Finite Group Databasebeta

Subgroups — #27158052

Order 12416.

220 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 62 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	#20261791	C₂	1	1	yes	maximal, characteristic, complemented
2	D₉₇⋊#108	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261650	C₂	1	4	yes	maximal, complemented
4	C₂×#15690833	C₂	1	1	yes	maximal, characteristic
5	D₉₇⋊M₁₆	—	2	1	no	—
6	#15690885	—	2	1	no	—
7	#15690884	—	2	1	no	—
8	#15690883	C₂²	1	2	yes	complemented
9	#15690881	C₂²	1	2	yes	complemented
10	#15690880	—	2	1	no	—
11	C₂×#13095512	—	2	1	no	—
12	#15690833	—	2	1	no	—
13	#15690833	—	2	1	no	—
14	#15690833	C₄	1	2	yes	—
15	#15690833	C₂²	1	2	yes	—
16	C₂×#13095498	—	2	1	no	—
17	C₂×D₇₇₆	C₄	1	1	yes	characteristic
18	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
19	C₂×Q₁₆×C₉₇	C₄	1	1	yes	characteristic
20	#13095521	—	2	2	no	—
21	#13095520	—	4	1	no	—
22	#13095519	—	4	1	no	—
23	#13095514	—	2	2	no	—
24	#13095512	—	2	2	no	—
25	#13095512	—	2	2	no	—
26	#13095512	—	4	1	no	—
27	#13095512	—	4	1	no	—
28	C₂×D₃₈₈	—	2	1	no	—
29	C₂×D₃₈₈	—	4	1	no	—
30	C₂×C₄×D₉₇	—	4	1	no	—
31	C₂×C₄×D₉₇	D₄	1	1	yes	characteristic
32	#13095498	—	2	2	no	—
33	#13095498	—	2	2	no	—
34	D₇₇₆	—	2	1	no	—
35	D₇₇₆	C₂×C₄	1	2	yes	—
36	C₂×#11715895	D₄	1	1	yes	characteristic
37	C₈×D₉₇	C₂×C₄	1	1	yes	characteristic
38	C₈×D₉₇	D₄	1	1	yes	characteristic
39	C₈×D₉₇	D₄	1	1	yes	characteristic
40	C₈×D₉₇	C₂³	1	1	yes	characteristic
41	C₂×Q₈×C₉₇	—	2	1	no	—
42	Q₁₆×C₉₇	—	2	1	no	—
43	Q₁₆×C₉₇	C₂×C₄	1	2	yes	—
44	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×D₉₇	—	4	1	no	—
46	C₂²×D₉₇	SD₁₆	1	1	yes	characteristic
47	Q₈×C₉₇	—	2	2	no	—
48	Q₈×C₉₇	—	4	1	no	—
49	C₂×C₄×C₉₇	—	4	1	no	—
50	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Dic₉₇	D₈	1	1	yes	characteristic
52	D₃₈₈	—	2	2	no	—
53	D₃₈₈	—	4	2	no	—
54	D₃₈₈	—	4	2	no	—
55	D₃₈₈	—	4	1	no	—
56	C₄×D₉₇	—	4	2	no	—
57	C₄×D₉₇	—	4	2	no	—
58	C₄×D₉₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₄×D₉₇	D₈	1	1	yes	characteristic
60	C₄×D₉₇	SD₁₆	1	1	yes	characteristic
61	C₄×D₉₇	C₂×D₄	1	1	yes	characteristic
62	#11715897	—	4	2	no	—
63	C₈×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₈×C₉₇	C₂²×C₄	1	1	yes	characteristic
65	#11715895	C₂²⋊C₄	1	1	yes	characteristic
66	#11715895	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₉₇	#102	1	1	yes	characteristic
68	C₂×D₉₇	—	4	1	no	—
69	C₂×D₉₇	—	4	2	no	—
70	C₂×D₉₇	—	8	2	no	—
71	C₂×D₉₇	#102	1	1	yes	characteristic
72	C₂×D₉₇	C₂×SD₁₆	1	1	yes	characteristic
73	C₄×C₉₇	—	4	2	no	—
74	C₄×C₉₇	#102	1	1	yes	characteristic
75	C₄×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	Dic₉₇	#102	1	1	yes	characteristic
77	Dic₉₇	C₂×D₈	1	1	yes	characteristic
78	C₂×C₉₇	—	2	1	no	—
79	C₂×C₉₇	C₂×#102	1	1	yes	characteristic
80	D₉₇	—	2	1	no	—
81	D₉₇	—	2	1	no	—
82	D₉₇	—	8	2	no	—
83	#2143	—	97	1	no	maximal, Sylow
84	C₉₇	#2143	1	1	yes	characteristic, complemented, Sylow
85	C₂×#135	—	97	1	no	—
86	C₂×M₃₂	—	97	1	no	—
87	C₂×#108	—	97	1	no	—
88	#359	—	97	4	no	—
89	C₂×#41	—	194	1	no	—
90	#135	—	97	2	no	—
91	#135	—	97	2	no	—
92	#135	—	194	1	no	—
93	#135	—	194	1	no	—
94	C₂×Q₁₆	#87604	1	1	yes	characteristic
95	C₂×SD₁₆	—	194	1	no	—
96	C₂×D₈	—	97	1	no	—
97	C₂×M₁₆	—	194	1	no	—
98	C₂²×C₈	—	97	1	no	—
99	M₃₂	—	194	1	no	—
100	M₃₂	—	194	1	no	—
101	C₂×C₁₆	—	97	2	no	—
102	#108	—	97	2	no	—
103	#108	—	194	1	no	—
104	#41	—	388	1	no	—
105	#41	—	388	1	no	—
106	#41	—	194	2	no	—
107	#41	—	194	2	no	—
108	C₂×Q₈	—	2	1	no	—
109	C₂×D₄	—	388	1	no	—
110	C₂×D₄	—	194	1	no	—
111	C₂²×C₄	—	97	1	no	—
112	C₂²×C₄	—	388	1	no	—
113	Q₁₆	—	2	1	no	—
114	Q₁₆	C₂×#87604	1	2	yes	—
115	SD₁₆	—	194	2	no	—
116	SD₁₆	—	194	2	no	—
117	D₈	—	97	2	no	—
118	D₈	—	194	1	no	—
119	M₁₆	—	388	1	no	—
120	M₁₆	—	194	2	no	—
121	C₂×C₈	—	97	1	no	—
122	C₂×C₈	—	97	1	no	—
123	C₂×C₈	—	97	1	no	—
124	C₂×C₈	—	97	1	no	—
125	C₂×C₈	—	97	1	no	—
126	C₂×C₈	—	388	1	no	—
127	C₂×C₈	C₂×#87604	1	1	yes	characteristic
128	C₁₆	—	194	2	no	—
129	C₂³	—	97	1	no	—
130	C₂³	—	388	1	no	—
131	Q₈	—	2	2	no	—
132	Q₈	—	4	1	no	—
133	D₄	—	388	2	no	—
134	D₄	—	388	2	no	—
135	D₄	—	388	1	no	—
136	D₄	—	194	2	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	97	1	no	—
139	C₂×C₄	—	97	1	no	—
140	C₂×C₄	—	97	1	no	—
141	C₂×C₄	—	97	1	no	—
142	C₂×C₄	—	97	1	no	—
143	C₂×C₄	—	388	2	no	—
144	C₂×C₄	—	388	2	no	—
145	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
146	C₈	—	97	1	no	—
147	C₈	—	97	1	no	—
148	C₈	—	388	2	no	—
149	C₈	C₂²⋊#87604	1	1	yes	characteristic
150	C₈	C₂²×#87604	1	1	yes	characteristic
151	C₂²	—	776	2	no	—
152	C₂²	—	97	1	no	—
153	C₂²	—	97	1	no	—
154	C₂²	—	388	1	no	—
155	C₂²	—	388	2	no	—
156	C₂²	#15690872	1	1	yes	characteristic
157	C₄	—	4	2	no	—
158	C₄	—	97	1	no	—
159	C₄	—	97	1	no	—
160	C₄	#15690872	1	1	yes	characteristic
161	C₄	C₂×C₂²⋊#87604	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	776	2	no	—
164	C₂	—	194	1	no	—
165	C₂	—	194	1	no	—
166	C₂	C₂×#15690872	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#15690872
commutator	C₈×C₉₇	C₂²×C₄
Frattini	C₈	C₂²×#87604
Fitting	C₂×Q₁₆×C₉₇	C₄
socle	C₂×C₉₇	C₂×#102
prime radical	C₂×#15690833	C₂
cosocle	C₈×D₉₇	C₂³
hypercenter	C₂×Q₁₆	#87604
hypocenter	C₉₇	#2143
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
1. #20261791	C₂	semidirect
2. D₉₇⋊#108	C₂	semidirect
3. #20261650	C₂	semidirect
8. #15690883	C₂²	semidirect
9. #15690881	C₂²	semidirect
84. C₉₇	#2143	semidirect

Maximal subgroups

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

subgroup	index
#20261791	2
D₉₇⋊#108	2
#20261650	2
C₂×#15690833	2
#2143	97