Subgroups — #27144143

Order 12416.

656 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690769	C₂	1	2	yes	maximal
2	#20260605	C₂	1	2	yes	maximal, complemented
3	#20260601	C₂	1	2	yes	maximal, complemented
4	#20260595	C₂	1	4	yes	maximal
5	C₄×C₄⋊Dic₉₇	C₂	1	4	yes	maximal, complemented
6	C₂²×C₄²×C₉₇	C₂	1	1	yes	maximal, characteristic
7	#15690769	—	2	8	no	—
8	C₂×C₄⋊Dic₉₇	C₂²	1	4	yes	—
9	C₂×C₄⋊Dic₉₇	C₂²	1	8	yes	—
10	C₂×C₂²⋊Dic₉₇	C₂²	1	4	yes	—
11	C₂×C₄×Dic₉₇	C₂²	1	4	yes	—
12	#15690717	C₂²	1	8	yes	—
13	C₂³×C₄×C₉₇	C₂²	1	2	yes	—
14	C₂³×C₄×C₉₇	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₉₇	C₄	1	8	yes	complemented
16	C₂×C₄²×C₉₇	C₂²	1	2	yes	—
17	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
19	C₂²×Dic₉₇	C₂³	1	8	yes	—
20	C₄⋊Dic₉₇	—	2	8	no	—
21	C₄⋊Dic₉₇	—	2	16	no	—
22	C₂²⋊Dic₉₇	—	2	8	no	—
23	C₄×Dic₉₇	—	2	8	no	—
24	C₂⁴×C₉₇	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₉₇	—	2	8	no	—
26	C₂²×C₄×C₉₇	C₂×C₄	1	8	yes	—
27	C₂²×C₄×C₉₇	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₉₇	D₄	1	4	yes	—
29	C₂²×C₄×C₉₇	Q₈	1	4	yes	—
30	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
31	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
33	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
34	C₄²×C₉₇	C₂×C₄	1	4	yes	—
35	C₄²×C₉₇	C₂×C₄	1	8	yes	—
36	C₄²×C₉₇	C₂×C₄	1	4	yes	—
37	C₂³×C₉₇	—	2	4	no	—
38	C₂³×C₉₇	C₂²×C₄	1	2	yes	—
39	C₂³×C₉₇	C₂×D₄	1	2	yes	—
40	C₂³×C₉₇	C₂×Q₈	1	2	yes	—
41	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
42	C₂×C₄×C₉₇	—	2	4	no	—
43	C₂×C₄×C₉₇	—	2	16	no	—
44	C₂×C₄×C₉₇	C₄⋊C₄	1	16	yes	complemented
45	C₂×C₄×C₉₇	C₂²×C₄	1	4	yes	—
46	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₉₇	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₉₇	#41	1	8	yes	—
54	C₂×Dic₉₇	—	2	8	no	—
55	C₂×Dic₉₇	—	2	16	no	—
56	C₂²×C₉₇	—	2	4	no	—
57	C₂²×C₉₇	—	2	4	no	—
58	C₂²×C₉₇	—	2	4	no	—
59	C₂²×C₉₇	C₂×C₄⋊C₄	1	4	yes	—
60	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
61	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
62	C₂²×C₉₇	C₂²×Q₈	1	1	yes	characteristic
63	C₂²×C₉₇	C₂×#41	1	4	yes	—
64	C₄×C₉₇	—	2	4	no	—
65	C₄×C₉₇	C₂×C₄⋊C₄	1	4	yes	—
66	C₄×C₉₇	C₂×C₄⋊C₄	1	4	yes	—
67	C₄×C₉₇	#117	1	8	yes	complemented
68	Dic₉₇	—	4	8	no	—
69	C₂×C₉₇	—	2	4	no	—
70	C₂×C₉₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₉₇	C₂×#117	1	2	yes	—
72	C₂×C₉₇	#532	1	2	yes	—
73	C₂×C₉₇	#533	1	2	yes	—
74	#2285	—	97	1	no	maximal, Sylow
75	C₉₇	#2285	1	1	yes	characteristic, complemented, Sylow
76	C₂×#117	—	97	2	no	—
77	C₂²×C₄²	D₉₇	1	1	yes	characteristic
78	#384	—	97	2	no	—
79	#382	—	97	2	no	—
80	#380	—	97	4	no	—
81	C₄×C₄⋊C₄	—	97	4	no	—
82	C₂³×C₄	C₂×D₉₇	1	2	yes	—
83	C₂³×C₄	C₂×D₉₇	1	1	yes	characteristic
84	#117	—	194	8	no	—
85	C₂×C₄⋊C₄	—	97	4	no	—
86	C₂×C₄⋊C₄	—	97	8	no	—
87	C₂×C₂²⋊C₄	—	97	4	no	—
88	C₂×C₄²	—	97	4	no	—
89	C₂×C₄²	Dic₉₇	1	8	yes	complemented
90	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
91	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
92	C₂×C₄²	C₂×D₉₇	1	2	yes	—
93	#95	—	97	8	no	—
94	C₂⁴	C₂²×D₉₇	1	1	yes	characteristic
95	C₂²×C₄	—	2	8	no	—
96	C₂²×C₄	—	97	8	no	—
97	C₂²×C₄	Dic₁₉₄	1	4	yes	—
98	C₂²×C₄	D₃₈₈	1	4	yes	—
99	C₂²×C₄	C₂×Dic₉₇	1	8	yes	—
100	C₂²×C₄	C₂×Dic₉₇	1	4	yes	—
101	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
102	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
104	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
105	C₄⋊C₄	—	194	8	no	—
106	C₄⋊C₄	—	194	16	no	—
107	C₂²⋊C₄	—	194	8	no	—
108	C₄²	—	194	8	no	—
109	C₄²	C₂×Dic₉₇	1	4	yes	—
110	C₄²	C₂×Dic₉₇	1	8	yes	—
111	C₄²	C₂×Dic₉₇	1	4	yes	—
112	C₂³	—	2	4	no	—
113	C₂³	C₂²×Dic₉₇	1	2	yes	—
114	C₂³	C₂×D₃₈₈	1	2	yes	—
115	C₂³	C₂×Dic₁₉₄	1	2	yes	—
116	C₂³	C₂³×D₉₇	1	1	yes	characteristic
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	16	no	—
119	C₂×C₄	—	194	8	no	—
120	C₂×C₄	—	194	16	no	—
121	C₂×C₄	C₄⋊Dic₉₇	1	16	yes	complemented
122	C₂×C₄	C₂²×Dic₉₇	1	4	yes	—
123	C₂×C₄	C₂²×Dic₉₇	1	2	yes	—
124	C₂×C₄	C₂²×Dic₉₇	1	2	yes	—
125	C₂×C₄	C₂²×Dic₉₇	1	4	yes	—
126	C₂×C₄	C₂×D₃₈₈	1	2	yes	—
127	C₂×C₄	C₂×D₃₈₈	1	2	yes	—
128	C₂×C₄	C₂×Dic₁₉₄	1	2	yes	—
129	C₂×C₄	C₂×Dic₁₉₄	1	2	yes	—
130	C₂×C₄	#13095511	1	8	yes	—
131	C₂²	—	2	4	no	—
132	C₂²	—	2	4	no	—
133	C₂²	—	2	4	no	—
134	C₂²	C₂×C₄⋊Dic₉₇	1	4	yes	—
135	C₂²	C₂³×Dic₉₇	1	1	yes	characteristic
136	C₂²	C₂²×D₃₈₈	1	1	yes	characteristic
137	C₂²	C₂²×Dic₁₉₄	1	1	yes	characteristic
138	C₂²	C₂×#13095511	1	4	yes	—
139	C₄	—	2	4	no	—
140	C₄	—	388	8	no	—
141	C₄	C₂×C₄⋊Dic₉₇	1	4	yes	—
142	C₄	C₂×C₄⋊Dic₉₇	1	4	yes	—
143	C₄	#15690769	1	8	yes	complemented
144	C₂	—	2	4	no	—
145	C₂	C₂²×C₄⋊Dic₉₇	1	1	yes	characteristic
146	C₂	C₂×#15690769	1	2	yes	—
147	C₂	#20261285	1	2	yes	—
148	C₂	#20261292	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉₇
commutator	C₂²×C₉₇	C₂³×C₄
Frattini	C₂³	C₂³×D₉₇
Fitting	C₂²×C₄²×C₉₇	C₂
socle, cosocle	C₂³×C₉₇	C₂⁴
prime radical	C₂⁴×C₉₇	C₂³
hypercenter	C₂²×C₄²	D₉₇
hypocenter	C₉₇	#2285
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. #20260605	C₂	semidirect
3. #20260601	C₂	semidirect
5. C₄×C₄⋊Dic₉₇	C₂	semidirect
15. C₂×C₄²×C₉₇	C₄	semidirect
44. C₂×C₄×C₉₇	C₄⋊C₄	semidirect
67. C₄×C₉₇	#117	semidirect
75. C₉₇	#2285	semidirect
89. C₂×C₄²	Dic₉₇	semidirect
121. C₂×C₄	C₄⋊Dic₉₇	semidirect
143. C₄	#15690769	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690769	2
#20260605	2
#20260601	2
#20260595	2
C₄×C₄⋊Dic₉₇	2
C₂²×C₄²×C₉₇	2
#2285	97