Subgroups — #38751275

Order 25216.

484 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15691342	C₂	1	1	yes	maximal, characteristic
2	C₂×#20263285	C₂	1	1	yes	maximal, characteristic
3	#27161107	C₂	1	4	yes	maximal, complemented
4	#27161105	C₂	1	4	yes	maximal, complemented
5	#27161102	C₂	1	4	yes	maximal, complemented
6	C₂²×M₁₆×C₁₉₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#15691342	—	2	4	no	—
8	C₂×#15691342	C₂²	1	2	yes	—
9	C₂×#15691342	C₂²	1	2	yes	—
10	C₂²×#13096802	C₂²	1	2	yes	—
11	#20263285	C₄	1	2	yes	—
12	#20263285	C₄	1	2	yes	—
13	#20263285	C₂²	1	2	yes	—
14	#20263285	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₁₉₇	C₄	1	2	yes	—
16	C₂×C₂²⋊Dic₁₉₇	C₄	1	2	yes	—
17	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
18	#20263255	C₂²	1	16	yes	complemented
19	C₂³×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₁₉₇	—	2	4	no	—
21	C₂×M₁₆×C₁₉₇	C₂²	1	2	yes	—
22	C₂×M₁₆×C₁₉₇	C₂²	1	2	yes	—
23	C₂²×C₈×C₁₉₇	C₂²	1	2	yes	—
24	C₂²×Dic₁₉₇	C₂×C₄	1	4	yes	—
25	#15691342	—	2	4	no	—
26	#15691342	—	2	4	no	—
27	C₂×#13096802	—	2	4	no	—
28	C₂×#13096802	C₂³	1	4	yes	—
29	C₄⋊Dic₁₉₇	C₂×C₄	1	4	yes	—
30	C₄⋊Dic₁₉₇	C₂×C₄	1	4	yes	—
31	C₂²⋊Dic₁₉₇	C₂×C₄	1	4	yes	—
32	C₂²⋊Dic₁₉₇	C₂×C₄	1	4	yes	—
33	C₄×Dic₁₉₇	C₂×C₄	1	4	yes	—
34	C₄×Dic₁₉₇	C₂³	1	4	yes	—
35	C₂⁴×C₁₉₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₉₇	D₄	1	4	yes	—
40	C₂²×C₄×C₁₉₇	Q₈	1	4	yes	—
41	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
44	M₁₆×C₁₉₇	—	2	4	no	—
45	M₁₆×C₁₉₇	—	2	4	no	—
46	C₂×C₈×C₁₉₇	—	2	4	no	—
47	C₂×C₈×C₁₉₇	C₂³	1	4	yes	—
48	C₂×Dic₁₉₇	—	4	4	no	—
49	C₂×Dic₁₉₇	C₂²×C₄	1	8	yes	—
50	#13096802	—	2	4	no	—
51	C₂³×C₁₉₇	—	4	2	no	—
52	C₂³×C₁₉₇	C₄⋊C₄	1	4	yes	—
53	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
54	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
55	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₉₇	C₄⋊C₄	1	4	yes	—
57	C₂×C₄×C₁₉₇	C₄⋊C₄	1	4	yes	—
58	C₂×C₄×C₁₉₇	C₄⋊C₄	1	4	yes	—
59	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₁₉₇	C₂×Q₈	1	2	yes	—
66	C₂×C₄×C₁₉₇	C₂×Q₈	1	2	yes	—
67	C₂×C₄×C₁₉₇	C₂×Q₈	1	2	yes	—
68	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
69	C₈×C₁₉₇	—	2	4	no	—
70	C₂²×C₁₉₇	—	2	2	no	—
71	C₂²×C₁₉₇	—	2	2	no	—
72	C₂²×C₁₉₇	—	2	2	no	—
73	C₂²×C₁₉₇	—	4	4	no	—
74	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	2	yes	—
75	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	2	yes	—
76	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	2	yes	—
77	C₂²×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
78	C₄×C₁₉₇	C₂×C₄⋊C₄	1	2	yes	—
79	C₄×C₁₉₇	C₂×C₄⋊C₄	1	2	yes	—
80	C₄×C₁₉₇	C₂×C₄⋊C₄	1	2	yes	—
81	C₄×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
82	C₄×C₁₉₇	C₂²×Q₈	1	1	yes	characteristic
83	Dic₁₉₇	—	2	8	no	—
84	C₂×C₁₉₇	—	2	2	no	—
85	C₂×C₁₉₇	—	2	2	no	—
86	C₂×C₁₉₇	—	2	2	no	—
87	C₂×C₁₉₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₁₉₇	#568	1	2	yes	—
89	C₁₉₇	#2905	1	1	yes	characteristic, complemented, Sylow
90	#2905	—	197	1	no	maximal, Sylow
91	C₂²×M₁₆	—	197	1	no	—
92	C₂²×M₁₆	D₁₉₇	1	1	yes	characteristic
93	C₂×#117	—	197	1	no	—
94	#424	—	197	4	no	—
95	#424	—	197	4	no	—
96	#423	—	197	4	no	—
97	C₂³×C₄	C₂×D₁₉₇	1	1	yes	characteristic
98	C₂×M₁₆	—	2	4	no	—
99	C₂×M₁₆	—	394	4	no	—
100	C₂×M₁₆	—	197	2	no	—
101	C₂×M₁₆	—	197	2	no	—
102	C₂×M₁₆	C₂×D₁₉₇	1	2	yes	—
103	C₂×M₁₆	C₂×D₁₉₇	1	2	yes	—
104	C₂²×C₈	—	197	2	no	—
105	C₂²×C₈	C₂×D₁₉₇	1	2	yes	—
106	#117	—	197	2	no	—
107	#117	—	197	2	no	—
108	#117	—	197	2	no	—
109	#117	—	197	2	no	—
110	C₂×C₄⋊C₄	—	197	2	no	—
111	C₂×C₂²⋊C₄	—	197	2	no	—
112	C₂×C₄²	—	197	2	no	—
113	C₄⋊C₈	—	197	16	no	—
114	C₂⁴	C₄×D₁₉₇	1	1	yes	characteristic
115	C₂²×C₄	—	197	4	no	—
116	C₂²×C₄	C₄×D₁₉₇	1	1	yes	characteristic
117	C₂²×C₄	C₄×D₁₉₇	1	1	yes	characteristic
118	C₂²×C₄	C₄×D₁₉₇	1	1	yes	characteristic
119	C₂²×C₄	C₂²⋊D₁₉₇	1	4	yes	—
120	C₂²×C₄	Dic₃₉₄	1	4	yes	—
121	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
122	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
124	M₁₆	—	2	4	no	—
125	M₁₆	—	2	4	no	—
126	M₁₆	—	394	4	no	—
127	M₁₆	—	394	4	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	394	4	no	—
130	C₂×C₈	—	197	4	no	—
131	C₂×C₈	C₂²×D₁₉₇	1	4	yes	—
132	C₄⋊C₄	—	197	4	no	—
133	C₄⋊C₄	—	197	4	no	—
134	C₂²⋊C₄	—	197	4	no	—
135	C₂²⋊C₄	—	197	4	no	—
136	C₄²	—	197	4	no	—
137	C₄²	—	197	4	no	—
138	C₂³	—	4	2	no	—
139	C₂³	#15691337	1	4	yes	—
140	C₂³	C₂×C₄×D₁₉₇	1	1	yes	characteristic
141	C₂³	C₂×C₄×D₁₉₇	1	1	yes	characteristic
142	C₂³	C₂×C₄×D₁₉₇	1	1	yes	characteristic
143	C₂×C₄	—	788	4	no	—
144	C₂×C₄	—	197	8	no	—
145	C₂×C₄	#15691337	1	4	yes	—
146	C₂×C₄	#15691337	1	4	yes	—
147	C₂×C₄	#15691337	1	4	yes	—
148	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
151	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
152	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
153	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
154	C₂×C₄	C₂×Dic₃₉₄	1	2	yes	—
155	C₂×C₄	C₂×Dic₃₉₄	1	2	yes	—
156	C₂×C₄	C₂×Dic₃₉₄	1	2	yes	—
157	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	394	4	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	4	4	no	—
164	C₂²	C₂×#15691337	1	2	yes	—
165	C₂²	C₂×#15691337	1	2	yes	—
166	C₂²	C₂×#15691337	1	2	yes	—
167	C₂²	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
168	C₄	—	394	8	no	—
169	C₄	C₂×#15691337	1	2	yes	—
170	C₄	C₂×#15691337	1	2	yes	—
171	C₄	C₂×#15691337	1	2	yes	—
172	C₄	C₂²×C₂²⋊D₁₉₇	1	1	yes	characteristic
173	C₄	C₂²×Dic₃₉₄	1	1	yes	characteristic
174	C₂	—	2	2	no	—
175	C₂	—	2	2	no	—
176	C₂	—	2	2	no	—
177	C₂	C₂²×#15691337	1	1	yes	characteristic
178	C₂	#27161845	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉₇
commutator, socle	C₂²×C₁₉₇	C₂³×C₄
Fitting	C₂²×M₁₆×C₁₉₇	C₂
prime radical	C₂⁴×C₁₉₇	C₂×C₄
cosocle	C₂×C₄×C₁₉₇	C₂⁴
hypercenter	C₂²×M₁₆	D₁₉₇
hypocenter	C₁₉₇	#2905
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #27161107	C₂	semidirect
4. #27161105	C₂	semidirect
5. #27161102	C₂	semidirect
18. #20263255	C₂²	semidirect
89. C₁₉₇	#2905	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#15691342	2
C₂×#20263285	2
#27161107	2
#27161105	2
#27161102	2
C₂²×M₁₆×C₁₉₇	2
#2905	197