Finite Group Databasebeta

Subgroups — #27143464

Order 12416.

220 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 89 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×#15690750	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15690746	C₂	1	1	yes	maximal, characteristic, complemented
3	#20260542	C₂	1	4	yes	maximal, complemented
4	C₂×#135×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13095495	—	2	1	no	—
6	#15690750	—	2	1	no	—
7	#15690750	—	2	1	no	—
8	C₂×#13095486	C₂²	1	2	yes	complemented
9	#15690746	—	2	1	no	—
10	#15690746	C₂²	1	2	yes	complemented
11	C₂×#41×C₉₇	—	2	1	no	—
12	#135×C₉₇	—	2	1	no	—
13	#135×C₉₇	—	2	1	no	—
14	#135×C₉₇	C₄	1	2	yes	—
15	#135×C₉₇	C₂²	1	2	yes	—
16	C₂×Q₁₆×C₉₇	C₄	1	1	yes	characteristic
17	C₂×SD₁₆×C₉₇	—	2	1	no	—
18	C₂×D₈×C₉₇	C₄	1	1	yes	characteristic
19	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
20	#13095495	—	2	2	no	—
21	#13095495	—	4	1	no	—
22	C₂×#11715895	—	4	1	no	—
23	#13095486	—	2	2	no	—
24	#41×C₉₇	—	2	2	no	—
25	#41×C₉₇	—	2	2	no	—
26	#41×C₉₇	—	4	1	no	—
27	#41×C₉₇	—	4	1	no	—
28	C₂×Q₈×C₉₇	—	2	1	no	—
29	C₂×D₄×C₉₇	—	2	1	no	—
30	C₂×D₄×C₉₇	—	4	1	no	—
31	C₂²×C₄×C₉₇	—	4	1	no	—
32	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
33	Q₁₆×C₉₇	—	2	1	no	—
34	Q₁₆×C₉₇	C₂×C₄	1	2	yes	—
35	SD₁₆×C₉₇	—	2	2	no	—
36	SD₁₆×C₉₇	—	2	2	no	—
37	D₈×C₉₇	—	2	1	no	—
38	D₈×C₉₇	C₂×C₄	1	2	yes	—
39	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
42	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
43	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
44	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
45	C₂³×C₉₇	—	4	1	no	—
46	C₂³×C₉₇	SD₁₆	1	1	yes	characteristic
47	Q₈×C₉₇	—	2	2	no	—
48	Q₈×C₉₇	—	4	1	no	—
49	D₄×C₉₇	—	2	2	no	—
50	D₄×C₉₇	—	4	2	no	—
51	D₄×C₉₇	—	4	2	no	—
52	D₄×C₉₇	—	4	1	no	—
53	C₂×C₄×C₉₇	—	4	1	no	—
54	C₂×C₄×C₉₇	—	4	2	no	—
55	C₂×C₄×C₉₇	—	4	2	no	—
56	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
59	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
60	C₂×C₄×C₉₇	SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
62	C₈×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₈×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₈×C₉₇	C₂²×C₄	1	1	yes	characteristic
65	C₈×C₉₇	C₂×D₄	1	1	yes	characteristic
66	#11715895	—	4	2	no	—
67	C₂²×C₉₇	—	4	1	no	—
68	C₂²×C₉₇	—	4	2	no	—
69	C₂²×C₉₇	—	8	2	no	—
70	C₂²×C₉₇	#102	1	1	yes	characteristic
71	C₂²×C₉₇	#102	1	1	yes	characteristic
72	C₂²×C₉₇	C₂×SD₁₆	1	1	yes	characteristic
73	C₄×C₉₇	—	4	2	no	—
74	C₄×C₉₇	#102	1	1	yes	characteristic
75	C₄×C₉₇	#102	1	1	yes	characteristic
76	C₄×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₄×C₉₇	C₂×D₈	1	1	yes	characteristic
78	C₂×C₉₇	—	2	1	no	—
79	C₂×C₉₇	—	2	1	no	—
80	C₂×C₉₇	—	2	1	no	—
81	C₂×C₉₇	—	8	2	no	—
82	C₂×C₉₇	C₂×#102	1	1	yes	characteristic
83	#2143	—	97	1	no	maximal, Sylow
84	C₉₇	#2143	1	1	yes	characteristic, complemented, Sylow
85	C₂×#135	D₉₇	1	1	yes	characteristic
86	C₂×M₃₂	—	97	1	no	—
87	C₂×#108	—	97	1	no	—
88	#359	—	97	4	no	—
89	C₂×#41	—	2	1	no	—
90	#135	—	2	1	no	—
91	#135	—	2	1	no	—
92	#135	Dic₉₇	1	2	yes	—
93	#135	C₂×D₉₇	1	2	yes	—
94	C₂×Q₁₆	Dic₉₇	1	1	yes	characteristic
95	C₂×SD₁₆	—	2	1	no	—
96	C₂×D₈	Dic₉₇	1	1	yes	characteristic
97	C₂×M₁₆	—	194	1	no	—
98	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
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	—	2	2	no	—
105	#41	—	2	2	no	—
106	#41	—	4	1	no	—
107	#41	—	4	1	no	—
108	C₂×Q₈	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	4	1	no	—
111	C₂²×C₄	—	4	1	no	—
112	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
113	Q₁₆	—	2	1	no	—
114	Q₁₆	C₂×Dic₉₇	1	2	yes	—
115	SD₁₆	—	2	2	no	—
116	SD₁₆	—	2	2	no	—
117	D₈	—	2	1	no	—
118	D₈	C₂×Dic₉₇	1	2	yes	—
119	M₁₆	—	388	1	no	—
120	M₁₆	—	194	2	no	—
121	C₂×C₈	—	388	1	no	—
122	C₂×C₈	C₂×Dic₉₇	1	1	yes	characteristic
123	C₂×C₈	C₂×Dic₉₇	1	1	yes	characteristic
124	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
125	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
126	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
127	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
128	C₁₆	—	194	2	no	—
129	C₂³	—	4	1	no	—
130	C₂³	#13095499	1	1	yes	characteristic
131	Q₈	—	2	2	no	—
132	Q₈	—	4	1	no	—
133	D₄	—	2	2	no	—
134	D₄	—	4	2	no	—
135	D₄	—	4	2	no	—
136	D₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
141	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
142	C₂×C₄	#13095496	1	1	yes	characteristic
143	C₂×C₄	#13095496	1	1	yes	characteristic
144	C₂×C₄	#13095499	1	1	yes	characteristic
145	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
146	C₈	—	388	2	no	—
147	C₈	C₂²⋊Dic₉₇	1	1	yes	characteristic
148	C₈	C₂²⋊Dic₉₇	1	1	yes	characteristic
149	C₈	C₂²×Dic₉₇	1	1	yes	characteristic
150	C₈	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
151	C₂²	—	4	1	no	—
152	C₂²	—	4	2	no	—
153	C₂²	—	8	2	no	—
154	C₂²	#15690731	1	1	yes	characteristic
155	C₂²	#15690731	1	1	yes	characteristic
156	C₂²	C₂×#13095499	1	1	yes	characteristic
157	C₄	—	4	2	no	—
158	C₄	#15690731	1	1	yes	characteristic
159	C₄	#15690731	1	1	yes	characteristic
160	C₄	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
161	C₄	C₂×#13095496	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	—	8	2	no	—
166	C₂	C₂×#15690731	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13095496
commutator	C₈×C₉₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₉₇
Fitting, prime radical	C₂×#135×C₉₇	C₂
socle	C₂×C₉₇	C₂×#102
cosocle	C₂×C₈×C₉₇	C₂³
hypercenter	C₂×#135	D₉₇
hypocenter	C₉₇	#2143
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#15690750	C₂	semidirect
2. C₂×#15690746	C₂	semidirect
3. #20260542	C₂	semidirect
8. C₂×#13095486	C₂²	semidirect
10. #15690746	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
C₂×#15690750	2
C₂×#15690746	2
#20260542	2
C₂×#135×C₉₇	2
#2143	97