Subgroups — #27142455

Order 12416.

260 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11715895	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11715895	C₂	1	2	yes	maximal
3	C₂×C₄×#11715895	C₂	1	1	yes	maximal, characteristic
4	#20260490	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11715895	C₂²	1	2	yes	—
7	C₂²×#11715895	C₂²	1	1	yes	characteristic
8	C₂²×#11715895	C₂²	1	1	yes	characteristic
9	C₄⋊#11715895	—	2	4	no	—
10	C₄⋊#11715895	—	2	2	no	—
11	C₄×#11715895	—	2	2	no	—
12	C₂×C₄⋊C₄×C₉₇	C₄	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₉₇	C₄	1	1	yes	characteristic
14	C₂×C₄²×C₉₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
16	#95×C₉₇	C₄	1	2	yes	—
17	C₂×#11715895	—	2	1	no	—
18	C₂×#11715895	—	2	1	no	—
19	C₂×#11715895	—	2	1	no	—
20	C₂×#11715895	—	2	1	no	—
21	C₂×#11715895	—	2	2	no	—
22	C₂×#11715895	—	2	2	no	—
23	C₂×#11715895	—	2	2	no	—
24	C₂×#11715895	D₄	1	2	yes	—
25	C₂×#11715895	Q₈	1	2	yes	—
26	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₉₇	—	2	2	no	—
32	C₄⋊C₄×C₉₇	C₈	1	4	yes	complemented
33	C₄²×C₉₇	—	2	2	no	—
34	C₄²×C₉₇	—	2	2	no	—
35	C₄²×C₉₇	D₄	1	2	yes	—
36	C₄²×C₉₇	Q₈	1	2	yes	—
37	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₉₇	—	2	2	no	—
39	C₂×C₄×C₉₇	—	2	1	no	—
40	C₂×C₄×C₉₇	—	2	1	no	—
41	C₂×C₄×C₉₇	—	2	2	no	—
42	C₂×C₄×C₉₇	—	2	2	no	—
43	C₂×C₄×C₉₇	—	2	2	no	—
44	C₂×C₄×C₉₇	—	2	2	no	—
45	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₉₇	C₂×C₈	1	4	yes	—
48	C₂×C₄×C₉₇	C₂×C₈	1	2	yes	—
49	C₂×C₄×C₉₇	M₁₆	1	2	yes	—
50	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
55	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
56	#11715895	—	2	2	no	—
57	#11715895	—	2	2	no	—
58	#11715895	—	4	2	no	—
59	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₉₇	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₉₇	C₄×Q₈	1	1	yes	characteristic
62	C₂²×C₉₇	C₂²×C₈	1	1	yes	characteristic
63	C₂²×C₉₇	C₂×M₁₆	1	1	yes	characteristic
64	C₂²×C₉₇	#131	1	1	yes	characteristic
65	C₂²×C₉₇	#131	1	1	yes	characteristic
66	C₄×C₉₇	—	2	2	no	—
67	C₄×C₉₇	—	2	4	no	—
68	C₄×C₉₇	—	4	2	no	—
69	C₄×C₉₇	C₄⋊C₈	1	4	yes	complemented
70	C₄×C₉₇	#121	1	1	yes	characteristic
71	C₄×C₉₇	C₂²⋊Q₈	1	1	yes	characteristic
72	C₄×C₉₇	#125	1	1	yes	characteristic
73	C₄×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₉₇	#389	1	1	yes	characteristic
75	C₂×C₉₇	C₂×C₄⋊C₈	1	1	yes	characteristic
76	C₂×C₉₇	#424	1	1	yes	characteristic
77	C₂×C₉₇	C₈×D₄	1	1	yes	characteristic
78	C₂×C₉₇	#436	1	1	yes	characteristic
79	C₂×C₉₇	C₈×Q₈	1	1	yes	characteristic
80	C₂×C₉₇	#446	1	1	yes	characteristic
81	#1939	—	97	1	no	maximal, Sylow
82	C₉₇	#1939	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₄⋊C₈	—	97	2	no	—
84	C₂×C₄⋊C₈	—	97	1	no	—
85	C₂×C₄×C₈	—	97	1	no	—
86	C₄×C₄⋊C₄	D₉₇	1	1	yes	characteristic
87	#336	—	97	2	no	—
88	C₂²×C₈	—	97	1	no	—
89	C₂²×C₈	—	97	1	no	—
90	C₂²×C₈	—	97	2	no	—
91	C₂×C₄⋊C₄	Dic₉₇	1	1	yes	characteristic
92	C₂×C₄⋊C₄	Dic₉₇	1	1	yes	characteristic
93	C₂×C₄²	C₂×D₉₇	1	2	yes	—
94	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
95	C₄⋊C₈	—	194	4	no	—
96	C₄⋊C₈	—	194	2	no	—
97	C₄×C₈	—	194	2	no	—
98	#95	Dic₉₇	1	2	yes	—
99	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
100	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
101	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
102	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
104	C₂×C₈	—	97	2	no	—
105	C₂×C₈	—	97	2	no	—
106	C₂×C₈	—	194	2	no	—
107	C₂×C₈	—	194	2	no	—
108	C₂×C₈	—	194	2	no	—
109	C₂×C₈	—	194	1	no	—
110	C₂×C₈	—	194	1	no	—
111	C₂×C₈	—	194	1	no	—
112	C₂×C₈	—	194	1	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	#11715895	1	4	yes	complemented
115	C₄²	—	2	2	no	—
116	C₄²	—	2	2	no	—
117	C₄²	Dic₁₉₄	1	2	yes	—
118	C₄²	D₃₈₈	1	2	yes	—
119	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	C₄⋊Dic₉₇	1	2	yes	—
128	C₂×C₄	C₄⋊Dic₉₇	1	2	yes	—
129	C₂×C₄	C₂×#11715895	1	4	yes	—
130	C₂×C₄	C₂×#11715895	1	2	yes	—
131	C₂×C₄	#13095495	1	2	yes	—
132	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
133	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
134	C₂×C₄	Q₈×D₉₇	1	1	yes	characteristic
135	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
136	C₂×C₄	#13095510	1	1	yes	characteristic
137	C₂×C₄	#13095512	1	1	yes	characteristic
138	C₈	—	388	2	no	—
139	C₈	—	194	2	no	—
140	C₈	—	194	2	no	—
141	C₂²	C₂×C₄⋊Dic₉₇	1	1	yes	characteristic
142	C₂²	D₄×Dic₉₇	1	1	yes	characteristic
143	C₂²	Q₈×Dic₉₇	1	1	yes	characteristic
144	C₂²	C₂²×#11715895	1	1	yes	characteristic
145	C₂²	C₂×#13095495	1	1	yes	characteristic
146	C₂²	#15690817	1	1	yes	characteristic
147	C₂²	#15690817	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	2	4	no	—
150	C₄	—	4	2	no	—
151	C₄	C₄⋊#11715895	1	4	yes	complemented
152	C₄	#15690785	1	1	yes	characteristic
153	C₄	#15690790	1	1	yes	characteristic
154	C₄	#15690801	1	1	yes	characteristic
155	C₄	#15690809	1	1	yes	characteristic
156	C₂	#20260634	1	1	yes	characteristic
157	C₂	C₂×C₄⋊#11715895	1	1	yes	characteristic
158	C₂	#20260762	1	1	yes	characteristic
159	C₂	D₄×#11715895	1	1	yes	characteristic
160	C₂	#20260813	1	1	yes	characteristic
161	C₂	Q₈×#11715895	1	1	yes	characteristic
162	C₂	#20260866	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₉₇
commutator	C₂²×C₉₇	C₂²×C₈
Fitting	C₄×C₄⋊C₄×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₄×C₄⋊C₄	D₉₇
hypocenter	C₉₇	#1939
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
32. C₄⋊C₄×C₉₇	C₈	semidirect
69. C₄×C₉₇	C₄⋊C₈	semidirect
82. C₉₇	#1939	semidirect
114. C₄⋊C₄	#11715895	semidirect
151. C₄	C₄⋊#11715895	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11715895	2
C₂×C₄×#11715895	2
#20260490	2
C₄×C₄⋊C₄×C₉₇	2
#1939	97