Subgroups — #27141429

Order 12416.

480 conjugacy classes of subgroups.

The subgroup lattice has 172 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	C₂²×C₄×Dic₉₇	C₂	1	1	yes	maximal, characteristic
3	#20260489	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₉₇	C₄	1	2	yes	—
6	C₂×#13095495	—	2	2	no	—
7	C₂×#13095495	—	2	2	no	—
8	C₂×#13095495	C₄	1	4	yes	complemented
9	C₂²×#11715895	C₂²	1	2	yes	—
10	C₂×C₄×Dic₉₇	—	2	2	no	—
11	C₂×C₄×Dic₉₇	—	2	2	no	—
12	C₂×C₄×Dic₉₇	C₄	1	2	yes	—
13	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
14	C₂³×C₄×C₉₇	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₉₇	—	2	2	no	—
16	C₂×M₁₆×C₉₇	—	2	2	no	—
17	C₂×M₁₆×C₉₇	C₄	1	4	yes	complemented
18	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
19	C₂²×Dic₉₇	—	2	4	no	—
20	C₂²×Dic₉₇	—	2	8	no	—
21	C₂²×Dic₉₇	C₂×C₄	1	4	yes	—
22	#13095495	—	2	8	no	—
23	C₂×#11715895	—	2	2	no	—
24	C₂×#11715895	—	2	2	no	—
25	C₂×#11715895	C₂×C₄	1	4	yes	complemented
26	C₄×Dic₉₇	—	2	4	no	—
27	C₄×Dic₉₇	—	2	2	no	—
28	C₄×Dic₉₇	—	2	2	no	—
29	C₂⁴×C₉₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₉₇	D₄	1	2	yes	—
34	C₂²×C₄×C₉₇	D₄	1	2	yes	—
35	C₂²×C₄×C₉₇	D₄	1	2	yes	—
36	C₂²×C₄×C₉₇	Q₈	1	2	yes	—
37	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
38	M₁₆×C₉₇	—	2	8	no	—
39	C₂×C₈×C₉₇	—	2	2	no	—
40	C₂×C₈×C₉₇	—	2	2	no	—
41	C₂×C₈×C₉₇	C₂×C₄	1	4	yes	complemented
42	C₂³×C₉₇	—	2	4	no	—
43	C₂³×C₉₇	C₄²	1	2	yes	—
44	C₂³×C₉₇	C₂²⋊C₄	1	2	yes	—
45	C₂³×C₉₇	C₄⋊C₄	1	2	yes	—
46	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₉₇	C₄²	1	2	yes	—
48	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₉₇	C₂²⋊C₄	1	4	yes	—
50	C₂×C₄×C₉₇	C₂²⋊C₄	1	4	yes	—
51	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
52	C₂×C₄×C₉₇	C₄⋊C₄	1	4	yes	—
53	C₂×C₄×C₉₇	C₄⋊C₄	1	4	yes	—
54	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
60	C₂×Dic₉₇	—	2	16	no	—
61	C₂×Dic₉₇	—	2	4	no	—
62	C₂×Dic₉₇	—	2	4	no	—
63	C₂×Dic₉₇	M₁₆	1	8	yes	complemented
64	C₈×C₉₇	—	2	4	no	—
65	#11715895	—	2	4	no	—
66	C₂²×C₉₇	—	2	4	no	—
67	C₂²×C₉₇	—	2	4	no	—
68	C₂²×C₉₇	—	2	4	no	—
69	C₂²×C₉₇	#95	1	4	yes	—
70	C₂²×C₉₇	C₂×C₄²	1	1	yes	characteristic
71	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₉₇	C₂×M₁₆	1	4	yes	—
74	C₄×C₉₇	#95	1	4	yes	—
75	C₄×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₄×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₄×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₄×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	Dic₉₇	—	2	8	no	—
80	C₂×C₉₇	—	2	4	no	—
81	C₂×C₉₇	C₂×#95	1	1	yes	characteristic
82	C₂×C₉₇	C₄×M₁₆	1	2	yes	—
83	C₂×C₉₇	#407	1	2	yes	—
84	C₂×C₉₇	#423	1	2	yes	—
85	#1728	—	97	1	no	maximal, Sylow
86	C₉₇	#1728	1	1	yes	characteristic, complemented, Sylow
87	C₂²×M₁₆	—	97	1	no	—
88	C₂²×M₁₆	D₉₇	1	1	yes	characteristic
89	C₂²×C₄²	—	97	1	no	—
90	#336	—	97	4	no	—
91	C₂³×C₄	—	97	2	no	—
92	C₂³×C₄	C₂×D₉₇	1	1	yes	characteristic
93	C₂×M₁₆	—	2	2	no	—
94	C₂×M₁₆	—	2	2	no	—
95	C₂×M₁₆	—	97	4	no	—
96	C₂×M₁₆	—	194	2	no	—
97	C₂×M₁₆	—	194	2	no	—
98	C₂×M₁₆	Dic₉₇	1	4	yes	complemented
99	C₂²×C₈	—	97	2	no	—
100	C₂²×C₈	C₂×D₉₇	1	2	yes	—
101	C₂×C₄²	—	97	2	no	—
102	C₂×C₄²	—	97	2	no	—
103	C₂×C₄²	—	194	2	no	—
104	C₂×C₄²	—	194	2	no	—
105	C₂⁴	C₄×D₉₇	1	1	yes	characteristic
106	C₂²×C₄	—	97	4	no	—
107	C₂²×C₄	—	194	8	no	—
108	C₂²×C₄	—	194	4	no	—
109	C₂²×C₄	Dic₁₉₄	1	2	yes	—
110	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
111	C₂²×C₄	C₄×D₉₇	1	2	yes	—
112	C₂²×C₄	D₃₈₈	1	2	yes	—
113	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
114	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
115	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
116	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
117	M₁₆	—	2	8	no	—
118	M₁₆	—	194	8	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	97	4	no	—
122	C₂×C₈	—	194	2	no	—
123	C₂×C₈	—	194	2	no	—
124	C₂×C₈	C₂×Dic₉₇	1	4	yes	complemented
125	C₄²	—	194	4	no	—
126	C₄²	—	194	2	no	—
127	C₄²	—	194	2	no	—
128	C₂³	—	2	4	no	—
129	C₂³	C₄×Dic₉₇	1	2	yes	—
130	C₂³	#13095488	1	2	yes	—
131	C₂³	#13095490	1	2	yes	—
132	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
133	C₂×C₄	—	97	8	no	—
134	C₂×C₄	—	194	16	no	—
135	C₂×C₄	—	194	4	no	—
136	C₂×C₄	—	194	4	no	—
137	C₂×C₄	C₄×Dic₉₇	1	2	yes	—
138	C₂×C₄	#13095488	1	4	yes	—
139	C₂×C₄	#13095488	1	2	yes	—
140	C₂×C₄	C₂²⋊Dic₉₇	1	4	yes	—
141	C₂×C₄	#13095490	1	4	yes	—
142	C₂×C₄	#13095490	1	2	yes	—
143	C₂×C₄	C₄⋊Dic₉₇	1	4	yes	—
144	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
145	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
147	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
148	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
149	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
150	C₈	—	2	4	no	—
151	C₈	—	194	4	no	—
152	C₂²	—	2	4	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	4	no	—
155	C₂²	#15690717	1	4	yes	—
156	C₂²	C₂×C₄×Dic₉₇	1	1	yes	characteristic
157	C₂²	C₂×#13095488	1	1	yes	characteristic
158	C₂²	C₂×#13095490	1	1	yes	characteristic
159	C₂²	M₁₆×D₉₇	1	4	yes	—
160	C₄	—	194	8	no	—
161	C₄	#15690717	1	4	yes	—
162	C₄	C₂×#13095488	1	1	yes	characteristic
163	C₄	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
164	C₄	C₂×#13095490	1	1	yes	characteristic
165	C₄	C₂×C₄⋊Dic₉₇	1	1	yes	characteristic
166	C₂	—	2	4	no	—
167	C₂	C₂×#15690717	1	1	yes	characteristic
168	C₂	M₁₆×Dic₉₇	1	2	yes	—
169	C₂	#20260698	1	2	yes	—
170	C₂	#20260758	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₉₇
commutator	C₂²×C₉₇	C₂×C₄²
Fitting	C₂²×M₁₆×C₉₇	C₂
socle	C₂³×C₉₇	C₂²×C₄
prime radical	C₂⁴×C₉₇	C₂×C₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂²×M₁₆	D₉₇
hypocenter	C₉₇	#1728
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20260489	C₂	semidirect
8. C₂×#13095495	C₄	semidirect
17. C₂×M₁₆×C₉₇	C₄	semidirect
25. C₂×#11715895	C₂×C₄	semidirect
41. C₂×C₈×C₉₇	C₂×C₄	semidirect
63. C₂×Dic₉₇	M₁₆	semidirect
86. C₉₇	#1728	semidirect
98. C₂×M₁₆	Dic₉₇	semidirect
124. 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
C₂²×C₄×Dic₉₇	2
#20260489	2
C₂²×M₁₆×C₉₇	2
#1728	97