Subgroups — #27150433

Order 12416.

520 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690817	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₈×D₉₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261242	C₂	1	1	yes	maximal, characteristic
4	#20260809	C₂	1	4	yes	maximal, complemented
5	#20260798	C₂	1	4	yes	maximal, complemented
6	#20260761	C₂	1	1	yes	maximal, characteristic, complemented
7	#20260698	C₂	1	1	yes	maximal, characteristic, complemented
8	#20260693	C₂	1	1	yes	maximal, characteristic, complemented
9	#408×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×D₉₇	C₂²	1	1	yes	characteristic
11	#15690817	—	2	4	no	—
12	C₂×#13095495	C₂²	1	1	yes	characteristic
13	C₂×#13095495	C₂²	1	2	yes	complemented
14	C₂²×#11715895	C₂²	1	1	yes	characteristic
15	C₂²×#11715895	C₂²	1	2	yes	complemented
16	C₂×C₈×D₉₇	—	2	4	no	—
17	C₂×C₈×D₉₇	C₂²	1	2	yes	—
18	C₂×C₈×D₉₇	C₂²	1	2	yes	—
19	#15690791	C₄	1	2	yes	—
20	#15690786	C₄	1	2	yes	—
21	#15690781	C₄	1	2	yes	—
22	#15690778	C₄	1	2	yes	—
23	C₄×C₂²⋊D₉₇	C₂²	1	4	yes	—
24	#15690769	C₂²	1	1	yes	characteristic
25	#15690739	C₂²	1	4	yes	complemented
26	#15690722	C₂²	1	4	yes	—
27	#15690720	C₂²	1	2	yes	complemented
28	C₈×Dic₉₇	C₂²	1	2	yes	complemented
29	C₂×#41×C₉₇	C₂²	1	1	yes	characteristic
30	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
31	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
32	C₂²⋊C₈×C₉₇	C₂²	1	4	yes	complemented
33	C₂³×D₉₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₂²⋊D₉₇	C₂×C₄	1	4	yes	—
35	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₉₇	—	2	2	no	—
37	C₂×C₄×D₉₇	—	2	2	no	—
38	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₉₇	C₂³	1	2	yes	—
40	#13095495	—	2	2	no	—
41	#13095495	—	2	4	no	—
42	C₂×#11715895	—	2	4	no	—
43	C₂×#11715895	D₄	1	4	yes	complemented
44	C₂×#11715895	C₂³	1	2	yes	—
45	C₂×#11715895	C₂³	1	1	yes	characteristic
46	C₂×#11715895	C₂³	1	1	yes	characteristic
47	C₈×D₉₇	—	2	4	no	—
48	C₈×D₉₇	—	2	4	no	—
49	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
50	#13095490	C₂×C₄	1	4	yes	—
51	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
52	C₂²⋊Dic₉₇	C₂×C₄	1	4	yes	—
53	#13095488	C₂×C₄	1	4	yes	—
54	C₄×Dic₉₇	C₂³	1	2	yes	—
55	#41×C₉₇	—	2	2	no	—
56	#41×C₉₇	—	2	2	no	—
57	C₂×Q₈×C₉₇	C₂×C₄	1	1	yes	characteristic
58	C₂×D₄×C₉₇	C₂×C₄	1	2	yes	—
59	C₂×D₄×C₉₇	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
61	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
62	M₁₆×C₉₇	—	2	2	no	—
63	C₂×C₈×C₉₇	—	2	2	no	—
64	C₂×C₈×C₉₇	C₂³	1	2	yes	—
65	C₂×C₈×C₉₇	C₂³	1	2	yes	—
66	C₂³×C₉₇	C₂²×C₄	1	2	yes	—
67	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×D₉₇	—	2	2	no	—
69	C₂²×D₉₇	—	2	4	no	—
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	8	no	—
82	C₂×Dic₉₇	—	2	2	no	—
83	C₂×Dic₉₇	C₂²×C₄	1	4	yes	—
84	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
85	C₄×D₉₇	—	2	4	no	—
86	C₄×D₉₇	—	2	2	no	—
87	C₄×D₉₇	—	2	2	no	—
88	C₈×C₉₇	—	2	2	no	—
89	C₈×C₉₇	—	2	2	no	—
90	#11715895	—	2	2	no	—
91	#11715895	C₂×D₄	1	2	yes	—
92	#11715895	C₂×D₄	1	2	yes	—
93	C₂²×C₉₇	—	2	2	no	—
94	C₂²×C₉₇	—	2	2	no	—
95	C₂²×C₉₇	—	2	4	no	—
96	C₂²×C₉₇	C₄×D₄	1	2	yes	—
97	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₉₇	—	2	8	no	—
99	C₂×D₉₇	—	2	2	no	—
100	C₂×D₉₇	—	2	2	no	—
101	C₂×D₉₇	#131	1	4	yes	complemented
102	C₄×C₉₇	—	2	2	no	—
103	C₄×C₉₇	C₄×D₄	1	2	yes	—
104	C₄×C₉₇	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
106	Dic₉₇	—	2	2	no	—
107	Dic₉₇	—	2	4	no	—
108	C₂×C₉₇	—	2	2	no	—
109	C₂×C₉₇	—	4	2	no	—
110	C₂×C₉₇	C₂×C₄×D₄	1	1	yes	characteristic
111	C₂×C₉₇	C₂×#131	1	2	yes	—
112	D₉₇	—	2	4	no	—
113	#2929	—	97	1	no	maximal, Sylow
114	C₉₇	#2929	1	1	yes	characteristic, complemented, Sylow
115	C₂×#131	—	97	1	no	—
116	C₂³×C₈	—	97	1	no	—
117	#525	—	97	1	no	—
118	#435	—	97	4	no	—
119	C₈×D₄	—	97	4	no	—
120	#424	—	97	1	no	—
121	#408	D₉₇	1	1	yes	characteristic, complemented
122	#407	—	97	1	no	—
123	#405	—	97	1	no	—
124	C₂×#41	C₂×D₉₇	1	1	yes	characteristic
125	C₂³×C₄	—	97	1	no	—
126	#131	—	194	4	no	—
127	C₂×M₁₆	—	97	1	no	—
128	C₂×M₁₆	—	97	2	no	—
129	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
130	C₂²×C₈	—	97	1	no	—
131	C₂²×C₈	—	97	2	no	—
132	C₂²×C₈	—	97	2	no	—
133	C₂²×C₈	—	97	2	no	—
134	C₂²×C₈	—	194	4	no	—
135	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
136	#123	—	97	2	no	—
137	C₂²⋊Q₈	—	97	2	no	—
138	#121	—	97	2	no	—
139	#120	—	97	2	no	—
140	C₄×D₄	—	97	4	no	—
141	#117	—	97	1	no	—
142	C₄⋊C₈	—	97	4	no	—
143	C₂²⋊C₈	—	97	4	no	—
144	C₂²⋊C₈	C₂×D₉₇	1	4	yes	complemented
145	#97	—	97	2	no	—
146	C₄×C₈	—	97	2	no	—
147	C₂⁴	—	97	1	no	—
148	#41	—	2	2	no	—
149	#41	—	2	2	no	—
150	C₂×Q₈	C₄×D₉₇	1	1	yes	characteristic
151	C₂×D₄	—	97	4	no	—
152	C₂×D₄	C₄×D₉₇	1	2	yes	—
153	C₂×D₄	C₄×D₉₇	1	1	yes	characteristic
154	C₂²×C₄	—	97	1	no	—
155	C₂²×C₄	—	97	2	no	—
156	C₂²×C₄	—	97	2	no	—
157	C₂²×C₄	—	194	2	no	—
158	C₂²×C₄	—	194	2	no	—
159	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
160	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	194	4	no	—
163	M₁₆	—	194	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	97	4	no	—
166	C₂×C₈	—	97	1	no	—
167	C₂×C₈	—	97	1	no	—
168	C₂×C₈	—	97	2	no	—
169	C₂×C₈	—	194	4	no	—
170	C₂×C₈	—	194	4	no	—
171	C₂×C₈	—	194	4	no	—
172	C₂×C₈	C₂²×D₉₇	1	2	yes	—
173	C₂×C₈	C₂²×D₉₇	1	2	yes	—
174	C₄⋊C₄	—	97	4	no	—
175	C₄⋊C₄	—	97	2	no	—
176	C₂²⋊C₄	—	97	2	no	—
177	C₂²⋊C₄	—	97	4	no	—
178	C₂²⋊C₄	—	97	4	no	—
179	C₄²	—	97	2	no	—
180	C₂³	—	97	2	no	—
181	C₂³	—	194	4	no	—
182	C₂³	—	194	2	no	—
183	C₂³	C₂×C₄×D₉₇	1	2	yes	—
184	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
185	Q₈	—	2	2	no	—
186	D₄	—	2	4	no	—
187	D₄	—	2	2	no	—
188	D₄	—	194	8	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	97	4	no	—
192	C₂×C₄	—	97	2	no	—
193	C₂×C₄	—	194	2	no	—
194	C₂×C₄	—	194	4	no	—
195	C₂×C₄	—	194	2	no	—
196	C₂×C₄	—	194	2	no	—
197	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
198	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
199	C₂×C₄	D₄×D₉₇	1	2	yes	—
200	C₂×C₄	#13095510	1	2	yes	—
201	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	2	2	no	—
204	C₈	—	97	2	no	—
205	C₈	—	97	2	no	—
206	C₈	—	194	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	97	4	no	—
211	C₂²	—	194	8	no	—
212	C₂²	—	194	2	no	—
213	C₂²	—	194	2	no	—
214	C₂²	#15690770	1	2	yes	—
215	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	194	2	no	—
218	C₄	—	194	4	no	—
219	C₄	#15690770	1	2	yes	—
220	C₄	C₂×D₄×D₉₇	1	1	yes	characteristic
221	C₄	C₂×#13095510	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	4	2	no	—
224	C₂	—	194	4	no	—
225	C₂	C₂×#15690770	1	1	yes	characteristic
226	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	#20261242	C₂
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#408	D₉₇
hypocenter	C₉₇	#2929
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×#15690817	C₂	semidirect
2. C₂²×C₈×D₉₇	C₂	semidirect
4. #20260809	C₂	semidirect
5. #20260798	C₂	semidirect
6. #20260761	C₂	semidirect
7. #20260698	C₂	semidirect
8. #20260693	C₂	semidirect
9. #408×C₉₇	C₂	semidirect
13. C₂×#13095495	C₂²	semidirect
15. C₂²×#11715895	C₂²	semidirect
25. #15690739	C₂²	semidirect
27. #15690720	C₂²	semidirect
28. C₈×Dic₉₇	C₂²	semidirect
32. C₂²⋊C₈×C₉₇	C₂²	semidirect
43. C₂×#11715895	D₄	semidirect
101. C₂×D₉₇	#131	semidirect
114. C₉₇	#2929	semidirect
121. #408	D₉₇	semidirect
144. C₂²⋊C₈	C₂×D₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690817	2
C₂²×C₈×D₉₇	2
#20261242	2
#20260809	2
#20260798	2
#20260761	2
#20260698	2
#20260693	2
#408×C₉₇	2
#2929	97