₂²	C₂×#15691337	1	4	yes	—
177	C₂²	#20263288	1	4	yes	—
178	C₂²	#20263291	1	4	yes	—
179	C₂²	#20263297	1	4	yes	—
180	C₂²	#20263302	1	4	yes	—
181	C₂²	#20263317	1	4	yes	—
182	C₂²	#20263325	1	4	yes	—
183	C₂²	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
184	C₂²	C₂×D₄×D₁₉₇	1	1	yes	characteristic
185	C₂²	C₂²×C₂²⋊D₁₉₇	1	1	yes	characteristic
186	C₂²	C₂×Q₈×D₁₉₇	1	1	yes	characteristic
187	C₂²	C₂²×Dic₃₉₄	1	1	yes	characteristic
188	C₂²	C₂×#15691357	1	1	yes	characteristic
189	C₂²	C₂×#15691359	1	1	yes	characteristic
190	C₄	—	4	4	no	—
191	C₄	—	394	8	no	—
192	C₄	—	394	4	no	—
193	C₄	—	394	4	no	—
194	C₄	C₂×#15691337	1	8	yes	complemented
195	C₂	#27160975	1	8	yes	direct product factor
196	C₂	C₂²×#15691337	1	1	yes	characteristic
197	C₂	C₂×#20263288	1	1	yes	characteristic
198	C₂	C₂×#20263291	1	1	yes	characteristic
199	C₂	C₂×#20263297	1	1	yes	characteristic
200	C₂	C₂×#20263302	1	1	yes	characteristic
201	C₂	C₂×#20263317	1	1	yes	characteristic
202	C₂	C₂×#20263325	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₁₉₇
commutator	C₂²×C₁₉₇	C₂³×C₄
Frattini	C₂³	C₂³×D₁₉₇
Fitting	C₂²×C₄⋊C₄×C₁₉₇	C₂
socle, prime radical	C₂⁴×C₁₉₇	C₂³
cosocle	C₂³×C₁₉₇	C₂⁴
hypercenter	C₂²×C₄⋊C₄	D₁₉₇
hypocenter	C₁₉₇	C₂×#389
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. #27160975	C₂	direct (N × H)
10. C₂×C₄⋊Dic₁₉₇	C₄	semidirect
33. C₄⋊Dic₁₉₇	C₂×C₄	semidirect
70. C₂×C₄×C₁₉₇	C₄⋊C₄	semidirect
89. C₄×C₁₉₇	C₂×C₄⋊C₄	semidirect
93. C₂×C₁₉₇	#389	semidirect
101. C₁₉₇	C₂×#389	semidirect
172. C₂×C₄	#15691337	semidirect
194. C₄	C₂×#15691337	semidirect
195. C₂	#27160975	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×C₄⋊Dic₁₉₇	2
C₂²×#15691337	2
C₂²×C₄×Dic₁₉₇	2
#27160975	2
C₂×#20263233	2
C₂²×C₄⋊C₄×C₁₉₇	2
C₂×#389	197

Subgroups — C2×#27160975 #38745161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27160975 #38745161