Subgroups — #27150908

Order 12416.

346 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261269	C₂	1	1	yes	maximal, characteristic
2	#20260863	C₂	1	2	yes	maximal, complemented
3	#20260810	C₂	1	2	yes	maximal, complemented
4	#20260809	C₂	1	2	yes	maximal, complemented
5	#20260808	C₂	1	2	yes	maximal, complemented
6	#20260795	C₂	1	2	yes	maximal, complemented
7	#20260760	C₂	1	2	yes	maximal, complemented
8	#20260759	C₂	1	1	yes	maximal, characteristic, complemented
9	#432×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13095495	C₂²	1	1	yes	characteristic
11	C₂×#13095495	C₂²	1	1	yes	characteristic
12	C₂×#13095494	C₂²	1	2	yes	—
13	C₂×C₈×D₉₇	C₂²	1	2	yes	—
14	#15690804	C₄	1	2	yes	—
15	#15690802	C₄	1	1	yes	characteristic
16	#15690798	C₄	1	1	yes	characteristic
17	#15690793	C₄	1	2	yes	—
18	#15690789	C₄	1	1	yes	characteristic
19	#15690784	C₄	1	1	yes	characteristic
20	C₄×Dic₁₉₄	C₂²	1	1	yes	characteristic
21	C₄×D₃₈₈	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₉₇	C₂²	1	2	yes	—
23	#15690767	C₂²	1	2	yes	—
24	C₄⋊#11715895	C₂²	1	2	yes	—
25	C₄⋊#11715895	C₂²	1	2	yes	—
26	#15690739	C₂²	1	2	yes	—
27	#15690739	C₂²	1	2	yes	—
28	#15690722	C₂²	1	2	yes	—
29	#15690722	C₂²	1	2	yes	—
30	#15690720	C₂²	1	2	yes	—
31	C₈×Dic₉₇	C₂²	1	2	yes	—
32	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
33	C₄⋊C₈×C₉₇	C₂²	1	1	yes	characteristic
34	C₄⋊C₈×C₉₇	C₂²	1	1	yes	characteristic
35	C₂²⋊C₈×C₉₇	C₂²	1	2	yes	—
36	#97×C₉₇	C₂²	1	2	yes	—
37	C₂×Dic₁₉₄	C₂×C₄	1	1	yes	characteristic
38	C₂×D₃₈₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₂²⋊D₉₇	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₉₇	C₂³	1	2	yes	—
41	#13095495	—	4	1	no	—
42	#13095495	—	4	1	no	—
43	#13095494	—	4	2	no	—
44	C₂×#11715895	C₂³	1	2	yes	—
45	C₂×#11715895	C₂³	1	2	yes	—
46	C₈×D₉₇	—	4	2	no	—
47	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
48	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
49	#13095490	C₂×C₄	1	2	yes	—
50	#13095490	C₂×C₄	1	2	yes	—
51	#13095490	C₂×C₄	1	2	yes	—
52	#13095490	C₂×C₄	1	2	yes	—
53	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
54	#13095488	C₂×C₄	1	2	yes	—
55	#13095488	C₂×C₄	1	2	yes	—
56	#13095488	C₂×C₄	1	2	yes	—
57	#13095488	C₂×C₄	1	2	yes	—
58	C₄×Dic₉₇	C₂³	1	2	yes	—
59	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₉₇	C₂³	1	2	yes	—
63	C₂×C₈×C₉₇	C₂³	1	2	yes	—
64	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
65	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
66	C₄²×C₉₇	C₂³	1	1	yes	characteristic
67	C₄²×C₉₇	C₂³	1	1	yes	characteristic
68	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×D₉₇	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₉₇	—	2	1	no	—
71	C₂×C₄×C₉₇	—	2	1	no	—
72	C₂×C₄×C₉₇	—	4	1	no	—
73	C₂×C₄×C₉₇	—	4	1	no	—
74	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
80	C₂²⋊D₉₇	—	4	2	no	—
81	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
82	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
83	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
84	D₃₈₈	—	4	1	no	—
85	C₄×D₉₇	—	2	2	no	—
86	C₄×D₉₇	—	2	2	no	—
87	Dic₁₉₄	—	4	1	no	—
88	C₈×C₉₇	—	2	2	no	—
89	C₈×C₉₇	—	2	2	no	—
90	#11715895	—	2	2	no	—
91	#11715895	—	2	2	no	—
92	C₂²×C₉₇	—	2	1	no	—
93	C₂²×C₉₇	—	2	1	no	—
94	C₂²×C₉₇	—	2	1	no	—
95	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
96	C₂×D₉₇	—	2	2	no	—
97	C₂×D₉₇	—	2	2	no	—
98	C₂×D₉₇	—	2	2	no	—
99	C₄×C₉₇	—	2	1	no	—
100	C₄×C₉₇	—	2	1	no	—
101	C₄×C₉₇	—	2	1	no	—
102	C₄×C₉₇	—	2	1	no	—
103	C₄×C₉₇	—	2	1	no	—
104	C₄×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
105	C₄×C₉₇	2₋¹⁺⁴	1	1	yes	characteristic
106	Dic₉₇	—	2	2	no	—
107	Dic₉₇	—	2	2	no	—
108	Dic₉₇	—	2	2	no	—
109	C₂×C₉₇	—	4	1	no	—
110	C₂×C₉₇	#520	1	1	yes	characteristic
111	C₂×C₉₇	#568	1	1	yes	characteristic
112	C₂×C₉₇	#568	1	1	yes	characteristic
113	D₉₇	—	4	2	no	—
114	#2995	—	97	1	no	maximal, Sylow
115	C₉₇	#2995	1	1	yes	characteristic, complemented, Sylow
116	#529	—	97	1	no	—
117	#446	—	97	2	no	—
118	#436	—	97	2	no	—
119	#435	—	97	2	no	—
120	#435	—	97	2	no	—
121	#433	—	97	2	no	—
122	#432	D₉₇	1	1	yes	characteristic, complemented
123	#424	—	97	2	no	—
124	#423	—	97	1	no	—
125	C₂×M₁₆	—	97	1	no	—
126	C₂×M₁₆	—	97	1	no	—
127	C₂×M₁₆	—	97	2	no	—
128	C₂²×C₈	—	97	2	no	—
129	#126	—	97	2	no	—
130	#125	—	97	1	no	—
131	#124	—	97	1	no	—
132	#123	—	97	2	no	—
133	C₂²⋊Q₈	—	97	1	no	—
134	#121	—	97	1	no	—
135	C₄×Q₈	—	97	1	no	—
136	C₄×D₄	—	97	2	no	—
137	C₄×D₄	—	97	1	no	—
138	#117	—	97	2	no	—
139	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
140	C₄⋊C₈	—	97	2	no	—
141	C₄⋊C₈	—	97	2	no	—
142	C₄⋊C₈	—	97	2	no	—
143	C₄⋊C₈	—	97	2	no	—
144	C₄⋊C₈	C₂×D₉₇	1	1	yes	characteristic
145	C₄⋊C₈	C₂×D₉₇	1	1	yes	characteristic
146	C₂²⋊C₈	—	97	2	no	—
147	C₂²⋊C₈	—	97	2	no	—
148	C₂²⋊C₈	C₂×D₉₇	1	2	yes	—
149	#97	—	97	2	no	—
150	#97	C₂×D₉₇	1	2	yes	—
151	C₄×C₈	—	97	2	no	—
152	C₂×Q₈	—	97	1	no	—
153	C₂×D₄	—	97	2	no	—
154	C₂×D₄	—	97	1	no	—
155	C₂²×C₄	—	97	2	no	—
156	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
157	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
159	M₁₆	—	388	2	no	—
160	M₁₆	—	388	1	no	—
161	M₁₆	—	388	1	no	—
162	C₂×C₈	—	97	2	no	—
163	C₂×C₈	—	97	2	no	—
164	C₂×C₈	—	388	2	no	—
165	C₂×C₈	C₂²×D₉₇	1	2	yes	—
166	C₂×C₈	C₂²×D₉₇	1	2	yes	—
167	C₄⋊C₄	—	97	2	no	—
168	C₄⋊C₄	—	97	2	no	—
169	C₄⋊C₄	—	97	2	no	—
170	C₄⋊C₄	—	97	2	no	—
171	C₄⋊C₄	—	97	1	no	—
172	C₄⋊C₄	—	97	1	no	—
173	C₂²⋊C₄	—	97	2	no	—
174	C₂²⋊C₄	—	97	2	no	—
175	C₂²⋊C₄	—	97	2	no	—
176	C₂²⋊C₄	—	97	2	no	—
177	C₂²⋊C₄	—	97	2	no	—
178	C₄²	—	97	2	no	—
179	C₄²	C₄×D₉₇	1	1	yes	characteristic
180	C₄²	C₄×D₉₇	1	1	yes	characteristic
181	C₄²	C₂²×D₉₇	1	1	yes	characteristic
182	C₄²	C₂²×D₉₇	1	1	yes	characteristic
183	C₂³	—	97	2	no	—
184	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
185	Q₈	—	388	1	no	—
186	D₄	—	388	1	no	—
187	D₄	—	388	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	97	2	no	—
193	C₂×C₄	—	97	2	no	—
194	C₂×C₄	—	97	2	no	—
195	C₂×C₄	—	194	2	no	—
196	C₂×C₄	—	194	2	no	—
197	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
199	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
200	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
202	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	2	2	no	—
205	C₈	—	194	2	no	—
206	C₈	—	194	2	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	194	2	no	—
211	C₂²	—	194	2	no	—
212	C₂²	—	194	2	no	—
213	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	194	2	no	—
220	C₄	—	194	2	no	—
221	C₄	—	194	2	no	—
222	C₄	#15690856	1	1	yes	characteristic
223	C₄	#15690857	1	1	yes	characteristic
224	C₂	—	4	1	no	—
225	C₂	—	388	2	no	—
226	C₂	#20261215	1	1	yes	characteristic
227	C₂	#20261501	1	1	yes	characteristic
228	C₂	#20261501	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	#432×C₉₇	C₂
prime radical	#15690784	C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#432	D₉₇
hypocenter	C₉₇	#2995
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. #20260863	C₂	semidirect
3. #20260810	C₂	semidirect
4. #20260809	C₂	semidirect
5. #20260808	C₂	semidirect
6. #20260795	C₂	semidirect
7. #20260760	C₂	semidirect
8. #20260759	C₂	semidirect
9. #432×C₉₇	C₂	semidirect
115. C₉₇	#2995	semidirect
122. #432	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
#20260863	2
#20260810	2
#20260809	2
#20260808	2
#20260795	2
#20260760	2
#20260759	2
#432×C₉₇	2
#2995	97