Subgroups — #27150432

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×#15690816	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₈×D₉₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261247	C₂	1	1	yes	maximal, characteristic
4	#20260808	C₂	1	4	yes	maximal, complemented
5	C₈×D₃₈₈	C₂	1	4	yes	maximal, complemented
6	#20260762	C₂	1	1	yes	maximal, characteristic, complemented
7	#20260703	C₂	1	1	yes	maximal, characteristic, complemented
8	#20260698	C₂	1	1	yes	maximal, characteristic, complemented
9	#405×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13095511	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₉₇	C₂²	1	1	yes	characteristic
12	#15690816	—	2	4	no	—
13	C₂×#13095495	C₂²	1	1	yes	characteristic
14	C₂×#13095494	C₂²	1	2	yes	complemented
15	C₂²×#11715895	C₂²	1	1	yes	characteristic
16	C₂×C₈×D₉₇	—	2	4	no	—
17	C₂×C₈×D₉₇	C₂²	1	2	yes	—
18	C₂×C₈×D₉₇	C₂²	1	2	yes	—
19	C₂×C₈×D₉₇	C₂²	1	2	yes	complemented
20	#15690794	C₄	1	2	yes	—
21	#15690790	C₄	1	2	yes	—
22	#15690785	C₄	1	2	yes	—
23	#15690780	C₄	1	2	yes	—
24	C₄×D₃₈₈	C₂²	1	4	yes	—
25	C₄⋊#11715895	C₂²	1	4	yes	complemented
26	#15690722	C₂²	1	4	yes	—
27	#15690722	C₂²	1	4	yes	complemented
28	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
29	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
30	#117×C₉₇	C₂²	1	1	yes	characteristic
31	#97×C₉₇	C₂²	1	2	yes	complemented
32	C₄×C₈×C₉₇	C₂²	1	2	yes	complemented
33	C₂³×D₉₇	C₂×C₄	1	1	yes	characteristic
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	4	yes	—
38	C₂×D₃₈₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₂²⋊D₉₇	C₂×C₄	1	2	yes	—
40	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₉₇	—	2	2	no	—
42	C₂×C₄×D₉₇	—	2	2	no	—
43	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₉₇	C₂³	1	2	yes	—
45	C₂×C₄×D₉₇	C₂³	1	2	yes	—
46	#13095495	—	2	2	no	—
47	#13095494	—	2	4	no	—
48	C₂×#11715895	—	2	2	no	—
49	C₂×#11715895	C₂³	1	2	yes	—
50	C₂×#11715895	C₂³	1	2	yes	—
51	C₈×D₉₇	—	2	4	no	—
52	C₈×D₉₇	—	2	4	no	—
53	C₈×D₉₇	—	2	4	no	—
54	C₄⋊Dic₉₇	C₂×C₄	1	4	yes	—
55	#13095488	C₂×C₄	1	4	yes	—
56	#13095488	C₂×C₄	1	4	yes	—
57	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
58	M₁₆×C₉₇	—	2	2	no	—
59	C₂×C₈×C₉₇	D₄	1	4	yes	complemented
60	C₂×C₈×C₉₇	C₂³	1	2	yes	—
61	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
64	C₂²⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
65	C₄²×C₉₇	C₂³	1	2	yes	—
66	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×D₉₇	—	2	2	no	—
68	C₂²×D₉₇	—	2	4	no	—
69	C₂²×D₉₇	C₂²×C₄	1	2	yes	—
70	C₂²×D₉₇	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₉₇	C₂²×C₄	1	4	yes	—
72	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₉₇	#41	1	2	yes	—
75	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
76	C₂²⋊D₉₇	—	2	4	no	—
77	C₂×Dic₉₇	—	2	2	no	—
78	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
79	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
80	D₃₈₈	—	2	8	no	—
81	D₃₈₈	—	2	2	no	—
82	C₄×D₉₇	—	2	4	no	—
83	C₄×D₉₇	—	2	2	no	—
84	C₄×D₉₇	—	2	2	no	—
85	C₄×D₉₇	—	2	2	no	—
86	C₄×D₉₇	—	2	2	no	—
87	Dic₁₉₄	—	2	2	no	—
88	C₈×C₉₇	—	2	2	no	—
89	C₈×C₉₇	C₂×D₄	1	2	yes	—
90	C₈×C₉₇	C₂×D₄	1	2	yes	—
91	#11715895	—	2	2	no	—
92	#11715895	—	2	2	no	—
93	C₂²×C₉₇	—	2	2	no	—
94	C₂²×C₉₇	C₄×D₄	1	2	yes	—
95	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
96	C₂×D₉₇	—	2	8	no	—
97	C₂×D₉₇	—	2	2	no	—
98	C₂×D₉₇	—	2	2	no	—
99	C₂×D₉₇	—	2	2	no	—
100	C₂×D₉₇	—	2	4	no	—
101	C₂×D₉₇	#131	1	4	yes	complemented
102	C₄×C₉₇	—	2	4	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	2	no	—
108	C₂×C₉₇	—	2	2	no	—
109	C₂×C₉₇	C₂×C₄×D₄	1	1	yes	characteristic
110	C₂×C₉₇	C₂×#131	1	2	yes	—
111	D₉₇	—	2	4	no	—
112	D₉₇	—	4	2	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	—	97	1	no	—
122	#407	—	97	1	no	—
123	#405	D₉₇	1	1	yes	characteristic, complemented
124	C₂×#41	—	97	1	no	—
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	C₂×D₉₇	1	1	yes	characteristic
142	C₄⋊C₈	—	97	4	no	—
143	C₂²⋊C₈	—	97	4	no	—
144	C₂²⋊C₈	—	97	4	no	—
145	#97	C₂×D₉₇	1	2	yes	complemented
146	C₄×C₈	C₂×D₉₇	1	2	yes	complemented
147	C₂⁴	—	97	1	no	—
148	#41	—	194	2	no	—
149	#41	—	194	2	no	—
150	C₂×Q₈	—	97	1	no	—
151	C₂×D₄	—	97	2	no	—
152	C₂×D₄	—	97	4	no	—
153	C₂×D₄	—	97	1	no	—
154	C₂²×C₄	—	97	1	no	—
155	C₂²×C₄	—	97	2	no	—
156	C₂²×C₄	—	97	2	no	—
157	C₂²×C₄	—	97	2	no	—
158	C₂²×C₄	—	194	2	no	—
159	C₂²×C₄	—	194	2	no	—
160	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	194	2	no	—
163	M₁₆	—	194	4	no	—
164	C₂×C₈	—	97	2	no	—
165	C₂×C₈	—	97	2	no	—
166	C₂×C₈	—	194	2	no	—
167	C₂×C₈	—	194	4	no	—
168	C₂×C₈	—	194	4	no	—
169	C₂×C₈	—	194	4	no	—
170	C₂×C₈	D₃₈₈	1	4	yes	complemented
171	C₂×C₈	C₂²×D₉₇	1	2	yes	—
172	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
174	C₄⋊C₄	—	97	4	no	—
175	C₄⋊C₄	C₄×D₉₇	1	2	yes	—
176	C₂²⋊C₄	—	97	4	no	—
177	C₂²⋊C₄	—	97	4	no	—
178	C₂²⋊C₄	C₄×D₉₇	1	2	yes	—
179	C₄²	C₂²×D₉₇	1	2	yes	—
180	C₂³	—	97	2	no	—
181	C₂³	—	97	2	no	—
182	C₂³	—	194	4	no	—
183	C₂³	—	194	2	no	—
184	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
185	Q₈	—	194	2	no	—
186	D₄	—	194	8	no	—
187	D₄	—	194	4	no	—
188	D₄	—	194	2	no	—
189	C₂×C₄	—	97	2	no	—
190	C₂×C₄	—	97	2	no	—
191	C₂×C₄	—	194	2	no	—
192	C₂×C₄	—	194	4	no	—
193	C₂×C₄	—	194	2	no	—
194	C₂×C₄	—	194	2	no	—
195	C₂×C₄	—	194	2	no	—
196	C₂×C₄	—	194	2	no	—
197	C₂×C₄	C₂×C₄×D₉₇	1	4	yes	—
198	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
199	C₂×C₄	C₂×D₃₈₈	1	2	yes	—
200	C₂×C₄	#13095511	1	2	yes	—
201	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	194	2	no	—
204	C₈	—	194	2	no	—
205	C₈	C₂×D₃₈₈	1	2	yes	—
206	C₈	C₂×D₃₈₈	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	—	97	4	no	—
209	C₂²	—	194	8	no	—
210	C₂²	—	194	2	no	—
211	C₂²	—	194	4	no	—
212	C₂²	—	194	2	no	—
213	C₂²	—	194	2	no	—
214	C₂²	C₄×D₃₈₈	1	2	yes	—
215	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
216	C₄	—	2	4	no	—
217	C₄	—	194	2	no	—
218	C₄	—	194	2	no	—
219	C₄	C₄×D₃₈₈	1	2	yes	—
220	C₄	C₂²×D₃₈₈	1	1	yes	characteristic
221	C₄	C₂×#13095511	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	388	2	no	—
224	C₂	—	194	4	no	—
225	C₂	C₂×C₄×D₃₈₈	1	1	yes	characteristic
226	C₂	#131×D₉₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	#405×C₉₇	C₂
prime radical	#20261247	C₂
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#405	D₉₇
hypocenter	C₉₇	#2929
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
1. C₂×#15690816	C₂	semidirect
2. C₂²×C₈×D₉₇	C₂	semidirect
4. #20260808	C₂	semidirect
5. C₈×D₃₈₈	C₂	semidirect
6. #20260762	C₂	semidirect
7. #20260703	C₂	semidirect
8. #20260698	C₂	semidirect
9. #405×C₉₇	C₂	semidirect
14. C₂×#13095494	C₂²	semidirect
19. C₂×C₈×D₉₇	C₂²	semidirect
25. C₄⋊#11715895	C₂²	semidirect
27. #15690722	C₂²	semidirect
31. #97×C₉₇	C₂²	semidirect
32. C₄×C₈×C₉₇	C₂²	semidirect
59. C₂×C₈×C₉₇	D₄	semidirect
101. C₂×D₉₇	#131	semidirect
114. C₉₇	#2929	semidirect
123. #405	D₉₇	semidirect
145. #97	C₂×D₉₇	semidirect
146. C₄×C₈	C₂×D₉₇	semidirect
170. C₂×C₈	D₃₈₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690816	2
C₂²×C₈×D₉₇	2
#20261247	2
#20260808	2
C₈×D₃₈₈	2
#20260762	2
#20260703	2
#20260698	2
#405×C₉₇	2
#2929	97