Subgroups — #38751644

Order 25216.

364 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27161632	C₂	1	1	yes	maximal, characteristic
2	#27161207	C₂	1	4	yes	maximal, complemented
3	C₈×Dic₃₉₄	C₂	1	4	yes	maximal, complemented
4	#27161107	C₂	1	1	yes	maximal, characteristic
5	#27161106	C₂	1	1	yes	maximal, characteristic
6	#27161102	C₂	1	1	yes	maximal, characteristic
7	#27161034	C₂	1	1	yes	maximal, characteristic
8	C₂×C₈×Dic₁₉₇	C₂	1	1	yes	maximal, characteristic
9	#405×C₁₉₇	C₂	1	1	yes	maximal, characteristic
10	C₂×#15691342	C₂²	1	1	yes	characteristic
11	C₂²×#13096802	C₂²	1	1	yes	characteristic
12	#20263325	C₄	1	2	yes	—
13	#20263317	C₄	1	2	yes	—
14	#20263303	C₄	1	4	yes	—
15	C₄×Dic₃₉₄	C₂²	1	4	yes	—
16	#20263285	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
18	C₄⋊#13096802	C₂²	1	4	yes	—
19	#20263255	C₂²	1	4	yes	—
20	#20263255	C₂²	1	4	yes	—
21	#20263236	C₂²	1	2	yes	—
22	C₈×Dic₁₉₇	C₂²	1	2	yes	—
23	C₈×Dic₁₉₇	C₂²	1	2	yes	—
24	C₈×Dic₁₉₇	C₂²	1	2	yes	—
25	C₂×M₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
26	C₂²×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
27	#117×C₁₉₇	C₂²	1	1	yes	characteristic
28	#97×C₁₉₇	C₂²	1	2	yes	—
29	C₄×C₈×C₁₉₇	C₂²	1	2	yes	—
30	C₂×Dic₃₉₄	C₂×C₄	1	4	yes	—
31	C₂²×Dic₁₉₇	C₂×C₄	1	2	yes	—
32	#15691342	—	2	2	no	—
33	C₂×#13096802	—	2	2	no	—
34	C₂×#13096802	C₂³	1	2	yes	—
35	C₂×#13096802	C₂³	1	2	yes	—
36	C₄⋊Dic₁₉₇	C₂×C₄	1	2	yes	—
37	C₄⋊Dic₁₉₇	C₂×C₄	1	4	yes	—
38	#15691337	C₂×C₄	1	4	yes	—
39	#15691337	C₂×C₄	1	4	yes	—
40	C₂²⋊Dic₁₉₇	C₂×C₄	1	2	yes	—
41	C₄×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
43	C₄×Dic₁₉₇	C₂³	1	2	yes	—
44	C₄×Dic₁₉₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
46	M₁₆×C₁₉₇	—	2	2	no	—
47	C₂×C₈×C₁₉₇	Q₈	1	4	yes	complemented
48	C₂×C₈×C₁₉₇	C₂³	1	2	yes	—
49	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₁₉₇	C₂×C₄	1	2	yes	—
52	C₂²⋊C₄×C₁₉₇	C₂×C₄	1	2	yes	—
53	C₄²×C₁₉₇	C₂³	1	2	yes	—
54	Dic₃₉₄	—	2	8	no	—
55	C₂×Dic₁₉₇	—	2	4	no	—
56	C₂×Dic₁₉₇	C₂²×C₄	1	2	yes	—
57	C₂×Dic₁₉₇	C₂²×C₄	1	4	yes	—
58	C₂×Dic₁₉₇	C₂²×C₄	1	2	yes	—
59	#13096802	—	2	2	no	—
60	#13096802	—	2	2	no	—
61	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₉₇	C₂²×C₄	1	4	yes	—
63	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₉₇	C₂×Q₈	1	2	yes	—
65	C₂×C₄×C₁₉₇	#41	1	2	yes	—
66	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
67	C₈×C₁₉₇	—	2	2	no	—
68	C₈×C₁₉₇	C₂×Q₈	1	2	yes	—
69	C₈×C₁₉₇	C₂×Q₈	1	2	yes	—
70	C₂²×C₁₉₇	—	2	2	no	—
71	C₂²×C₁₉₇	C₄×Q₈	1	2	yes	—
72	C₂²×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
73	C₄×C₁₉₇	—	2	4	no	—
74	C₄×C₁₉₇	C₄×Q₈	1	2	yes	—
75	C₄×C₁₉₇	C₂²×Q₈	1	1	yes	characteristic
76	C₄×C₁₉₇	C₂×#41	1	1	yes	characteristic
77	Dic₁₉₇	—	2	2	no	—
78	Dic₁₉₇	—	2	4	no	—
79	Dic₁₉₇	#131	1	4	yes	—
80	C₂×C₁₉₇	—	2	2	no	—
81	C₂×C₁₉₇	C₂×C₄×Q₈	1	1	yes	characteristic
82	C₂×C₁₉₇	C₂×#131	1	2	yes	—
83	C₁₉₇	#2960	1	1	yes	characteristic, complemented, Sylow
84	#2960	—	197	1	no	maximal, Sylow
85	#533	—	197	1	no	—
86	#446	—	197	4	no	—
87	C₈×Q₈	—	197	4	no	—
88	#424	—	197	1	no	—
89	#424	—	197	1	no	—
90	#423	—	197	1	no	—
91	#405	—	197	1	no	—
92	#405	D₁₉₇	1	1	yes	characteristic
93	C₂×C₄×C₈	—	197	1	no	—
94	C₂×M₁₆	—	197	1	no	—
95	C₂×M₁₆	C₂×D₁₉₇	1	1	yes	characteristic
96	C₂²×C₈	—	197	1	no	—
97	C₂²×C₈	C₂×D₁₉₇	1	1	yes	characteristic
98	C₄⋊Q₈	—	197	2	no	—
99	#125	—	197	2	no	—
100	C₂²⋊Q₈	—	197	4	no	—
101	C₄×Q₈	—	197	4	no	—
102	#117	—	197	1	no	—
103	#117	C₂×D₁₉₇	1	1	yes	characteristic
104	C₂×C₄²	—	197	1	no	—
105	C₄⋊C₈	—	197	4	no	—
106	C₄⋊C₈	—	197	4	no	—
107	C₄⋊C₈	—	197	4	no	—
108	#97	—	197	2	no	—
109	#97	C₂×D₁₉₇	1	2	yes	—
110	C₄×C₈	—	197	2	no	—
111	C₄×C₈	—	197	2	no	—
112	C₄×C₈	—	197	2	no	—
113	C₄×C₈	C₂×D₁₉₇	1	2	yes	—
114	C₂×Q₈	—	197	4	no	—
115	C₂²×C₄	—	197	2	no	—
116	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
117	M₁₆	—	2	2	no	—
118	M₁₆	—	394	2	no	—
119	C₂×C₈	—	394	2	no	—
120	C₂×C₈	—	197	2	no	—
121	C₂×C₈	—	197	2	no	—
122	C₂×C₈	Dic₃₉₄	1	4	yes	complemented
123	C₂×C₈	C₂²×D₁₉₇	1	2	yes	—
124	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
125	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
126	C₄⋊C₄	—	197	4	no	—
127	C₄⋊C₄	—	197	4	no	—
128	C₄⋊C₄	—	197	2	no	—
129	C₄⋊C₄	—	197	4	no	—
130	C₄⋊C₄	C₄×D₁₉₇	1	2	yes	—
131	C₂²⋊C₄	—	197	2	no	—
132	C₂²⋊C₄	C₄×D₁₉₇	1	2	yes	—
133	C₄²	—	197	1	no	—
134	C₄²	—	197	1	no	—
135	C₄²	—	197	2	no	—
136	C₄²	—	197	2	no	—
137	C₄²	C₂²×D₁₉₇	1	2	yes	—
138	C₂³	C₂×C₄×D₁₉₇	1	1	yes	characteristic
139	Q₈	—	394	8	no	—
140	C₂×C₄	—	394	4	no	—
141	C₂×C₄	—	197	2	no	—
142	C₂×C₄	—	197	4	no	—
143	C₂×C₄	—	197	2	no	—
144	C₂×C₄	C₂×C₄×D₁₉₇	1	4	yes	—
145	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
146	C₂×C₄	C₂×Dic₃₉₄	1	2	yes	—
147	C₂×C₄	#15691358	1	2	yes	—
148	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	394	2	no	—
151	C₈	—	394	2	no	—
152	C₈	C₂×Dic₃₉₄	1	2	yes	—
153	C₈	C₂×Dic₃₉₄	1	2	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	C₄×Dic₃₉₄	1	2	yes	—
156	C₂²	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
157	C₄	—	2	4	no	—
158	C₄	—	394	2	no	—
159	C₄	—	394	4	no	—
160	C₄	—	197	4	no	—
161	C₄	C₄×Dic₃₉₄	1	2	yes	—
162	C₄	C₂²×Dic₃₉₄	1	1	yes	characteristic
163	C₄	C₂×#15691358	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	C₂×C₄×Dic₃₉₄	1	1	yes	characteristic
166	C₂	#131×D₁₉₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₉₇
commutator, socle	C₂²×C₁₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting	#405×C₁₉₇	C₂
prime radical	C₂³×C₁₉₇	C₂²×C₄
cosocle	C₂×C₄×C₁₉₇	C₂⁴
hypercenter	#405	D₁₉₇
hypocenter	C₁₉₇	#2960
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #27161207	C₂	semidirect
3. C₈×Dic₃₉₄	C₂	semidirect
47. C₂×C₈×C₁₉₇	Q₈	semidirect
83. C₁₉₇	#2960	semidirect
122. C₂×C₈	Dic₃₉₄	semidirect

Maximal subgroups

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

subgroup	index
#27161632	2
#27161207	2
C₈×Dic₃₉₄	2
#27161107	2
#27161106	2
#27161102	2
#27161034	2
C₂×C₈×Dic₁₉₇	2
#405×C₁₉₇	2
#2960	197