Finite Group Databasebeta

Subgroups — #27157978

Order 12416.

220 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690884	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15690884	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261633	C₂	1	4	yes	maximal, complemented
4	C₂×#15690817	C₂	1	1	yes	maximal, characteristic
5	#15690884	—	2	1	no	—
6	#15690884	—	2	1	no	—
7	#15690884	—	2	1	no	—
8	#15690884	C₂²	1	2	yes	complemented
9	C₂×#13095514	—	2	1	no	—
10	C₂×#13095514	C₂²	1	2	yes	complemented
11	#15690817	—	2	1	no	—
12	#15690817	—	2	1	no	—
13	#15690817	C₄	1	2	yes	—
14	#15690817	C₂²	1	2	yes	—
15	C₂×#13095495	—	2	1	no	—
16	C₂×#13095495	C₄	1	1	yes	characteristic
17	C₂²×#11715895	—	2	1	no	—
18	C₂²×#11715895	C₂²	1	1	yes	characteristic
19	C₂×#41×C₉₇	C₄	1	1	yes	characteristic
20	#13095514	—	2	2	no	—
21	#13095514	—	2	2	no	—
22	#13095495	—	2	1	no	—
23	#13095495	—	2	2	no	—
24	#13095495	—	2	2	no	—
25	#13095495	C₂×C₄	1	2	yes	—
26	C₂×#11715895	—	2	1	no	—
27	C₂×#11715895	—	2	2	no	—
28	C₂×#11715895	—	2	2	no	—
29	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
30	C₂×#11715895	D₄	1	1	yes	characteristic
31	C₂×#11715895	D₄	1	1	yes	characteristic
32	C₂×#11715895	D₄	1	1	yes	characteristic
33	C₂×#11715895	D₄	1	1	yes	characteristic
34	C₂×#11715895	C₂³	1	1	yes	characteristic
35	#41×C₉₇	—	2	1	no	—
36	#41×C₉₇	—	2	1	no	—
37	#41×C₉₇	C₈	1	2	yes	—
38	#41×C₉₇	C₂×C₄	1	2	yes	—
39	C₂×Q₈×C₉₇	C₈	1	1	yes	characteristic
40	C₂×D₄×C₉₇	—	2	1	no	—
41	C₂×D₄×C₉₇	C₈	1	1	yes	characteristic
42	C₂²×C₄×C₉₇	—	2	1	no	—
43	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₉₇	—	2	1	no	—
45	C₂³×C₉₇	M₁₆	1	1	yes	characteristic
46	Q₈×C₉₇	—	2	1	no	—
47	Q₈×C₉₇	C₂×C₈	1	2	yes	—
48	D₄×C₉₇	—	2	2	no	—
49	D₄×C₉₇	—	2	2	no	—
50	D₄×C₉₇	—	2	1	no	—
51	D₄×C₉₇	C₂×C₈	1	2	yes	—
52	C₂×C₄×C₉₇	—	2	1	no	—
53	C₂×C₄×C₉₇	—	2	2	no	—
54	C₂×C₄×C₉₇	—	2	2	no	—
55	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
58	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
59	C₂×C₄×C₉₇	M₁₆	1	1	yes	characteristic
60	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
61	#11715895	—	2	2	no	—
62	#11715895	C₂²⋊C₄	1	1	yes	characteristic
63	#11715895	C₂²⋊C₄	1	1	yes	characteristic
64	#11715895	C₂×D₄	1	1	yes	characteristic
65	#11715895	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₉₇	—	2	2	no	—
67	C₂²×C₉₇	—	4	1	no	—
68	C₂²×C₉₇	—	4	2	no	—
69	C₂²×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
70	C₂²×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
71	C₂²×C₉₇	C₂×M₁₆	1	1	yes	characteristic
72	C₄×C₉₇	—	2	2	no	—
73	C₄×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
74	C₄×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
75	C₄×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₄×C₉₇	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₉₇	—	2	1	no	—
78	C₂×C₉₇	—	2	1	no	—
79	C₂×C₉₇	—	2	1	no	—
80	C₂×C₉₇	—	4	2	no	—
81	C₂×C₉₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
82	#2117	—	97	1	no	maximal, Sylow
83	C₉₇	#2117	1	1	yes	characteristic, complemented, Sylow
84	C₂×#131	—	97	1	no	—
85	C₂×M₃₂	—	97	1	no	—
86	C₂×M₃₂	—	97	1	no	—
87	#350	—	97	4	no	—
88	C₂×#41	#87604	1	1	yes	characteristic
89	#131	—	97	2	no	—
90	#131	—	97	2	no	—
91	#131	—	194	1	no	—
92	#131	—	194	1	no	—
93	C₂×M₁₆	—	97	1	no	—
94	C₂×M₁₆	—	194	1	no	—
95	C₂²×C₈	—	97	1	no	—
96	C₂²×C₈	—	194	1	no	—
97	M₃₂	—	97	2	no	—
98	M₃₂	—	194	1	no	—
99	M₃₂	—	194	1	no	—
100	M₃₂	—	194	1	no	—
101	C₂×C₁₆	—	97	2	no	—
102	C₂×C₁₆	—	194	1	no	—
103	#41	—	2	1	no	—
104	#41	—	2	1	no	—
105	#41	#11715897	1	2	yes	—
106	#41	C₂×#87604	1	2	yes	—
107	C₂×Q₈	#11715897	1	1	yes	characteristic
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	#11715897	1	1	yes	characteristic
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
112	M₁₆	—	97	2	no	—
113	M₁₆	—	194	2	no	—
114	M₁₆	—	194	2	no	—
115	M₁₆	—	194	1	no	—
116	C₂×C₈	—	97	1	no	—
117	C₂×C₈	—	97	1	no	—
118	C₂×C₈	—	97	1	no	—
119	C₂×C₈	—	97	1	no	—
120	C₂×C₈	—	97	1	no	—
121	C₂×C₈	—	97	1	no	—
122	C₂×C₈	—	194	2	no	—
123	C₂×C₈	—	194	2	no	—
124	C₂×C₈	—	194	1	no	—
125	C₁₆	—	194	2	no	—
126	C₁₆	—	194	2	no	—
127	C₂³	—	2	1	no	—
128	C₂³	#13095520	1	1	yes	characteristic
129	Q₈	—	2	1	no	—
130	Q₈	C₂×#11715897	1	2	yes	—
131	D₄	—	2	2	no	—
132	D₄	—	2	2	no	—
133	D₄	—	2	1	no	—
134	D₄	C₂×#11715897	1	2	yes	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
139	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
140	C₂×C₄	C₂×#11715897	1	1	yes	characteristic
141	C₂×C₄	C₂×#11715897	1	1	yes	characteristic
142	C₂×C₄	#13095520	1	1	yes	characteristic
143	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
144	C₈	—	97	1	no	—
145	C₈	—	97	1	no	—
146	C₈	—	97	1	no	—
147	C₈	—	97	1	no	—
148	C₈	—	194	2	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	4	1	no	—
151	C₂²	—	4	2	no	—
152	C₂²	C₂²⋊#11715897	1	1	yes	characteristic
153	C₂²	C₂²⋊#11715897	1	1	yes	characteristic
154	C₂²	C₂×#13095520	1	1	yes	characteristic
155	C₄	—	2	2	no	—
156	C₄	C₂²⋊#11715897	1	1	yes	characteristic
157	C₄	C₂²⋊#11715897	1	1	yes	characteristic
158	C₄	C₂×C₂²⋊#87604	1	1	yes	characteristic
159	C₄	C₂²×#11715897	1	1	yes	characteristic
160	C₂	—	2	1	no	—
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	—	4	2	no	—
164	C₂	C₂×C₂²⋊#11715897	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₂²⋊#87604
commutator	C₄×C₉₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×#87604
Fitting, prime radical	C₂×#41×C₉₇	C₄
socle	C₂×C₉₇	C₂×C₂²⋊C₈
cosocle	C₂×#11715895	C₂³
hypercenter	C₂×#41	#87604
hypocenter	C₉₇	#2117
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#15690884	C₂	semidirect
2. C₂×#15690884	C₂	semidirect
3. #20261633	C₂	semidirect
8. #15690884	C₂²	semidirect
10. C₂×#13095514	C₂²	semidirect
83. C₉₇	#2117	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690884	2
#20261633	2
C₂×#15690817	2
#2117	97