Subgroups — #27150444

Order 12416.

500 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690818	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15690817	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261281	C₂	1	1	yes	maximal, characteristic
4	#20260810	C₂	1	4	yes	maximal, complemented
5	#20260798	C₂	1	4	yes	maximal, complemented
6	#20260758	C₂	1	1	yes	maximal, characteristic, complemented
7	#20260701	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×Dic₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#408×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13095511	C₂²	1	1	yes	characteristic
11	#15690818	—	2	4	no	—
12	#15690817	—	2	4	no	—
13	C₂×#13095495	C₂²	1	1	yes	characteristic
14	C₂×#13095495	C₂²	1	2	yes	—
15	C₂×#13095494	C₂²	1	2	yes	—
16	C₂²×#11715895	C₂²	1	2	yes	—
17	C₂²×#11715895	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₉₇	C₂²	1	2	yes	—
19	#15690808	C₄	1	1	yes	characteristic
20	#15690806	C₄	1	1	yes	characteristic
21	#15690799	C₄	1	1	yes	characteristic
22	#15690797	C₄	1	1	yes	characteristic
23	#15690782	C₄	1	4	yes	—
24	C₄×C₂²⋊D₉₇	C₂²	1	4	yes	—
25	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
26	#15690739	C₂²	1	4	yes	—
27	#15690722	C₂²	1	4	yes	—
28	C₈×Dic₉₇	C₂²	1	2	yes	—
29	C₈×Dic₉₇	C₂²	1	2	yes	—
30	C₂×#41×C₉₇	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
32	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
33	C₂²⋊C₈×C₉₇	C₂²	1	4	yes	—
34	#13095511	—	2	2	no	—
35	#13095511	—	2	2	no	—
36	C₂×Dic₁₉₄	C₂×C₄	1	1	yes	characteristic
37	C₂×D₃₈₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₂²⋊D₉₇	C₂×C₄	1	4	yes	—
39	C₂×C₂²⋊D₉₇	C₂×C₄	1	2	yes	—
40	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
41	C₂×C₄×D₉₇	C₂³	1	2	yes	—
42	#13095495	—	2	2	no	—
43	#13095495	—	2	4	no	—
44	#13095494	—	2	4	no	—
45	C₂×#11715895	—	2	4	no	—
46	C₂×#11715895	D₄	1	4	yes	complemented
47	C₂×#11715895	C₂³	1	2	yes	—
48	C₂×#11715895	C₂³	1	1	yes	characteristic
49	C₂×#11715895	C₂³	1	1	yes	characteristic
50	C₈×D₉₇	—	2	4	no	—
51	#13095490	C₂×C₄	1	4	yes	—
52	C₂²⋊Dic₉₇	C₂×C₄	1	4	yes	—
53	#13095488	C₂×C₄	1	4	yes	—
54	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
55	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
56	C₄×Dic₉₇	C₂³	1	2	yes	—
57	#41×C₉₇	—	2	2	no	—
58	#41×C₉₇	—	2	2	no	—
59	C₂×Q₈×C₉₇	C₂×C₄	1	1	yes	characteristic
60	C₂×D₄×C₉₇	C₂×C₄	1	2	yes	—
61	C₂×D₄×C₉₇	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
63	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
64	M₁₆×C₉₇	—	2	2	no	—
65	C₂×C₈×C₉₇	—	2	2	no	—
66	C₂×C₈×C₉₇	C₂³	1	2	yes	—
67	C₂×C₈×C₉₇	C₂³	1	2	yes	—
68	C₂³×C₉₇	C₂²×C₄	1	2	yes	—
69	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×D₉₇	C₂²×C₄	1	2	yes	—
71	Q₈×C₉₇	—	2	2	no	—
72	D₄×C₉₇	—	2	4	no	—
73	D₄×C₉₇	—	2	2	no	—
74	C₂×C₄×C₉₇	—	2	2	no	—
75	C₂×C₄×C₉₇	—	2	2	no	—
76	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₉₇	#41	1	2	yes	—
80	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
81	C₂²⋊D₉₇	—	2	4	no	—
82	C₂²⋊D₉₇	—	2	8	no	—
83	C₂×Dic₉₇	—	2	4	no	—
84	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
85	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
86	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
87	D₃₈₈	—	2	2	no	—
88	C₄×D₉₇	—	2	2	no	—
89	C₄×D₉₇	—	2	2	no	—
90	Dic₁₉₄	—	2	2	no	—
91	C₈×C₉₇	—	2	2	no	—
92	C₈×C₉₇	—	2	2	no	—
93	#11715895	—	2	2	no	—
94	#11715895	C₂×D₄	1	2	yes	—
95	#11715895	C₂×D₄	1	2	yes	—
96	C₂²×C₉₇	—	2	2	no	—
97	C₂²×C₉₇	—	2	2	no	—
98	C₂²×C₉₇	—	2	4	no	—
99	C₂²×C₉₇	C₄×D₄	1	2	yes	—
100	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₉₇	—	2	2	no	—
102	C₂×D₉₇	—	2	4	no	—
103	C₄×C₉₇	—	2	2	no	—
104	C₄×C₉₇	C₄×D₄	1	2	yes	—
105	C₄×C₉₇	C₂²×D₄	1	1	yes	characteristic
106	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
107	Dic₉₇	—	2	2	no	—
108	Dic₉₇	—	2	2	no	—
109	Dic₉₇	#131	1	4	yes	complemented
110	C₂×C₉₇	—	2	2	no	—
111	C₂×C₉₇	—	4	2	no	—
112	C₂×C₉₇	C₂×C₄×D₄	1	1	yes	characteristic
113	C₂×C₉₇	C₂×#131	1	2	yes	—
114	D₉₇	—	4	2	no	—
115	#2931	—	97	1	no	maximal, Sylow
116	C₉₇	#2931	1	1	yes	characteristic, complemented, Sylow
117	C₂×#131	—	97	1	no	—
118	C₂×#131	—	97	1	no	—
119	#532	—	97	1	no	—
120	#436	—	97	4	no	—
121	C₈×D₄	—	97	4	no	—
122	#423	—	97	1	no	—
123	#408	—	97	1	no	—
124	#408	D₉₇	1	1	yes	characteristic, complemented
125	C₂×C₄×C₈	—	97	1	no	—
126	C₂×#41	—	97	1	no	—
127	C₂×#41	C₂×D₉₇	1	1	yes	characteristic
128	#131	—	194	4	no	—
129	#131	—	194	4	no	—
130	C₂×M₁₆	—	97	1	no	—
131	C₂×M₁₆	—	97	2	no	—
132	C₂×M₁₆	—	97	2	no	—
133	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
134	C₂²×C₈	—	97	1	no	—
135	C₂²×C₈	—	97	2	no	—
136	C₂²×C₈	—	97	2	no	—
137	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
138	C₄⋊Q₈	—	97	1	no	—
139	#127	—	97	1	no	—
140	#124	—	97	1	no	—
141	#124	—	97	1	no	—
142	#121	—	97	4	no	—
143	C₄×D₄	—	97	4	no	—
144	C₂×C₄²	—	97	1	no	—
145	C₄⋊C₈	—	97	4	no	—
146	C₂²⋊C₈	—	97	4	no	—
147	C₂²⋊C₈	C₂×D₉₇	1	4	yes	—
148	C₄×C₈	—	97	2	no	—
149	C₄×C₈	—	97	2	no	—
150	#41	—	2	2	no	—
151	#41	—	2	2	no	—
152	#41	—	194	2	no	—
153	#41	—	194	2	no	—
154	C₂×Q₈	—	97	1	no	—
155	C₂×Q₈	C₄×D₉₇	1	1	yes	characteristic
156	C₂×D₄	—	97	4	no	—
157	C₂×D₄	—	97	2	no	—
158	C₂×D₄	—	97	1	no	—
159	C₂×D₄	C₄×D₉₇	1	2	yes	—
160	C₂×D₄	C₄×D₉₇	1	1	yes	characteristic
161	C₂²×C₄	—	97	2	no	—
162	C₂²×C₄	—	97	2	no	—
163	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
164	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
165	M₁₆	—	2	2	no	—
166	M₁₆	—	194	4	no	—
167	M₁₆	—	194	4	no	—
168	M₁₆	—	194	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	97	4	no	—
171	C₂×C₈	—	97	1	no	—
172	C₂×C₈	—	97	1	no	—
173	C₂×C₈	—	97	2	no	—
174	C₂×C₈	—	194	4	no	—
175	C₂×C₈	—	194	4	no	—
176	C₂×C₈	C₂²×D₉₇	1	2	yes	—
177	C₂×C₈	C₂²×D₉₇	1	2	yes	—
178	C₄⋊C₄	—	97	4	no	—
179	C₂²⋊C₄	—	97	4	no	—
180	C₂²⋊C₄	—	97	4	no	—
181	C₄²	—	97	1	no	—
182	C₄²	—	97	1	no	—
183	C₄²	—	97	2	no	—
184	C₂³	—	97	2	no	—
185	C₂³	C₂×C₄×D₉₇	1	2	yes	—
186	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
187	Q₈	—	2	2	no	—
188	Q₈	—	194	2	no	—
189	D₄	—	2	4	no	—
190	D₄	—	2	2	no	—
191	D₄	—	194	8	no	—
192	D₄	—	194	4	no	—
193	D₄	—	194	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	97	2	no	—
197	C₂×C₄	—	97	2	no	—
198	C₂×C₄	—	97	2	no	—
199	C₂×C₄	—	194	4	no	—
200	C₂×C₄	—	194	2	no	—
201	C₂×C₄	—	194	2	no	—
202	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
204	C₂×C₄	D₄×D₉₇	1	2	yes	—
205	C₂×C₄	#13095510	1	2	yes	—
206	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	2	2	no	—
209	C₈	—	97	2	no	—
210	C₈	—	97	2	no	—
211	C₈	—	194	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	194	2	no	—
216	C₂²	—	194	4	no	—
217	C₂²	#15690770	1	2	yes	—
218	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
219	C₄	—	2	2	no	—
220	C₄	—	97	4	no	—
221	C₄	—	194	2	no	—
222	C₄	—	194	2	no	—
223	C₄	#15690770	1	2	yes	—
224	C₄	C₂×D₄×D₉₇	1	1	yes	characteristic
225	C₄	C₂×#13095510	1	1	yes	characteristic
226	C₂	—	2	2	no	—
227	C₂	—	4	2	no	—
228	C₂	—	388	2	no	—
229	C₂	C₂×#15690770	1	1	yes	characteristic
230	C₂	#131×D₉₇	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	#408×C₉₇	C₂
prime radical	#20261281	C₂
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#408	D₉₇
hypocenter	C₉₇	#2931
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
1. C₂×#15690818	C₂	semidirect
2. C₂×#15690817	C₂	semidirect
4. #20260810	C₂	semidirect
5. #20260798	C₂	semidirect
6. #20260758	C₂	semidirect
7. #20260701	C₂	semidirect
8. C₂×C₈×Dic₉₇	C₂	semidirect
9. #408×C₉₇	C₂	semidirect
46. C₂×#11715895	D₄	semidirect
109. Dic₉₇	#131	semidirect
116. C₉₇	#2931	semidirect
124. #408	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690818	2
C₂×#15690817	2
#20261281	2
#20260810	2
#20260798	2
#20260758	2
#20260701	2
C₂×C₈×Dic₉₇	2
#408×C₉₇	2
#2931	97