Subgroups — C₂×#27161650 #38756026

Order 25216.

1512 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₃₉₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#27161650	C₂	1	16	yes	maximal, direct product factor
3	C₂×#20263324	C₂	1	2	yes	maximal, complemented
4	C₂×#20263313	C₂	1	2	yes	maximal, complemented
5	C₂×#20263307	C₂	1	4	yes	maximal, complemented
6	C₂×#20263305	C₂	1	4	yes	maximal, complemented
7	C₂×#20263285	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#41×C₁₉₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₃₉₄	—	2	8	no	—
10	C₂²×Dic₃₉₄	C₂²	1	2	yes	—
11	C₂²×Dic₃₉₄	C₂²	1	2	yes	—
12	C₂²×Dic₃₉₄	C₂²	1	8	yes	complemented
13	C₂³×Dic₁₉₇	C₂²	1	2	yes	—
14	#20263324	C₂²	1	16	yes	complemented
15	#20263313	C₂²	1	16	yes	complemented
16	#20263307	C₂²	1	32	yes	complemented
17	#20263305	C₂²	1	32	yes	complemented
18	#20263285	C₂²	1	8	yes	complemented
19	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
20	C₂×#15691337	C₂²	1	8	yes	—
21	C₂×C₂²⋊Dic₁₉₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₁₉₇	C₂²	1	8	yes	—
23	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
24	C₂×#41×C₁₉₇	—	2	8	no	—
25	C₂×#41×C₁₉₇	C₂²	1	8	yes	—
26	C₂²×Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
27	C₂²×D₄×C₁₉₇	C₂²	1	2	yes	—
28	C₂²×D₄×C₁₉₇	C₂²	1	1	yes	characteristic
29	C₂³×C₄×C₁₉₇	C₂²	1	2	yes	—
30	C₂³×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
31	C₂×Dic₃₉₄	—	2	8	no	—
32	C₂×Dic₃₉₄	—	2	8	no	—
33	C₂×Dic₃₉₄	—	2	32	no	—
34	C₂×Dic₃₉₄	C₂³	1	8	yes	—
35	C₂×Dic₃₉₄	C₂³	1	8	yes	—
36	C₂²×Dic₁₉₇	—	2	8	no	—
37	C₂²×Dic₁₉₇	C₂³	1	4	yes	—
38	C₂²×Dic₁₉₇	C₂³	1	4	yes	—
39	C₂²×Dic₁₉₇	C₂³	1	8	yes	—
40	C₄⋊Dic₁₉₇	C₂³	1	8	yes	—
41	#15691337	C₂³	1	32	yes	—
42	C₂²⋊Dic₁₉₇	C₂³	1	8	yes	—
43	C₂²⋊Dic₁₉₇	C₂³	1	32	yes	—
44	C₄×Dic₁₉₇	C₂³	1	8	yes	—
45	C₂⁴×C₁₉₇	C₂³	1	2	yes	—
46	C₂⁴×C₁₉₇	C₂³	1	1	yes	characteristic
47	#41×C₁₉₇	—	2	32	no	—
48	C₂×Q₈×C₁₉₇	—	2	4	no	—
49	C₂×Q₈×C₁₉₇	C₂³	1	4	yes	—
50	C₂×D₄×C₁₉₇	—	2	8	no	—
51	C₂×D₄×C₁₉₇	—	2	4	no	—
52	C₂×D₄×C₁₉₇	C₂³	1	8	yes	—
53	C₂×D₄×C₁₉₇	C₂³	1	4	yes	—
54	C₂²×C₄×C₁₉₇	—	2	8	no	—
55	C₂²×C₄×C₁₉₇	D₄	1	8	yes	—
56	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
57	C₂²×C₄×C₁₉₇	C₂³	1	8	yes	—
58	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₉₇	C₂³	1	4	yes	—
61	Dic₃₉₄	—	2	16	no	—
62	Dic₃₉₄	—	2	16	no	—
63	C₂×Dic₁₉₇	—	2	16	no	—
64	C₂×Dic₁₉₇	—	2	8	no	—
65	C₂×Dic₁₉₇	—	2	8	no	—
66	C₂×Dic₁₉₇	C₂⁴	1	8	yes	—
67	C₂×Dic₁₉₇	C₂⁴	1	8	yes	—
68	C₂³×C₁₉₇	—	2	4	no	—
69	C₂³×C₁₉₇	—	2	4	no	—
70	C₂³×C₁₉₇	—	2	8	no	—
71	C₂³×C₁₉₇	C₂×D₄	1	4	yes	—
72	C₂³×C₁₉₇	C₂⁴	1	4	yes	—
73	C₂³×C₁₉₇	C₂⁴	1	2	yes	—
74	C₂³×C₁₉₇	C₂⁴	1	1	yes	characteristic
75	Q₈×C₁₉₇	—	2	8	no	—
76	D₄×C₁₉₇	—	2	16	no	—
77	D₄×C₁₉₇	—	2	8	no	—
78	C₂×C₄×C₁₉₇	—	2	4	no	—
79	C₂×C₄×C₁₉₇	—	2	16	no	—
80	C₂×C₄×C₁₉₇	C₂×D₄	1	4	yes	—
81	C₂×C₄×C₁₉₇	C₂×D₄	1	4	yes	—
82	C₂×C₄×C₁₉₇	C₂×D₄	1	16	yes	—
83	C₂×C₄×C₁₉₇	C₂⁴	1	4	yes	—
84	C₂×C₄×C₁₉₇	C₂⁴	1	2	yes	—
85	C₂×C₄×C₁₉₇	C₂⁴	1	2	yes	—
86	C₂²×C₁₉₇	—	2	8	no	—
87	C₂²×C₁₉₇	—	2	4	no	—
88	C₂²×C₁₉₇	—	2	4	no	—
89	C₂²×C₁₉₇	—	2	8	no	—
90	C₂²×C₁₉₇	—	4	8	no	—
91	C₂²×C₁₉₇	C₂²×D₄	1	4	yes	—
92	C₂²×C₁₉₇	C₂²×D₄	1	2	yes	—
93	C₂²×C₁₉₇	2₋¹⁺⁴	1	4	yes	—
94	C₂²×C₁₉₇	C₂⁵	1	1	yes	characteristic
95	C₄×C₁₉₇	—	2	4	no	—
96	C₄×C₁₉₇	C₂²×D₄	1	4	yes	—
97	C₄×C₁₉₇	C₂²×D₄	1	4	yes	—
98	Dic₁₉₇	—	2	8	no	—
99	Dic₁₉₇	—	2	8	no	—
100	C₂×C₁₉₇	—	2	4	no	—
101	C₂×C₁₉₇	—	4	4	no	—
102	C₂×C₁₉₇	#536	1	4	yes	complemented
103	C₂×C₁₉₇	C₂³×D₄	1	1	yes	characteristic
104	C₂×C₁₉₇	C₂×2₋¹⁺⁴	1	2	yes	—
105	C₁₉₇	C₂×#536	1	1	yes	characteristic, complemented, Sylow
106	C₂×#536	—	197	1	no	maximal, Sylow
107	C₂²×#41	D₁₉₇	1	1	yes	characteristic
108	C₂³×Q₈	—	197	1	no	—
109	#536	—	197	16	no	—
110	C₂×C₄⋊Q₈	—	197	2	no	—
111	C₂×#124	—	197	2	no	—
112	C₂×#123	—	197	4	no	—
113	C₂×C₂²⋊Q₈	—	197	4	no	—
114	C₂×#117	—	197	1	no	—
115	C₂×#41	—	2	8	no	—
116	C₂×#41	C₂×D₁₉₇	1	8	yes	—
117	C₂²×Q₈	—	394	8	no	—
118	C₂²×Q₈	—	197	2	no	—
119	C₂²×Q₈	—	197	2	no	—
120	C₂²×Q₈	—	197	8	no	—
121	C₂²×Q₈	C₂×D₁₉₇	1	1	yes	characteristic
122	C₂²×D₄	C₂×D₁₉₇	1	2	yes	—
123	C₂²×D₄	C₂×D₁₉₇	1	1	yes	characteristic
124	C₂³×C₄	—	197	2	no	—
125	C₂³×C₄	C₂×D₁₉₇	1	2	yes	—
126	C₂³×C₄	C₂×D₁₉₇	1	1	yes	characteristic
127	C₄⋊Q₈	—	197	16	no	—
128	#124	—	197	16	no	—
129	#123	—	197	32	no	—
130	C₂²⋊Q₈	—	197	32	no	—
131	#117	—	197	8	no	—
132	C₂×C₄⋊C₄	—	197	2	no	—
133	C₂×C₄⋊C₄	—	197	8	no	—
134	C₂×C₂²⋊C₄	—	197	2	no	—
135	C₂×C₂²⋊C₄	—	197	8	no	—
136	C₂×C₄²	—	197	2	no	—
137	C₂⁴	C₂²×D₁₉₇	1	2	yes	—
138	C₂⁴	C₂²×D₁₉₇	1	1	yes	characteristic
139	#41	—	2	32	no	—
140	C₂×Q₈	—	2	4	no	—
141	C₂×Q₈	—	394	8	no	—
142	C₂×Q₈	—	394	8	no	—
143	C₂×Q₈	—	394	32	no	—
144	C₂×Q₈	—	197	8	no	—
145	C₂×Q₈	—	197	8	no	—
146	C₂×Q₈	C₂²×D₁₉₇	1	4	yes	—
147	C₂×D₄	—	2	8	no	—
148	C₂×D₄	—	2	4	no	—
149	C₂×D₄	C₂²×D₁₉₇	1	8	yes	—
150	C₂×D₄	C₂²×D₁₉₇	1	4	yes	—
151	C₂²×C₄	—	2	8	no	—
152	C₂²×C₄	—	394	8	no	—
153	C₂²×C₄	—	197	8	no	—
154	C₂²×C₄	—	197	4	no	—
155	C₂²×C₄	—	197	4	no	—
156	C₂²×C₄	C₂²⋊D₁₉₇	1	8	yes	—
157	C₂²×C₄	C₂²×D₁₉₇	1	8	yes	—
158	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
159	C₂²×C₄	C₂²×D₁₉₇	1	4	yes	—
160	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
161	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
162	C₄⋊C₄	—	197	32	no	—
163	C₄⋊C₄	—	197	8	no	—
164	C₂²⋊C₄	—	197	8	no	—
165	C₂²⋊C₄	—	197	32	no	—
166	C₄²	—	197	8	no	—
167	C₂³	—	2	4	no	—
168	C₂³	—	2	4	no	—
169	C₂³	—	2	8	no	—
170	C₂³	C₂×C₂²⋊D₁₉₇	1	4	yes	—
171	C₂³	C₂³×D₁₉₇	1	4	yes	—
172	C₂³	C₂³×D₁₉₇	1	2	yes	—
173	C₂³	C₂³×D₁₉₇	1	1	yes	characteristic
174	Q₈	—	2	8	no	—
175	Q₈	—	394	16	no	—
176	Q₈	—	394	16	no	—
177	D₄	—	2	16	no	—
178	D₄	—	2	8	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	16	no	—
181	C₂×C₄	—	394	16	no	—
182	C₂×C₄	—	394	8	no	—
183	C₂×C₄	—	394	8	no	—
184	C₂×C₄	—	197	8	no	—
185	C₂×C₄	—	197	8	no	—
186	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	4	yes	—
187	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	4	yes	—
188	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	16	yes	—
189	C₂×C₄	C₂³×D₁₉₇	1	4	yes	—
190	C₂×C₄	C₂³×D₁₉₇	1	2	yes	—
191	C₂×C₄	C₂³×D₁₉₇	1	2	yes	—
192	C₂²	—	2	8	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	2	4	no	—
195	C₂²	—	2	8	no	—
196	C₂²	—	4	8	no	—
197	C₂²	C₂²×C₂²⋊D₁₉₇	1	4	yes	—
198	C₂²	C₂²×C₂²⋊D₁₉₇	1	2	yes	—
199	C₂²	#20263373	1	4	yes	—
200	C₂²	C₂⁴×D₁₉₇	1	1	yes	characteristic
201	C₄	—	2	4	no	—
202	C₄	—	394	8	no	—
203	C₄	—	394	8	no	—
204	C₄	C₂²×C₂²⋊D₁₉₇	1	4	yes	—
205	C₄	C₂²×C₂²⋊D₁₉₇	1	4	yes	—
206	C₂	—	2	4	no	—
207	C₂	—	4	4	no	—
208	C₂	#27161650	1	4	yes	direct product factor
209	C₂	C₂³×C₂²⋊D₁₉₇	1	1	yes	characteristic
210	C₂	C₂×#20263373	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₁₉₇
commutator, cosocle	C₂²×C₁₉₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₉₇
Fitting, prime radical	C₂²×#41×C₁₉₇	C₂
socle	C₂³×C₁₉₇	C₂⁴
hypercenter	C₂²×#41	D₁₉₇
hypocenter	C₁₉₇	C₂×#536
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂³ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂³×Dic₃₉₄	C₂	semidirect
2. #27161650	C₂	direct (N × H)
3. C₂×#20263324	C₂	semidirect
4. C₂×#20263313	C₂	semidirect
5. C₂×#20263307	C₂	semidirect
6. C₂×#20263305	C₂	semidirect
7. C₂×#20263285	C₂	semidirect
12. C₂²×Dic₃₉₄	C₂²	semidirect
14. #20263324	C₂²	semidirect
15. #20263313	C₂²	semidirect
16. #20263307	C₂²	semidirect
17. #20263305	C₂²	semidirect
18. #20263285	C₂²	semidirect
102. C₂×C₁₉₇	#536	semidirect
105. C₁₉₇	C₂×#536	semidirect
208. C₂	#27161650	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₃₉₄	2
#27161650	2
C₂×#20263324	2
C₂×#20263313	2
C₂×#20263307	2
C₂×#20263305	2
C₂×#20263285	2
C₂²×#41×C₁₉₇	2
C₂×#536	197

Subgroups — C2×#27161650 #38756026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27161650 #38756026