Subgroups — #27152306

Order 12416.

380 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261269	C₂	1	2	yes	maximal, complemented
2	#20261137	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261119	C₂	1	1	yes	maximal, characteristic, complemented
4	#20261090	C₂	1	2	yes	maximal, complemented
5	#20260976	C₂	1	1	yes	maximal, characteristic, complemented
6	#20260971	C₂	1	1	yes	maximal, characteristic
7	#20260964	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₃₈₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13095500	C₂	1	2	yes	maximal, complemented
10	C₄×D₇₇₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×C₈×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₃₈₈	C₂²	1	1	yes	characteristic
13	C₂×#13095500	C₂²	1	2	yes	—
14	C₂×D₇₇₆	C₂²	1	1	yes	characteristic
15	#15690804	—	2	2	no	—
16	#15690802	C₂²	1	2	yes	—
17	#15690798	C₂²	1	2	yes	—
18	#15690793	—	2	2	no	—
19	#15690789	C₂²	1	2	yes	—
20	#15690784	C₂²	1	2	yes	—
21	C₄×Dic₁₉₄	C₂²	1	2	yes	—
22	C₄×D₃₈₈	C₂²	1	2	yes	—
23	C₄×C₂²⋊D₉₇	—	2	2	no	—
24	#15690767	—	2	2	no	—
25	#15690744	C₂²	1	1	yes	characteristic
26	#15690744	C₂²	1	1	yes	characteristic
27	#15690742	C₂²	1	2	yes	—
28	#15690735	C₂²	1	2	yes	—
29	#15690735	C₂²	1	2	yes	—
30	#15690732	C₂²	1	2	yes	—
31	#15690732	C₂²	1	2	yes	—
32	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
33	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
35	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
36	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
37	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
39	C₂×Dic₁₉₄	C₂³	1	2	yes	—
40	C₂×D₃₈₈	C₂³	1	2	yes	—
41	C₂×C₂²⋊D₉₇	—	2	2	no	—
42	C₂×C₄×D₉₇	—	2	2	no	—
43	Dic₃₈₈	—	2	2	no	—
44	#13095500	—	2	4	no	—
45	D₇₇₆	—	2	2	no	—
46	C₄⋊Dic₉₇	C₂³	1	2	yes	—
47	C₄⋊Dic₉₇	C₂³	1	2	yes	—
48	#13095490	—	2	2	no	—
49	#13095490	—	2	2	no	—
50	#13095490	—	2	2	no	—
51	#13095490	—	2	2	no	—
52	C₂²⋊Dic₉₇	—	2	2	no	—
53	#13095488	—	2	2	no	—
54	#13095488	—	2	2	no	—
55	#13095488	—	2	2	no	—
56	#13095488	—	2	2	no	—
57	C₄×Dic₉₇	—	2	2	no	—
58	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
59	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₉₇	—	2	2	no	—
62	C₂×C₈×C₉₇	—	2	2	no	—
63	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
67	C₄²×C₉₇	D₄	1	1	yes	characteristic
68	C₄²×C₉₇	D₄	1	1	yes	characteristic
69	C₄²×C₉₇	C₂³	1	1	yes	characteristic
70	C₄²×C₉₇	C₂³	1	1	yes	characteristic
71	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×D₉₇	—	2	2	no	—
73	C₂×C₄×C₉₇	—	2	2	no	—
74	C₂×C₄×C₉₇	—	2	2	no	—
75	C₂×C₄×C₉₇	—	2	1	no	—
76	C₂×C₄×C₉₇	—	2	1	no	—
77	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
83	C₂²⋊D₉₇	—	4	4	no	—
84	C₂×Dic₉₇	—	2	2	no	—
85	C₂×Dic₉₇	—	2	2	no	—
86	C₂×Dic₉₇	—	2	2	no	—
87	D₃₈₈	—	2	2	no	—
88	D₃₈₈	—	2	2	no	—
89	C₄×D₉₇	—	4	4	no	—
90	Dic₁₉₄	—	2	2	no	—
91	Dic₁₉₄	—	2	2	no	—
92	C₈×C₉₇	—	2	1	no	—
93	C₈×C₉₇	—	2	1	no	—
94	C₈×C₉₇	#41	1	2	yes	—
95	C₈×C₉₇	#41	1	2	yes	—
96	C₂²×C₉₇	—	2	1	no	—
97	C₂²×C₉₇	—	2	1	no	—
98	C₂²×C₉₇	#135	1	2	yes	—
99	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
100	C₂×D₉₇	—	4	2	no	—
101	C₂×D₉₇	—	4	2	no	—
102	C₂×D₉₇	—	4	2	no	—
103	C₄×C₉₇	—	2	1	no	—
104	C₄×C₉₇	—	2	1	no	—
105	C₄×C₉₇	—	2	1	no	—
106	C₄×C₉₇	—	2	1	no	—
107	C₄×C₉₇	#135	1	2	yes	—
108	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
109	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
110	Dic₉₇	—	4	2	no	—
111	Dic₉₇	—	4	2	no	—
112	Dic₉₇	—	4	2	no	—
113	C₂×C₉₇	—	2	2	no	—
114	C₂×C₉₇	#532	1	1	yes	characteristic
115	C₂×C₉₇	C₂×#135	1	1	yes	characteristic
116	C₂×C₉₇	C₂×#135	1	1	yes	characteristic
117	D₉₇	—	8	2	no	—
118	#3168	—	97	1	no	maximal, Sylow
119	C₉₇	#3168	1	1	yes	characteristic, complemented, Sylow
120	#529	—	97	2	no	—
121	#499	—	97	1	no	—
122	#495	—	97	1	no	—
123	#488	—	97	2	no	—
124	#467	—	97	1	no	—
125	#466	—	97	1	no	—
126	#465	—	97	2	no	—
127	C₄×Q₁₆	—	97	1	no	—
128	C₄×SD₁₆	—	97	2	no	—
129	C₄×D₈	—	97	1	no	—
130	C₂×C₄×C₈	D₉₇	1	1	yes	characteristic, complemented
131	C₂×Q₁₆	—	97	1	no	—
132	C₂×SD₁₆	—	97	2	no	—
133	C₂×D₈	—	97	1	no	—
134	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
135	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
136	#126	—	194	2	no	—
137	#125	—	97	2	no	—
138	#124	—	97	2	no	—
139	#123	—	194	2	no	—
140	C₂²⋊Q₈	—	97	2	no	—
141	#121	—	97	2	no	—
142	C₄×Q₈	—	97	2	no	—
143	C₄×D₄	—	97	2	no	—
144	C₄×D₄	—	194	2	no	—
145	#117	—	194	2	no	—
146	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
147	#107	—	97	1	no	—
148	#107	—	97	1	no	—
149	#106	—	97	2	no	—
150	#103	—	97	2	no	—
151	#103	—	97	2	no	—
152	#102	—	97	2	no	—
153	#102	—	97	2	no	—
154	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
155	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
156	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
157	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
158	C₂×Q₈	—	97	2	no	—
159	C₂×D₄	—	97	2	no	—
160	C₂×D₄	—	194	2	no	—
161	C₂²×C₄	—	194	2	no	—
162	C₂²×C₄	D₃₈₈	1	1	yes	characteristic
163	C₂²×C₄	D₃₈₈	1	1	yes	characteristic
164	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
165	Q₁₆	—	194	2	no	—
166	SD₁₆	—	194	4	no	—
167	D₈	—	194	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
174	C₄⋊C₄	—	97	2	no	—
175	C₄⋊C₄	—	97	2	no	—
176	C₄⋊C₄	—	194	2	no	—
177	C₄⋊C₄	—	194	2	no	—
178	C₄⋊C₄	—	194	2	no	—
179	C₄⋊C₄	—	194	2	no	—
180	C₂²⋊C₄	—	194	2	no	—
181	C₂²⋊C₄	—	194	2	no	—
182	C₂²⋊C₄	—	194	2	no	—
183	C₂²⋊C₄	—	194	2	no	—
184	C₂²⋊C₄	—	194	2	no	—
185	C₄²	—	194	2	no	—
186	C₄²	D₃₈₈	1	1	yes	characteristic
187	C₄²	D₃₈₈	1	1	yes	characteristic
188	C₄²	C₂²×D₉₇	1	1	yes	characteristic
189	C₄²	C₂²×D₉₇	1	1	yes	characteristic
190	C₂³	—	194	2	no	—
191	C₂³	C₂×D₃₈₈	1	1	yes	characteristic
192	Q₈	—	194	2	no	—
193	Q₈	—	194	2	no	—
194	D₄	—	388	4	no	—
195	D₄	—	194	2	no	—
196	D₄	—	194	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	388	4	no	—
202	C₂×C₄	—	194	2	no	—
203	C₂×C₄	—	194	2	no	—
204	C₂×C₄	—	194	2	no	—
205	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
206	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
207	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
208	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
209	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
210	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
211	C₈	—	2	1	no	—
212	C₈	—	2	1	no	—
213	C₈	#13095511	1	2	yes	—
214	C₈	#13095511	1	2	yes	—
215	C₂²	—	2	1	no	—
216	C₂²	—	2	1	no	—
217	C₂²	—	388	2	no	—
218	C₂²	—	388	2	no	—
219	C₂²	—	388	2	no	—
220	C₂²	#15690832	1	2	yes	—
221	C₂²	C₂²×D₃₈₈	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	388	2	no	—
227	C₄	—	388	2	no	—
228	C₄	—	388	2	no	—
229	C₄	#15690832	1	2	yes	—
230	C₄	C₂×#13095511	1	1	yes	characteristic
231	C₄	C₂×#13095511	1	1	yes	characteristic
232	C₂	—	2	2	no	—
233	C₂	—	776	2	no	—
234	C₂	#20261285	1	1	yes	characteristic
235	C₂	C₂×#15690832	1	1	yes	characteristic
236	C₂	C₂×#15690832	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃₈₈
commutator, cosocle	C₂×C₄×C₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	C₂×C₄×C₈×C₉₇	C₂
socle	C₂²×C₉₇	C₂²×D₄
prime radical	#20260971	C₂
hypercenter	C₂×C₄×C₈	D₉₇
hypocenter	C₉₇	#3168
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. #20261269	C₂	semidirect
2. #20261137	C₂	semidirect
3. #20261119	C₂	semidirect
4. #20261090	C₂	semidirect
5. #20260976	C₂	semidirect
7. #20260964	C₂	semidirect
8. C₄×Dic₃₈₈	C₂	semidirect
9. C₄×#13095500	C₂	semidirect
10. C₄×D₇₇₆	C₂	semidirect
11. C₂×C₄×C₈×C₉₇	C₂	semidirect
119. C₉₇	#3168	semidirect
130. C₂×C₄×C₈	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261269	2
#20261137	2
#20261119	2
#20261090	2
#20260976	2
#20260971	2
#20260964	2
C₄×Dic₃₈₈	2
C₄×#13095500	2
C₄×D₇₇₆	2
C₂×C₄×C₈×C₉₇	2
#3168	97