Subgroups — #27150154

Order 12416.

564 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13095495	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690769	C₂	1	1	yes	maximal, characteristic
3	#20260693	C₂	1	4	yes	maximal, complemented
4	#20260690	C₂	1	4	yes	maximal, complemented
5	M₁₆×Dic₉₇	C₂	1	4	yes	maximal, complemented
6	C₂²×M₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#13095495	C₄	1	8	yes	complemented
8	C₂×#13095495	C₂²	1	2	yes	—
9	C₂×#13095495	C₂²	1	2	yes	—
10	C₂²×#11715895	C₂²	1	2	yes	—
11	#15690769	C₄	1	2	yes	—
12	#15690769	C₄	1	2	yes	—
13	#15690769	C₂²	1	2	yes	—
14	#15690769	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₉₇	C₄	1	2	yes	—
16	C₂×C₂²⋊Dic₉₇	C₄	1	2	yes	—
17	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
18	#15690720	C₂²	1	8	yes	complemented
19	C₈×Dic₉₇	C₂²	1	8	yes	complemented
20	C₂³×C₄×C₉₇	C₂²	1	1	yes	characteristic
21	C₂×M₁₆×C₉₇	C₄	1	8	yes	complemented
22	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
23	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
24	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
25	C₂²×Dic₉₇	C₂×C₄	1	4	yes	—
26	#13095495	C₂×C₄	1	8	yes	—
27	#13095495	C₂×C₄	1	8	yes	complemented
28	C₂×#11715895	C₂×C₄	1	8	yes	complemented
29	C₂×#11715895	C₂³	1	4	yes	—
30	C₄⋊Dic₉₇	C₂×C₄	1	4	yes	—
31	C₄⋊Dic₉₇	C₂×C₄	1	4	yes	—
32	C₂²⋊Dic₉₇	C₂×C₄	1	4	yes	—
33	C₂²⋊Dic₉₇	C₂×C₄	1	4	yes	—
34	C₄×Dic₉₇	C₂×C₄	1	4	yes	—
35	C₄×Dic₉₇	C₂³	1	4	yes	—
36	C₂⁴×C₉₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₉₇	C₂×C₄	1	4	yes	—
41	C₂²×C₄×C₉₇	C₂×C₄	1	4	yes	—
42	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
45	M₁₆×C₉₇	C₂×C₄	1	8	yes	—
46	M₁₆×C₉₇	C₂×C₄	1	8	yes	complemented
47	C₂×C₈×C₉₇	C₂×C₄	1	8	yes	complemented
48	C₂×C₈×C₉₇	C₂³	1	4	yes	—
49	C₂³×C₉₇	—	4	2	no	—
50	C₂³×C₉₇	C₄²	1	4	yes	—
51	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
52	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
53	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₉₇	C₄²	1	4	yes	—
55	C₂×C₄×C₉₇	C₄²	1	4	yes	—
56	C₂×C₄×C₉₇	C₄²	1	4	yes	—
57	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
67	C₂×Dic₉₇	—	4	4	no	—
68	C₂×Dic₉₇	C₂²×C₄	1	8	yes	—
69	C₈×C₉₇	C₂²×C₄	1	8	yes	—
70	#11715895	C₂²×C₄	1	8	yes	—
71	C₂²×C₉₇	—	2	2	no	—
72	C₂²×C₉₇	—	2	2	no	—
73	C₂²×C₉₇	—	2	2	no	—
74	C₂²×C₉₇	—	4	4	no	—
75	C₂²×C₉₇	C₂×C₄²	1	2	yes	—
76	C₂²×C₉₇	C₂×C₄²	1	2	yes	—
77	C₂²×C₉₇	C₂×C₄²	1	2	yes	—
78	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
79	C₄×C₉₇	C₂×C₄²	1	2	yes	—
80	C₄×C₉₇	C₂×C₄²	1	2	yes	—
81	C₄×C₉₇	C₂×C₄²	1	2	yes	—
82	C₄×C₉₇	C₂³×C₄	1	1	yes	characteristic
83	C₄×C₉₇	C₂³×C₄	1	1	yes	characteristic
84	Dic₉₇	—	2	8	no	—
85	C₂×C₉₇	—	2	2	no	—
86	C₂×C₉₇	—	2	2	no	—
87	C₂×C₉₇	—	2	2	no	—
88	C₂×C₉₇	C₂²×C₄²	1	1	yes	characteristic
89	C₂×C₉₇	#568	1	2	yes	—
90	#2873	—	97	1	no	maximal, Sylow
91	C₉₇	#2873	1	1	yes	characteristic, complemented, Sylow
92	C₂²×M₁₆	—	97	1	no	—
93	C₂²×M₁₆	D₉₇	1	1	yes	characteristic
94	C₂×#117	—	97	1	no	—
95	#405	—	97	4	no	—
96	#405	—	97	4	no	—
97	C₄×M₁₆	—	97	4	no	—
98	C₂³×C₄	C₂×D₉₇	1	1	yes	characteristic
99	C₂×M₁₆	—	97	8	no	—
100	C₂×M₁₆	—	97	2	no	—
101	C₂×M₁₆	—	97	2	no	—
102	C₂×M₁₆	Dic₉₇	1	8	yes	complemented
103	C₂×M₁₆	C₂×D₉₇	1	2	yes	—
104	C₂×M₁₆	C₂×D₉₇	1	2	yes	—
105	C₂²×C₈	—	97	2	no	—
106	C₂²×C₈	C₂×D₉₇	1	2	yes	—
107	#117	—	97	2	no	—
108	#117	—	97	2	no	—
109	#117	—	97	2	no	—
110	#117	—	97	2	no	—
111	C₂×C₄⋊C₄	—	97	2	no	—
112	C₂×C₂²⋊C₄	—	97	2	no	—
113	C₂×C₄²	—	97	2	no	—
114	#97	—	97	8	no	—
115	C₄×C₈	—	97	8	no	—
116	C₂⁴	C₄×D₉₇	1	1	yes	characteristic
117	C₂²×C₄	—	97	4	no	—
118	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
119	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
120	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
121	C₂²×C₄	C₄×D₉₇	1	4	yes	—
122	C₂²×C₄	C₂×Dic₉₇	1	4	yes	—
123	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
124	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
126	M₁₆	—	97	8	no	—
127	M₁₆	—	97	8	no	—
128	M₁₆	C₂×Dic₉₇	1	8	yes	—
129	M₁₆	C₂×Dic₉₇	1	8	yes	complemented
130	C₂×C₈	—	97	8	no	—
131	C₂×C₈	—	97	4	no	—
132	C₂×C₈	C₂×Dic₉₇	1	8	yes	complemented
133	C₂×C₈	C₂²×D₉₇	1	4	yes	—
134	C₄⋊C₄	—	97	4	no	—
135	C₄⋊C₄	—	97	4	no	—
136	C₂²⋊C₄	—	97	4	no	—
137	C₂²⋊C₄	—	97	4	no	—
138	C₄²	—	97	4	no	—
139	C₄²	—	97	4	no	—
140	C₂³	—	4	2	no	—
141	C₂³	C₄×Dic₉₇	1	4	yes	—
142	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
143	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
144	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
145	C₂×C₄	—	97	8	no	—
146	C₂×C₄	—	388	4	no	—
147	C₂×C₄	C₄×Dic₉₇	1	4	yes	—
148	C₂×C₄	C₄×Dic₉₇	1	4	yes	—
149	C₂×C₄	C₄×Dic₉₇	1	4	yes	—
150	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
154	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
155	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
156	C₂×C₄	C₂²×Dic₉₇	1	2	yes	—
157	C₂×C₄	C₂²×Dic₉₇	1	2	yes	—
158	C₂×C₄	C₂²×Dic₉₇	1	2	yes	—
159	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
160	C₈	—	97	8	no	—
161	C₈	C₂²×Dic₉₇	1	8	yes	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	4	4	no	—
166	C₂²	C₂×C₄×Dic₉₇	1	2	yes	—
167	C₂²	C₂×C₄×Dic₉₇	1	2	yes	—
168	C₂²	C₂×C₄×Dic₉₇	1	2	yes	—
169	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
170	C₄	—	194	8	no	—
171	C₄	C₂×C₄×Dic₉₇	1	2	yes	—
172	C₄	C₂×C₄×Dic₉₇	1	2	yes	—
173	C₄	C₂×C₄×Dic₉₇	1	2	yes	—
174	C₄	C₂²×C₄×D₉₇	1	1	yes	characteristic
175	C₄	C₂³×Dic₉₇	1	1	yes	characteristic
176	C₂	—	2	2	no	—
177	C₂	—	2	2	no	—
178	C₂	—	2	2	no	—
179	C₂	C₂²×C₄×Dic₉₇	1	1	yes	characteristic
180	C₂	#20261501	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator	C₂×C₉₇	C₂²×C₄²
Fitting	C₂²×M₁₆×C₉₇	C₂
socle	C₂²×C₉₇	C₂³×C₄
prime radical	C₂⁴×C₉₇	C₂×C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₂²×M₁₆	D₉₇
hypocenter	C₉₇	#2873
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20260693	C₂	semidirect
4. #20260690	C₂	semidirect
5. M₁₆×Dic₉₇	C₂	semidirect
7. C₂×#13095495	C₄	semidirect
18. #15690720	C₂²	semidirect
19. C₈×Dic₉₇	C₂²	semidirect
21. C₂×M₁₆×C₉₇	C₄	semidirect
27. #13095495	C₂×C₄	semidirect
28. C₂×#11715895	C₂×C₄	semidirect
46. M₁₆×C₉₇	C₂×C₄	semidirect
47. C₂×C₈×C₉₇	C₂×C₄	semidirect
91. C₉₇	#2873	semidirect
102. C₂×M₁₆	Dic₉₇	semidirect
129. M₁₆	C₂×Dic₉₇	semidirect
132. C₂×C₈	C₂×Dic₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13095495	2
C₂×#15690769	2
#20260693	2
#20260690	2
M₁₆×Dic₉₇	2
C₂²×M₁₆×C₉₇	2
#2873	97