Finite Group Databasebeta

Subgroups — #27141619

Order 12416.

224 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 105 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	#20260693	C₂	1	1	yes	maximal, characteristic
4	#20260691	C₂	1	1	yes	maximal, characteristic
5	C₂×#15690721	C₂	1	1	yes	maximal, characteristic
6	#20260495	C₂	1	1	yes	maximal, characteristic
7	#339×C₉₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#13095495	C₂²	1	1	yes	characteristic
9	C₂²×#11715895	C₂²	1	1	yes	characteristic
10	C₂²×#11715895	C₂²	1	1	yes	characteristic
11	#15690769	C₂²	1	1	yes	characteristic
12	#15690745	C₄	1	4	yes	complemented
13	#15690743	C₄	1	2	yes	complemented
14	#15690741	C₄	1	2	yes	complemented
15	#15690721	—	2	1	no	—
16	#15690721	C₄	1	4	yes	complemented
17	#15690720	—	2	1	no	—
18	C₄×#11715895	—	2	1	no	—
19	C₈×Dic₉₇	—	2	1	no	—
20	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
21	#117×C₉₇	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
23	#13095495	—	2	1	no	—
24	#13095495	C₂×C₄	1	2	yes	—
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	C₂×C₄	1	2	yes	—
30	C₂×#11715895	C₂×C₄	1	2	yes	—
31	C₂×#11715895	D₄	1	1	yes	characteristic
32	C₂×#11715895	D₄	1	1	yes	characteristic
33	C₂×#11715895	Q₈	1	2	yes	—
34	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
35	C₂²⋊Dic₉₇	—	2	1	no	—
36	C₄×Dic₉₇	—	2	1	no	—
37	C₂²×C₄×C₉₇	—	2	1	no	—
38	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
39	M₁₆×C₉₇	—	2	1	no	—
40	M₁₆×C₉₇	C₂×C₄	1	2	yes	—
41	C₂×C₈×C₉₇	—	2	1	no	—
42	C₄⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
43	C₂²⋊C₄×C₉₇	—	2	1	no	—
44	C₄²×C₉₇	—	2	1	no	—
45	C₄²×C₉₇	—	2	1	no	—
46	C₄²×C₉₇	C₂×C₄	1	2	yes	—
47	C₂³×C₉₇	#41	1	1	yes	characteristic
48	C₂×C₄×C₉₇	—	2	2	no	—
49	C₂×C₄×C₉₇	—	2	2	no	—
50	C₂×C₄×C₉₇	—	4	2	no	—
51	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
57	C₂×Dic₉₇	—	2	2	no	—
58	C₈×C₉₇	—	2	2	no	—
59	#11715895	—	2	2	no	—
60	#11715895	C₄²	1	2	yes	—
61	#11715895	C₄²	1	2	yes	—
62	#11715895	C₄⋊C₄	1	2	yes	—
63	#11715895	C₄⋊C₄	1	2	yes	—
64	C₂²×C₉₇	—	2	2	no	—
65	C₂²×C₉₇	#117	1	1	yes	characteristic
66	C₂²×C₉₇	C₄×D₄	1	1	yes	characteristic
67	C₂²×C₉₇	C₄×Q₈	1	1	yes	characteristic
68	C₄×C₉₇	—	4	2	no	—
69	C₄×C₉₇	—	4	2	no	—
70	C₄×C₉₇	C₂×C₄²	1	1	yes	characteristic
71	C₄×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₄×C₉₇	C₄×D₄	1	1	yes	characteristic
73	C₄×C₉₇	C₄×Q₈	1	1	yes	characteristic
74	Dic₉₇	—	4	2	no	—
75	C₂×C₉₇	—	2	1	no	—
76	C₂×C₉₇	—	2	1	no	—
77	C₂×C₉₇	C₄×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₉₇	#444	1	2	yes	—
79	#1773	—	97	1	no	maximal, Sylow
80	C₉₇	#1773	1	1	yes	characteristic, complemented, Sylow
81	C₂×#108	—	97	1	no	—
82	#427	—	97	1	no	—
83	#405	—	97	1	no	—
84	#405	—	97	1	no	—
85	C₂×#97	—	97	1	no	—
86	#339	—	97	1	no	—
87	#339	D₉₇	1	1	yes	characteristic
88	C₂×M₁₆	—	97	1	no	—
89	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
90	C₂²×C₈	—	97	1	no	—
91	C₂²×C₈	—	97	1	no	—
92	#117	—	97	1	no	—
93	#117	C₂×D₉₇	1	1	yes	characteristic
94	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
95	#108	—	97	4	no	—
96	#107	—	97	2	no	—
97	#106	—	97	2	no	—
98	#97	—	97	4	no	—
99	#97	—	194	1	no	—
100	#97	—	194	1	no	—
101	C₄×C₈	—	194	1	no	—
102	C₄×C₈	—	194	1	no	—
103	C₂²×C₄	—	2	1	no	—
104	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
105	M₁₆	—	2	1	no	—
106	M₁₆	—	97	2	no	—
107	M₁₆	—	194	1	no	—
108	M₁₆	C₄×D₉₇	1	2	yes	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	97	2	no	—
111	C₂×C₈	—	97	2	no	—
112	C₂×C₈	—	97	1	no	—
113	C₂×C₈	—	97	1	no	—
114	C₂×C₈	—	97	2	no	—
115	C₂×C₈	—	97	1	no	—
116	C₂×C₈	—	97	1	no	—
117	C₂×C₈	—	97	2	no	—
118	C₂×C₈	—	194	1	no	—
119	C₄⋊C₄	—	97	2	no	—
120	C₄⋊C₄	C₄×D₉₇	1	2	yes	—
121	C₂²⋊C₄	—	2	1	no	—
122	C₂²⋊C₄	—	194	1	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	2	1	no	—
125	C₄²	—	194	1	no	—
126	C₄²	C₄×D₉₇	1	2	yes	—
127	C₂³	#13095510	1	1	yes	characteristic
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	194	2	no	—
132	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
133	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
135	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
136	C₂×C₄	Q₈×D₉₇	1	1	yes	characteristic
137	C₂×C₄	#13095510	1	1	yes	characteristic
138	C₈	—	2	2	no	—
139	C₈	—	97	2	no	—
140	C₈	—	97	2	no	—
141	C₈	—	97	2	no	—
142	C₈	—	97	2	no	—
143	C₈	—	194	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	#15690766	1	1	yes	characteristic
146	C₂²	#15690770	1	1	yes	characteristic
147	C₂²	#15690775	1	1	yes	characteristic
148	C₄	—	4	2	no	—
149	C₄	—	4	2	no	—
150	C₄	—	388	2	no	—
151	C₄	C₄²×D₉₇	1	1	yes	characteristic
152	C₄	C₄⋊C₄×D₉₇	1	1	yes	characteristic
153	C₄	#15690770	1	1	yes	characteristic
154	C₄	#15690775	1	1	yes	characteristic
155	C₂	—	2	1	no	—
156	C₂	—	2	1	no	—
157	C₂	#20260586	1	1	yes	characteristic
158	C₂	#20260854	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×D₉₇
commutator	C₄×C₉₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#339×C₉₇	C₂
socle	C₂²×C₉₇	C₄×D₄
prime radical	C₂³×C₉₇	#41
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#339	D₉₇
hypocenter	C₉₇	#1773
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

Complements

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

N	H ≅ G/N	type
12. #15690745	C₄	semidirect
13. #15690743	C₄	semidirect
14. #15690741	C₄	semidirect
16. #15690721	C₄	semidirect
80. C₉₇	#1773	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
#20260693	2
#20260691	2
C₂×#15690721	2
#20260495	2
#339×C₉₇	2
#1773	97