Finite Group Databasebeta

Subgroups — #27141858

Order 12416.

172 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 73 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×#15690745	C₂	1	1	yes	maximal, characteristic
2	#20260771	C₂	1	1	yes	maximal, characteristic
3	#20260691	C₂	1	1	yes	maximal, characteristic
4	#20260507	C₂	1	1	yes	maximal, characteristic
5	#20260505	C₂	1	1	yes	maximal, characteristic
6	#20260491	C₂	1	1	yes	maximal, characteristic
7	H(3,ℤ/4)×C₉₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#13095495	C₂²	1	1	yes	characteristic
9	C₂×#13095495	C₂²	1	1	yes	characteristic
10	C₂²×#11715895	C₂²	1	1	yes	characteristic
11	#15690769	C₂²	1	1	yes	characteristic
12	#15690745	—	2	1	no	—
13	#15690745	—	2	1	no	—
14	#15690743	—	2	1	no	—
15	#15690741	—	2	1	no	—
16	#15690721	C₄	1	2	yes	complemented
17	C₄×#11715895	C₄	1	2	yes	complemented
18	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
19	#117×C₉₇	C₂²	1	1	yes	characteristic
20	#117×C₉₇	C₂²	1	1	yes	characteristic
21	#13095495	—	2	1	no	—
22	#13095495	—	2	1	no	—
23	#13095495	—	2	1	no	—
24	#13095495	—	2	1	no	—
25	C₂×#11715895	—	2	1	no	—
26	C₂×#11715895	C₂×C₄	1	2	yes	—
27	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
28	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
29	C₂×#11715895	D₄	1	1	yes	characteristic
30	C₂×#11715895	D₄	1	1	yes	characteristic
31	C₂×#11715895	Q₈	1	1	yes	characteristic
32	C₂×#11715895	Q₈	1	1	yes	characteristic
33	C₄⋊Dic₉₇	—	2	1	no	—
34	C₂²⋊Dic₉₇	—	2	1	no	—
35	C₄×Dic₉₇	—	2	1	no	—
36	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
37	M₁₆×C₉₇	—	2	1	no	—
38	M₁₆×C₉₇	—	2	1	no	—
39	C₂×C₈×C₉₇	—	2	1	no	—
40	C₄⋊C₄×C₉₇	—	2	1	no	—
41	C₄⋊C₄×C₉₇	—	2	1	no	—
42	C₂²⋊C₄×C₉₇	—	2	1	no	—
43	C₂²⋊C₄×C₉₇	—	2	1	no	—
44	C₄²×C₉₇	—	2	1	no	—
45	C₄²×C₉₇	C₂×C₄	1	2	yes	—
46	C₂³×C₉₇	#41	1	1	yes	characteristic
47	C₂×C₄×C₉₇	—	2	2	no	—
48	C₂×C₄×C₉₇	—	4	1	no	—
49	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
53	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
54	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
55	C₂×Dic₉₇	—	4	1	no	—
56	C₈×C₉₇	—	4	1	no	—
57	#11715895	—	2	2	no	—
58	#11715895	—	4	1	no	—
59	#11715895	C₄⋊C₄	1	1	yes	characteristic
60	#11715895	C₄⋊C₄	1	1	yes	characteristic
61	#11715895	C₄⋊C₄	1	1	yes	characteristic
62	#11715895	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₉₇	—	4	1	no	—
64	C₂²×C₉₇	#117	1	1	yes	characteristic
65	C₂²×C₉₇	#124	1	1	yes	characteristic
66	C₂²×C₉₇	#125	1	1	yes	characteristic
67	C₄×C₉₇	—	4	2	no	—
68	C₄×C₉₇	—	8	1	no	—
69	C₄×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₄×C₉₇	#117	1	1	yes	characteristic
71	C₄×C₉₇	#124	1	1	yes	characteristic
72	C₄×C₉₇	#125	1	1	yes	characteristic
73	Dic₉₇	—	8	1	no	—
74	C₂×C₉₇	—	2	1	no	—
75	C₂×C₉₇	—	2	1	no	—
76	C₂×C₉₇	—	2	1	no	—
77	C₂×C₉₇	#382	1	1	yes	characteristic
78	#1833	—	97	1	no	maximal, Sylow
79	C₉₇	#1833	1	1	yes	characteristic, complemented, Sylow
80	C₂×#108	—	97	1	no	—
81	#427	—	97	1	no	—
82	#405	—	97	1	no	—
83	#344	—	97	1	no	—
84	#344	—	97	1	no	—
85	H(3,ℤ/4)	—	97	1	no	—
86	H(3,ℤ/4)	D₉₇	1	1	yes	characteristic
87	C₂×M₁₆	—	97	1	no	—
88	C₂×M₁₆	—	97	1	no	—
89	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
90	C₂²×C₈	—	97	1	no	—
91	#117	—	97	1	no	—
92	#117	C₂×D₉₇	1	1	yes	characteristic
93	#117	C₂×D₉₇	1	1	yes	characteristic
94	#108	—	194	1	no	—
95	#108	—	194	1	no	—
96	#107	—	194	1	no	—
97	#106	—	194	1	no	—
98	#97	—	97	2	no	—
99	C₄×C₈	—	97	2	no	—
100	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
101	M₁₆	—	2	1	no	—
102	M₁₆	—	2	1	no	—
103	M₁₆	—	194	1	no	—
104	M₁₆	—	194	1	no	—
105	M₁₆	—	194	1	no	—
106	M₁₆	—	194	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	97	1	no	—
109	C₂×C₈	—	97	1	no	—
110	C₂×C₈	—	97	1	no	—
111	C₂×C₈	—	97	1	no	—
112	C₂×C₈	—	97	1	no	—
113	C₂×C₈	—	97	1	no	—
114	C₂×C₈	—	97	2	no	—
115	C₂×C₈	—	194	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	194	1	no	—
119	C₂²⋊C₄	—	2	1	no	—
120	C₂²⋊C₄	—	2	1	no	—
121	C₂²⋊C₄	—	194	1	no	—
122	C₄²	—	2	1	no	—
123	C₄²	—	194	1	no	—
124	C₄²	C₄×D₉₇	1	2	yes	—
125	C₂³	#13095510	1	1	yes	characteristic
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	388	1	no	—
129	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
130	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
131	C₂×C₄	Q₈×D₉₇	1	1	yes	characteristic
132	C₂×C₄	#13095510	1	1	yes	characteristic
133	C₂×C₄	#13095511	1	1	yes	characteristic
134	C₂×C₄	#13095511	1	1	yes	characteristic
135	C₈	—	4	1	no	—
136	C₈	—	97	1	no	—
137	C₈	—	97	1	no	—
138	C₈	—	97	1	no	—
139	C₈	—	97	1	no	—
140	C₈	—	388	1	no	—
141	C₈	—	194	2	no	—
142	C₂²	—	4	1	no	—
143	C₂²	#15690766	1	1	yes	characteristic
144	C₂²	#15690796	1	1	yes	characteristic
145	C₂²	#15690800	1	1	yes	characteristic
146	C₄	—	4	2	no	—
147	C₄	—	8	1	no	—
148	C₄	—	776	1	no	—
149	C₄	C₄⋊C₄×D₉₇	1	1	yes	characteristic
150	C₄	#15690767	1	1	yes	characteristic
151	C₄	#15690796	1	1	yes	characteristic
152	C₄	#15690800	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#20260599	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15690767
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	H(3,ℤ/4)×C₉₇	C₂
socle	C₂×C₉₇	#382
prime radical	C₂³×C₉₇	#41
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	H(3,ℤ/4)	D₉₇
hypocenter	C₉₇	#1833
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ #117 ◁ H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
16. #15690721	C₄	semidirect
17. C₄×#11715895	C₄	semidirect
79. C₉₇	#1833	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690745	2
#20260771	2
#20260691	2
#20260507	2
#20260505	2
#20260491	2
H(3,ℤ/4)×C₉₇	2
#1833	97