Subgroups — C₂×#27161075 #38751208

Order 25216.

792 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#13096802	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#20263285	C₂	1	1	yes	maximal, characteristic, complemented
3	#27161075	C₂	1	8	yes	maximal, direct product factor
4	C₂×#20263250	C₂	1	2	yes	maximal, complemented
5	C₂×#20263247	C₂	1	2	yes	maximal, complemented
6	C₂²×#41×C₁₉₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#13096802	—	2	4	no	—
8	C₂²×#13096802	C₂²	1	2	yes	—
9	C₂²×#13096802	C₂²	1	4	yes	complemented
10	#20263285	—	2	2	no	—
11	#20263285	C₂²	1	4	yes	complemented
12	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
13	C₂×C₂²⋊Dic₁₉₇	—	2	1	no	—
14	C₂×C₄×Dic₁₉₇	—	2	1	no	—
15	#20263250	C₂²	1	8	yes	complemented
16	#20263247	C₂²	1	8	yes	complemented
17	C₂×#41×C₁₉₇	—	2	4	no	—
18	C₂×#41×C₁₉₇	—	2	2	no	—
19	C₂×#41×C₁₉₇	C₄	1	8	yes	complemented
20	C₂×#41×C₁₉₇	C₂²	1	4	yes	—
21	C₂²×Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
22	C₂²×D₄×C₁₉₇	—	2	1	no	—
23	C₂²×D₄×C₁₉₇	C₂²	1	1	yes	characteristic
24	C₂³×C₄×C₁₉₇	—	2	1	no	—
25	C₂³×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₉₇	—	2	2	no	—
27	C₂×#13096802	—	2	4	no	—
28	C₂×#13096802	—	2	8	no	—
29	C₂×#13096802	C₂³	1	4	yes	—
30	C₄⋊Dic₁₉₇	—	2	2	no	—
31	C₄⋊Dic₁₉₇	C₂³	1	4	yes	—
32	C₂²⋊Dic₁₉₇	—	2	4	no	—
33	C₄×Dic₁₉₇	—	2	4	no	—
34	C₂⁴×C₁₉₇	—	2	1	no	—
35	C₂⁴×C₁₉₇	D₄	1	1	yes	characteristic
36	#41×C₁₉₇	—	2	16	no	—
37	#41×C₁₉₇	—	2	8	no	—
38	#41×C₁₉₇	C₂×C₄	1	16	yes	complemented
39	C₂×Q₈×C₁₉₇	—	2	2	no	—
40	C₂×Q₈×C₁₉₇	—	2	1	no	—
41	C₂×Q₈×C₁₉₇	C₂×C₄	1	4	yes	—
42	C₂×Q₈×C₁₉₇	C₂³	1	2	yes	—
43	C₂×D₄×C₁₉₇	—	2	8	no	—
44	C₂×D₄×C₁₉₇	—	2	4	no	—
45	C₂×D₄×C₁₉₇	—	2	1	no	—
46	C₂×D₄×C₁₉₇	—	2	2	no	—
47	C₂×D₄×C₁₉₇	C₂×C₄	1	4	yes	—
48	C₂×D₄×C₁₉₇	C₂³	1	2	yes	—
49	C₂²×C₄×C₁₉₇	—	2	1	no	—
50	C₂²×C₄×C₁₉₇	—	2	8	no	—
51	C₂²×C₄×C₁₉₇	—	2	4	no	—
52	C₂²×C₄×C₁₉₇	C₂×C₄	1	4	yes	—
53	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
54	C₂²×C₄×C₁₉₇	D₄	1	4	yes	—
55	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
58	C₂×Dic₁₉₇	—	2	4	no	—
59	C₂×Dic₁₉₇	—	4	4	no	—
60	#13096802	—	2	4	no	—
61	C₂³×C₁₉₇	—	2	4	no	—
62	C₂³×C₁₉₇	—	2	4	no	—
63	C₂³×C₁₉₇	—	2	2	no	—
64	C₂³×C₁₉₇	—	4	4	no	—
65	C₂³×C₁₉₇	C₂²⋊C₄	1	4	yes	—
66	C₂³×C₁₉₇	C₂×D₄	1	2	yes	—
67	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
68	Q₈×C₁₉₇	—	2	4	no	—
69	Q₈×C₁₉₇	—	2	2	no	—
70	Q₈×C₁₉₇	C₂²×C₄	1	4	yes	—
71	D₄×C₁₉₇	—	2	16	no	—
72	D₄×C₁₉₇	—	2	4	no	—
73	D₄×C₁₉₇	—	2	2	no	—
74	D₄×C₁₉₇	C₂²×C₄	1	4	yes	—
75	C₂×C₄×C₁₉₇	—	2	4	no	—
76	C₂×C₄×C₁₉₇	—	2	2	no	—
77	C₂×C₄×C₁₉₇	—	2	16	no	—
78	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	4	yes	—
79	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	8	yes	—
80	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₁₉₇	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
83	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₁₉₇	C₂×D₄	1	4	yes	—
86	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
87	C₂²×C₁₉₇	—	2	4	no	—
88	C₂²×C₁₉₇	—	2	8	no	—
89	C₂²×C₁₉₇	—	2	4	no	—
90	C₂²×C₁₉₇	—	4	4	no	—
91	C₂²×C₁₉₇	—	4	8	no	—
92	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	4	yes	—
93	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	2	yes	—
94	C₂²×C₁₉₇	#135	1	4	yes	—
95	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
96	C₄×C₁₉₇	—	2	4	no	—
97	C₄×C₁₉₇	C₂×C₂²⋊C₄	1	4	yes	—
98	C₄×C₁₉₇	C₂×C₂²⋊C₄	1	2	yes	—
99	C₄×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
100	C₄×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
101	Dic₁₉₇	—	4	4	no	—
102	C₂×C₁₉₇	—	2	4	no	—
103	C₂×C₁₉₇	—	4	4	no	—
104	C₂×C₁₉₇	#416	1	4	yes	complemented
105	C₂×C₁₉₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₁₉₇	C₂×#135	1	2	yes	—
107	C₁₉₇	C₂×#416	1	1	yes	characteristic, complemented, Sylow
108	C₂×#416	—	197	1	no	maximal, Sylow
109	C₂²×#41	D₁₉₇	1	1	yes	characteristic
110	C₂³×C₈	—	197	1	no	—
111	C₂×#117	—	197	1	no	—
112	#416	—	197	8	no	—
113	C₂×#103	—	197	2	no	—
114	C₂×#102	—	197	2	no	—
115	C₂×#41	—	2	4	no	—
116	C₂×#41	—	2	2	no	—
117	C₂×#41	Dic₁₉₇	1	8	yes	complemented
118	C₂×#41	C₂×D₁₉₇	1	4	yes	—
119	C₂²×Q₈	C₂×D₁₉₇	1	1	yes	characteristic
120	C₂²×D₄	—	2	1	no	—
121	C₂²×D₄	C₂×D₁₉₇	1	1	yes	characteristic
122	C₂³×C₄	—	2	1	no	—
123	C₂³×C₄	C₂×D₁₉₇	1	1	yes	characteristic
124	C₂²×C₈	—	394	4	no	—
125	C₂²×C₈	—	197	2	no	—
126	C₂²×C₈	—	197	4	no	—
127	#117	—	394	2	no	—
128	#117	—	197	4	no	—
129	C₂×C₄⋊C₄	—	197	2	no	—
130	C₂×C₂²⋊C₄	—	394	1	no	—
131	C₂×C₄²	—	394	1	no	—
132	#103	—	197	8	no	—
133	#102	—	197	8	no	—
134	C₂⁴	—	2	1	no	—
135	C₂⁴	C₂²⋊D₁₉₇	1	1	yes	characteristic
136	#41	—	2	16	no	—
137	#41	—	2	8	no	—
138	#41	C₂×Dic₁₉₇	1	16	yes	complemented
139	C₂×Q₈	—	2	2	no	—
140	C₂×Q₈	—	2	1	no	—
141	C₂×Q₈	C₂×Dic₁₉₇	1	4	yes	—
142	C₂×Q₈	C₂²×D₁₉₇	1	2	yes	—
143	C₂×D₄	—	2	8	no	—
144	C₂×D₄	—	2	4	no	—
145	C₂×D₄	—	2	1	no	—
146	C₂×D₄	—	2	2	no	—
147	C₂×D₄	C₂×Dic₁₉₇	1	4	yes	—
148	C₂×D₄	C₂²×D₁₉₇	1	2	yes	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	2	8	no	—
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	—	394	2	no	—
153	C₂²×C₄	C₂×Dic₁₉₇	1	4	yes	—
154	C₂²×C₄	C₂²⋊D₁₉₇	1	2	yes	—
155	C₂²×C₄	C₂²⋊D₁₉₇	1	4	yes	—
156	C₂²×C₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
157	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
159	C₂×C₈	—	394	4	no	—
160	C₂×C₈	—	394	8	no	—
161	C₂×C₈	—	197	4	no	—
162	C₄⋊C₄	—	394	2	no	—
163	C₄⋊C₄	—	197	4	no	—
164	C₂²⋊C₄	—	394	4	no	—
165	C₄²	—	394	4	no	—
166	C₂³	—	2	4	no	—
167	C₂³	—	2	4	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	4	4	no	—
170	C₂³	C₂²⋊Dic₁₉₇	1	4	yes	—
171	C₂³	C₂×C₂²⋊D₁₉₇	1	2	yes	—
172	C₂³	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
173	Q₈	—	2	4	no	—
174	Q₈	—	2	2	no	—
175	Q₈	C₂²×Dic₁₉₇	1	4	yes	—
176	D₄	—	2	16	no	—
177	D₄	—	2	4	no	—
178	D₄	—	2	2	no	—
179	D₄	C₂²×Dic₁₉₇	1	4	yes	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	16	no	—
183	C₂×C₄	—	788	4	no	—
184	C₂×C₄	—	394	4	no	—
185	C₂×C₄	C₂²⋊Dic₁₉₇	1	4	yes	—
186	C₂×C₄	C₂²⋊Dic₁₉₇	1	8	yes	—
187	C₂×C₄	C₂²×Dic₁₉₇	1	2	yes	—
188	C₂×C₄	C₂²×Dic₁₉₇	1	4	yes	—
189	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
190	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
191	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
192	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	4	yes	—
193	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
194	C₈	—	394	4	no	—
195	C₂²	—	2	4	no	—
196	C₂²	—	2	8	no	—
197	C₂²	—	2	4	no	—
198	C₂²	—	4	4	no	—
199	C₂²	—	4	8	no	—
200	C₂²	C₂×C₂²⋊Dic₁₉₇	1	4	yes	—
201	C₂²	C₂×C₂²⋊Dic₁₉₇	1	2	yes	—
202	C₂²	#20263346	1	4	yes	—
203	C₂²	C₂²×C₂²⋊D₁₉₇	1	1	yes	characteristic
204	C₄	—	2	4	no	—
205	C₄	—	788	4	no	—
206	C₄	C₂×C₂²⋊Dic₁₉₇	1	4	yes	—
207	C₄	C₂×C₂²⋊Dic₁₉₇	1	2	yes	—
208	C₄	C₂³×Dic₁₉₇	1	1	yes	characteristic
209	C₄	C₂²×C₂²⋊D₁₉₇	1	1	yes	characteristic
210	C₂	—	2	4	no	—
211	C₂	—	4	4	no	—
212	C₂	#27161075	1	4	yes	direct product factor
213	C₂	C₂²×C₂²⋊Dic₁₉₇	1	1	yes	characteristic
214	C₂	C₂×#20263346	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊D₁₉₇
commutator	C₄×C₁₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting, prime radical	C₂²×#41×C₁₉₇	C₂
socle	C₂³×C₁₉₇	C₂×D₄
cosocle	C₂×C₄×C₁₉₇	C₂⁴
hypercenter	C₂²×#41	D₁₉₇
hypocenter	C₁₉₇	C₂×#416
LCS	G ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂³×#13096802	C₂	semidirect
2. C₂×#20263285	C₂	semidirect
3. #27161075	C₂	direct (N × H)
4. C₂×#20263250	C₂	semidirect
5. C₂×#20263247	C₂	semidirect
9. C₂²×#13096802	C₂²	semidirect
11. #20263285	C₂²	semidirect
15. #20263250	C₂²	semidirect
16. #20263247	C₂²	semidirect
19. C₂×#41×C₁₉₇	C₄	semidirect
38. #41×C₁₉₇	C₂×C₄	semidirect
104. C₂×C₁₉₇	#416	semidirect
107. C₁₉₇	C₂×#416	semidirect
117. C₂×#41	Dic₁₉₇	semidirect
138. #41	C₂×Dic₁₉₇	semidirect
212. C₂	#27161075	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×#13096802	2
C₂×#20263285	2
#27161075	2
C₂×#20263250	2
C₂×#20263247	2
C₂²×#41×C₁₉₇	2
C₂×#416	197

Subgroups — C2×#27161075 #38751208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27161075 #38751208