Subgroups — #38742428

Order 25216.

424 conjugacy classes of subgroups.

The subgroup lattice has 174 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
2	C₂×#20263285	C₂	1	1	yes	maximal, characteristic
3	#27160841	C₂	1	4	yes	maximal, complemented
4	C₂×#117×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13096802	—	2	2	no	—
6	C₂²×#13096802	—	2	2	no	—
7	C₂²×#13096802	C₄	1	4	yes	complemented
8	C₂²×#13096802	C₂²	1	2	yes	—
9	#20263285	—	2	2	no	—
10	#20263285	C₄	1	4	yes	complemented
11	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
12	C₂×C₂²⋊Dic₁₉₇	—	2	1	no	—
13	C₂×C₄×Dic₁₉₇	—	2	1	no	—
14	C₂³×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
15	#117×C₁₉₇	—	2	2	no	—
16	#117×C₁₉₇	C₄	1	4	yes	complemented
17	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₁₉₇	—	2	1	no	—
19	C₂×C₄²×C₁₉₇	—	2	1	no	—
20	C₂²×Dic₁₉₇	—	2	2	no	—
21	C₂×#13096802	—	2	8	no	—
22	C₂×#13096802	—	2	2	no	—
23	C₂×#13096802	—	2	2	no	—
24	C₂×#13096802	C₂×C₄	1	4	yes	—
25	C₄⋊Dic₁₉₇	—	2	2	no	—
26	C₄⋊Dic₁₉₇	C₂×C₄	1	4	yes	—
27	C₂²⋊Dic₁₉₇	—	2	4	no	—
28	C₄×Dic₁₉₇	—	2	4	no	—
29	C₂⁴×C₁₉₇	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₉₇	—	2	2	no	—
31	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
35	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
37	C₂²×C₄×C₁₉₇	Q₈	1	2	yes	—
38	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₁₉₇	—	2	2	no	—
40	C₄⋊C₄×C₁₉₇	C₂×C₄	1	4	yes	—
41	C₂²⋊C₄×C₁₉₇	—	2	4	no	—
42	C₄²×C₁₉₇	—	2	4	no	—
43	C₂×Dic₁₉₇	—	2	4	no	—
44	C₂×Dic₁₉₇	—	4	4	no	—
45	#13096802	—	2	4	no	—
46	C₂³×C₁₉₇	—	2	4	no	—
47	C₂³×C₁₉₇	C₂²⋊C₄	1	2	yes	—
48	C₂³×C₁₉₇	C₂²⋊C₄	1	2	yes	—
49	C₂³×C₁₉₇	C₄⋊C₄	1	2	yes	—
50	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₉₇	—	2	4	no	—
52	C₂×C₄×C₁₉₇	—	4	4	no	—
53	C₂×C₄×C₁₉₇	C₄²	1	4	yes	—
54	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
55	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	4	yes	—
56	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄×C₁₉₇	C₄⋊C₄	1	2	yes	—
58	C₂×C₄×C₁₉₇	C₄⋊C₄	1	4	yes	—
59	C₂×C₄×C₁₉₇	C₄⋊C₄	1	4	yes	—
60	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₁₉₇	—	2	4	no	—
67	C₂²×C₁₉₇	—	2	4	no	—
68	C₂²×C₁₉₇	—	2	4	no	—
69	C₂²×C₁₉₇	#95	1	4	yes	—
70	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₁₉₇	#135	1	4	yes	—
74	C₄×C₁₉₇	—	4	4	no	—
75	C₄×C₁₉₇	#95	1	4	yes	—
76	C₄×C₁₉₇	C₂×C₄²	1	1	yes	characteristic
77	C₄×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₄×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₄×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
80	Dic₁₉₇	—	4	4	no	—
81	C₂×C₁₉₇	—	2	4	no	—
82	C₂×C₁₉₇	C₂×#95	1	1	yes	characteristic
83	C₂×C₁₉₇	#416	1	2	yes	—
84	C₂×C₁₉₇	#416	1	2	yes	—
85	C₂×C₁₉₇	#427	1	2	yes	—
86	C₁₉₇	#1735	1	1	yes	characteristic, complemented, Sylow
87	#1735	—	197	1	no	maximal, Sylow
88	C₂³×C₈	—	197	1	no	—
89	C₂×#117	—	197	1	no	—
90	C₂×#117	D₁₉₇	1	1	yes	characteristic
91	#340	—	197	4	no	—
92	C₂³×C₄	C₂×D₁₉₇	1	1	yes	characteristic
93	C₂²×C₈	—	394	2	no	—
94	C₂²×C₈	—	394	2	no	—
95	C₂²×C₈	—	197	4	no	—
96	C₂²×C₈	—	197	2	no	—
97	#117	—	2	2	no	—
98	#117	—	394	2	no	—
99	#117	—	197	4	no	—
100	#117	Dic₁₉₇	1	4	yes	complemented
101	C₂×C₄⋊C₄	—	197	2	no	—
102	C₂×C₄⋊C₄	C₂×D₁₉₇	1	2	yes	—
103	C₂×C₂²⋊C₄	—	2	1	no	—
104	C₂×C₂²⋊C₄	—	394	1	no	—
105	C₂×C₄²	—	2	1	no	—
106	C₂×C₄²	—	394	1	no	—
107	C₂⁴	C₂²⋊D₁₉₇	1	1	yes	characteristic
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	394	2	no	—
110	C₂²×C₄	C₄×D₁₉₇	1	2	yes	—
111	C₂²×C₄	C₄×D₁₉₇	1	2	yes	—
112	C₂²×C₄	C₂×Dic₁₉₇	1	2	yes	—
113	C₂²×C₄	C₂²⋊D₁₉₇	1	2	yes	—
114	C₂²×C₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
115	C₂²×C₄	D₇₈₈	1	2	yes	—
116	C₂²×C₄	Dic₃₉₄	1	2	yes	—
117	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
118	C₂×C₈	—	394	8	no	—
119	C₂×C₈	—	394	2	no	—
120	C₂×C₈	—	394	2	no	—
121	C₂×C₈	—	197	4	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	394	2	no	—
124	C₄⋊C₄	—	197	4	no	—
125	C₄⋊C₄	C₂×Dic₁₉₇	1	4	yes	—
126	C₂²⋊C₄	—	2	4	no	—
127	C₂²⋊C₄	—	394	4	no	—
128	C₄²	—	2	4	no	—
129	C₄²	—	394	4	no	—
130	C₂³	—	2	4	no	—
131	C₂³	#15691335	1	2	yes	—
132	C₂³	C₂²⋊Dic₁₉₇	1	2	yes	—
133	C₂³	#15691337	1	2	yes	—
134	C₂³	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	—	788	4	no	—
138	C₂×C₄	—	394	4	no	—
139	C₂×C₄	C₄×Dic₁₉₇	1	4	yes	—
140	C₂×C₄	#15691335	1	2	yes	—
141	C₂×C₄	#15691335	1	4	yes	—
142	C₂×C₄	C₂²⋊Dic₁₉₇	1	2	yes	—
143	C₂×C₄	#15691337	1	2	yes	—
144	C₂×C₄	#15691337	1	4	yes	—
145	C₂×C₄	C₄⋊Dic₁₉₇	1	4	yes	—
146	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
147	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
148	C₂×C₄	C₂²×Dic₁₉₇	1	1	yes	characteristic
149	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
150	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
151	C₂×C₄	C₂×Dic₃₉₄	1	1	yes	characteristic
152	C₈	—	394	4	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	4	no	—
155	C₂²	—	2	4	no	—
156	C₂²	#20263233	1	4	yes	—
157	C₂²	C₂×#15691335	1	1	yes	characteristic
158	C₂²	C₂×C₂²⋊Dic₁₉₇	1	1	yes	characteristic
159	C₂²	C₂×#15691337	1	1	yes	characteristic
160	C₂²	#20263346	1	4	yes	—
161	C₄	—	4	4	no	—
162	C₄	—	788	4	no	—
163	C₄	#20263233	1	4	yes	—
164	C₄	C₂×C₄×Dic₁₉₇	1	1	yes	characteristic
165	C₄	C₂×#15691335	1	1	yes	characteristic
166	C₄	C₂×#15691337	1	1	yes	characteristic
167	C₄	C₂×C₄⋊Dic₁₉₇	1	1	yes	characteristic
168	C₂	—	2	4	no	—
169	C₂	C₂×#20263233	1	1	yes	characteristic
170	C₂	#27161073	1	2	yes	—
171	C₂	#27161075	1	2	yes	—
172	C₂	#27161115	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊D₁₉₇
commutator	C₄×C₁₉₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	C₂×#117×C₁₉₇	C₂
socle	C₂³×C₁₉₇	C₂×D₄
prime radical	C₂⁴×C₁₉₇	D₄
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	C₂×#117	D₁₉₇
hypocenter	C₁₉₇	#1735
LCS	G ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #27160841	C₂	semidirect
7. C₂²×#13096802	C₄	semidirect
10. #20263285	C₄	semidirect
16. #117×C₁₉₇	C₄	semidirect
86. C₁₉₇	#1735	semidirect
100. #117	Dic₁₉₇	semidirect

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
#27160841	2
C₂×#117×C₁₉₇	2
#1735	197