Finite Group Databasebeta

Subgroups — #27158906

Order 12416.

228 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 71 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×#15690906	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13095523	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261832	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#13095523	C₂	1	2	yes	maximal, complemented
5	C₂×#15690897	C₂	1	1	yes	maximal, characteristic
6	#15690906	—	2	2	no	—
7	C₂×#13095523	—	2	2	no	—
8	C₂×#13095523	C₂²	1	2	yes	—
9	C₂×#13095523	C₂²	1	2	yes	—
10	#15690897	—	2	2	no	—
11	#15690897	—	2	2	no	—
12	C₂×#13095521	C₄	1	1	yes	characteristic
13	C₂×#13095520	C₄	1	1	yes	characteristic
14	C₂×#13095520	C₄	1	1	yes	characteristic
15	C₂×#13095519	C₂²	1	1	yes	characteristic
16	C₂²×#11715897	C₂²	1	1	yes	characteristic
17	C₂²×#11715897	C₂²	1	1	yes	characteristic
18	C₂×#13095510	C₄	1	1	yes	characteristic
19	#13095523	—	2	2	no	—
20	#13095523	—	2	2	no	—
21	#13095521	—	2	2	no	—
22	#13095520	—	2	2	no	—
23	#13095520	—	2	2	no	—
24	#13095519	—	2	1	no	—
25	#13095519	—	2	1	no	—
26	C₂×#11715897	—	2	1	no	—
27	C₂×#11715897	—	2	1	no	—
28	C₂×#11715897	C₂×C₄	1	1	yes	characteristic
29	C₂×#11715897	C₂×C₄	1	1	yes	characteristic
30	C₂×#11715897	C₂×C₄	1	1	yes	characteristic
31	C₂×#11715897	D₄	1	2	yes	—
32	C₂×#11715897	D₄	1	2	yes	—
33	C₂×#11715897	C₂³	1	1	yes	characteristic
34	#13095510	—	2	2	no	—
35	#13095510	—	2	2	no	—
36	C₂×Dic₁₉₄	C₈	1	1	yes	characteristic
37	C₂×C₂²⋊D₉₇	C₈	1	1	yes	characteristic
38	C₂×C₂²⋊D₉₇	C₈	1	1	yes	characteristic
39	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₉₇	C₈	1	1	yes	characteristic
43	C₂³×C₉₇	C₂×C₈	1	1	yes	characteristic
44	C₂³×C₉₇	C₂×C₈	1	1	yes	characteristic
45	C₂²×D₉₇	C₂×C₈	1	1	yes	characteristic
46	D₄×C₉₇	—	2	2	no	—
47	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
48	C₂²⋊D₉₇	—	2	2	no	—
49	C₂²⋊D₉₇	—	2	2	no	—
50	C₂×Dic₉₇	—	2	1	no	—
51	C₂×Dic₉₇	—	2	1	no	—
52	C₂×Dic₉₇	C₂²⋊C₄	1	2	yes	—
53	C₂×Dic₉₇	C₂×C₈	1	1	yes	characteristic
54	C₂×Dic₉₇	C₂×C₈	1	1	yes	characteristic
55	C₂×Dic₉₇	M₁₆	1	2	yes	—
56	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
57	C₄×D₉₇	—	2	1	no	—
58	C₄×D₉₇	—	2	1	no	—
59	Dic₁₉₄	—	2	2	no	—
60	#11715897	—	2	1	no	—
61	#11715897	—	2	1	no	—
62	#11715897	C₂²⋊C₄	1	2	yes	—
63	#11715897	C₂×D₄	1	1	yes	characteristic
64	#11715897	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₉₇	—	2	2	no	—
66	C₂²×C₉₇	—	2	1	no	—
67	C₂²×C₉₇	—	2	2	no	—
68	C₂²×C₉₇	C₂²⋊C₈	1	2	yes	—
69	C₂²×C₉₇	C₂²×C₈	1	1	yes	characteristic
70	C₂×D₉₇	—	2	1	no	—
71	C₂×D₉₇	—	2	2	no	—
72	C₄×C₉₇	—	2	1	no	—
73	Dic₉₇	—	2	1	no	—
74	Dic₉₇	C₂²⋊C₈	1	2	yes	—
75	Dic₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	Dic₉₇	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₉₇	—	2	2	no	—
78	C₂×C₉₇	—	4	1	no	—
79	C₂×C₉₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₉₇	#504	1	2	yes	—
81	D₉₇	—	4	1	no	—
82	#2113	—	97	1	no	maximal, Sylow
83	C₉₇	#2113	1	1	yes	characteristic, complemented, Sylow
84	C₂×#131	—	97	1	no	—
85	C₂×M₃₂	—	97	1	no	—
86	C₂²×C₁₆	—	97	1	no	—
87	C₂²⋊C₁₆	—	97	2	no	—
88	C₂²⋊C₁₆	—	97	2	no	—
89	C₂×#41	—	97	1	no	—
90	#131	—	194	2	no	—
91	#131	—	194	2	no	—
92	C₂×M₁₆	—	97	1	no	—
93	C₂×M₁₆	—	97	1	no	—
94	C₂×M₁₆	—	97	1	no	—
95	C₂²×C₈	—	97	1	no	—
96	C₂²×C₈	—	97	1	no	—
97	C₂²×C₈	—	97	1	no	—
98	M₃₂	—	194	2	no	—
99	C₂×C₁₆	—	97	2	no	—
100	C₂×C₁₆	—	97	2	no	—
101	C₂×C₁₆	—	194	2	no	—
102	#41	—	194	2	no	—
103	#41	—	194	2	no	—
104	C₂×Q₈	—	97	1	no	—
105	C₂×D₄	—	97	1	no	—
106	C₂×D₄	—	97	1	no	—
107	C₂×D₄	#11715906	1	1	yes	characteristic
108	C₂²×C₄	—	97	1	no	—
109	C₂²×C₄	—	97	1	no	—
110	C₂²×C₄	—	97	1	no	—
111	M₁₆	—	194	2	no	—
112	M₁₆	—	194	2	no	—
113	M₁₆	—	194	2	no	—
114	C₂×C₈	—	97	1	no	—
115	C₂×C₈	—	97	2	no	—
116	C₂×C₈	—	97	2	no	—
117	C₂×C₈	—	97	1	no	—
118	C₂×C₈	—	97	1	no	—
119	C₂×C₈	—	97	1	no	—
120	C₂×C₈	—	194	1	no	—
121	C₂×C₈	—	194	1	no	—
122	C₂×C₈	—	194	1	no	—
123	C₂×C₈	—	194	1	no	—
124	C₁₆	—	194	2	no	—
125	C₁₆	—	194	2	no	—
126	C₂³	—	97	1	no	—
127	C₂³	C₂×#11715906	1	1	yes	characteristic
128	C₂³	C₂×#11715906	1	1	yes	characteristic
129	Q₈	—	194	2	no	—
130	D₄	—	2	2	no	—
131	D₄	—	194	2	no	—
132	D₄	—	194	2	no	—
133	C₂×C₄	—	97	2	no	—
134	C₂×C₄	—	97	2	no	—
135	C₂×C₄	—	97	1	no	—
136	C₂×C₄	—	97	1	no	—
137	C₂×C₄	—	97	1	no	—
138	C₂×C₄	—	194	1	no	—
139	C₂×C₄	—	194	1	no	—
140	C₂×C₄	—	194	1	no	—
141	C₂×C₄	—	194	1	no	—
142	C₂×C₄	C₂×#11715906	1	1	yes	characteristic
143	C₈	—	97	1	no	—
144	C₈	—	97	1	no	—
145	C₈	—	97	2	no	—
146	C₈	—	194	1	no	—
147	C₈	—	194	1	no	—
148	C₂²	—	2	2	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	2	no	—
151	C₂²	—	194	1	no	—
152	C₂²	—	194	2	no	—
153	C₂²	C₂²⋊#11715906	1	2	yes	—
154	C₂²	C₂²×#11715906	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	97	1	no	—
157	C₄	—	97	1	no	—
158	C₄	—	97	2	no	—
159	C₄	—	194	1	no	—
160	C₂	—	2	2	no	—
161	C₂	—	4	1	no	—
162	C₂	—	388	1	no	—
163	C₂	C₂×C₂²⋊#11715906	1	1	yes	characteristic
164	C₂	#20261856	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11715906
commutator, socle	C₂²×C₉₇	C₂²×C₈
Fitting	C₂×D₄×C₉₇	C₈
prime radical	C₂×#13095510	C₄
cosocle	C₂×#11715897	C₂³
hypercenter	C₂×D₄	#11715906
hypocenter	C₉₇	#2113
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#15690906	C₂	semidirect
2. C₂²×#13095523	C₂	semidirect
3. #20261832	C₂	semidirect
4. C₂²⋊#13095523	C₂	semidirect
83. C₉₇	#2113	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690906	2
C₂²×#13095523	2
#20261832	2
C₂²⋊#13095523	2
C₂×#15690897	2
#2113	97