Subgroups — #27150836

Order 12416.

330 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	#20261289	C₂	1	1	yes	maximal, characteristic
2	#20260866	C₂	1	1	yes	maximal, characteristic, complemented
3	#20260865	C₂	1	1	yes	maximal, characteristic, complemented
4	#20260864	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₁₉₄	C₂	1	2	yes	maximal, complemented
6	#20260859	C₂	1	1	yes	maximal, characteristic, complemented
7	#20260795	C₂	1	2	yes	maximal
8	#20260794	C₂	1	2	yes	maximal
9	#20260787	C₂	1	1	yes	maximal, characteristic
10	#20260786	C₂	1	1	yes	maximal, characteristic
11	C₈×Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#15690809	C₄	1	1	yes	characteristic
13	#15690808	C₄	1	1	yes	characteristic
14	#15690800	C₄	1	2	yes	—
15	#15690790	C₄	1	1	yes	characteristic
16	#15690788	C₄	1	2	yes	—
17	#15690786	C₄	1	1	yes	characteristic
18	C₄×Dic₁₉₄	C₂²	1	2	yes	—
19	C₄×Dic₁₉₄	C₂²	1	1	yes	characteristic
20	#15690767	C₂²	1	2	yes	—
21	C₄²×D₉₇	C₂²	1	1	yes	characteristic
22	C₄⋊#11715895	C₂²	1	1	yes	characteristic
23	C₄⋊#11715895	C₂²	1	2	yes	—
24	#15690739	C₂²	1	1	yes	characteristic
25	#15690739	C₂²	1	2	yes	—
26	#15690739	C₂²	1	2	yes	—
27	#15690739	C₂²	1	1	yes	characteristic
28	#15690722	C₂²	1	1	yes	characteristic
29	#15690722	C₂²	1	1	yes	characteristic
30	#15690722	C₂²	1	2	yes	—
31	#15690721	C₂²	1	2	yes	—
32	#15690720	C₂²	1	2	yes	—
33	C₄×#11715895	C₂²	1	1	yes	characteristic
34	C₈×Dic₉₇	C₂²	1	2	yes	—
35	C₈×Dic₉₇	C₂²	1	2	yes	—
36	C₄×Q₈×C₉₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₉₇	C₂²	1	2	yes	—
38	C₄⋊C₈×C₉₇	C₂²	1	1	yes	characteristic
39	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₉₇	C₂²	1	2	yes	—
41	C₂×Dic₁₉₄	C₂×C₄	1	2	yes	—
42	C₂×Dic₁₉₄	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
46	C₂×#11715895	C₂³	1	1	yes	characteristic
47	C₂×#11715895	C₂³	1	2	yes	—
48	C₂×#11715895	C₂³	1	1	yes	characteristic
49	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
50	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
51	#13095490	C₂×C₄	1	2	yes	—
52	#13095490	C₂×C₄	1	2	yes	—
53	#13095490	C₂×C₄	1	2	yes	—
54	#13095490	C₂×C₄	1	2	yes	—
55	#13095490	C₂×C₄	1	2	yes	—
56	#13095488	C₂×C₄	1	2	yes	—
57	#13095488	C₂×C₄	1	2	yes	—
58	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
59	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
61	C₄×Dic₉₇	C₂³	1	2	yes	—
62	C₂×Q₈×C₉₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₉₇	C₂³	1	2	yes	—
65	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
67	C₄⋊C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
68	C₄²×C₉₇	C₂³	1	1	yes	characteristic
69	C₄²×C₉₇	C₂³	1	2	yes	—
70	C₂²×D₉₇	C₂²×C₄	1	1	yes	characteristic
71	Q₈×C₉₇	—	2	2	no	—
72	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
77	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
80	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
81	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
82	C₄×D₉₇	—	2	2	no	—
83	C₄×D₉₇	—	2	1	no	—
84	C₄×D₉₇	—	2	1	no	—
85	C₄×D₉₇	—	2	2	no	—
86	Dic₁₉₄	—	2	2	no	—
87	Dic₁₉₄	—	2	4	no	—
88	C₈×C₉₇	—	2	2	no	—
89	C₈×C₉₇	—	2	1	no	—
90	C₈×C₉₇	—	2	1	no	—
91	#11715895	—	2	1	no	—
92	#11715895	—	2	1	no	—
93	#11715895	—	2	2	no	—
94	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
95	C₂×D₉₇	—	2	1	no	—
96	C₂×D₉₇	—	2	1	no	—
97	C₂×D₉₇	—	2	1	no	—
98	C₄×C₉₇	—	2	2	no	—
99	C₄×C₉₇	—	2	1	no	—
100	C₄×C₉₇	—	2	2	no	—
101	C₄×C₉₇	#131	1	2	yes	—
102	C₄×C₉₇	2₋¹⁺⁴	1	1	yes	characteristic
103	C₄×C₉₇	2₋¹⁺⁴	1	1	yes	characteristic
104	Dic₉₇	—	2	1	no	—
105	Dic₉₇	—	2	2	no	—
106	Dic₉₇	—	2	2	no	—
107	Dic₉₇	—	2	1	no	—
108	Dic₉₇	#131	1	2	yes	—
109	C₂×C₉₇	#519	1	1	yes	characteristic
110	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
111	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
112	D₉₇	—	4	1	no	—
113	#2986	—	97	1	no	maximal, Sylow
114	C₉₇	#2986	1	1	yes	characteristic, complemented, Sylow
115	#533	—	97	1	no	—
116	#446	—	97	1	no	—
117	#446	—	97	2	no	—
118	#446	—	97	1	no	—
119	C₈×Q₈	—	97	1	no	—
120	C₈×Q₈	—	97	2	no	—
121	C₈×Q₈	D₉₇	1	1	yes	characteristic, complemented
122	#433	—	97	2	no	—
123	#433	—	97	2	no	—
124	#431	—	97	1	no	—
125	#431	—	97	1	no	—
126	C₄⋊Q₈	—	97	1	no	—
127	C₄⋊Q₈	—	97	1	no	—
128	#125	—	97	2	no	—
129	C₂²⋊Q₈	—	97	1	no	—
130	C₂²⋊Q₈	—	97	2	no	—
131	C₂²⋊Q₈	—	97	1	no	—
132	C₄×Q₈	—	97	2	no	—
133	C₄×Q₈	—	97	1	no	—
134	C₄×Q₈	C₂×D₉₇	1	1	yes	characteristic
135	#117	—	97	2	no	—
136	C₂×C₄²	—	97	1	no	—
137	C₄⋊C₈	—	97	1	no	—
138	C₄⋊C₈	—	97	1	no	—
139	C₄⋊C₈	—	97	2	no	—
140	C₄⋊C₈	—	97	2	no	—
141	C₄⋊C₈	—	97	1	no	—
142	C₄⋊C₈	—	97	2	no	—
143	C₄⋊C₈	C₂×D₉₇	1	1	yes	characteristic
144	C₄⋊C₈	C₂×D₉₇	1	2	yes	—
145	C₂²⋊C₈	—	97	1	no	—
146	C₂²⋊C₈	—	97	2	no	—
147	C₂²⋊C₈	—	97	1	no	—
148	#97	—	97	2	no	—
149	#97	—	97	2	no	—
150	C₄×C₈	—	97	2	no	—
151	C₄×C₈	—	97	2	no	—
152	C₄×C₈	—	97	1	no	—
153	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
154	C₄×C₈	C₂×D₉₇	1	2	yes	—
155	C₂×Q₈	—	97	2	no	—
156	C₂×Q₈	—	97	1	no	—
157	C₂×Q₈	C₄×D₉₇	1	1	yes	characteristic
158	C₂²×C₄	—	97	1	no	—
159	C₂²×C₄	—	97	1	no	—
160	C₂²×C₄	—	97	1	no	—
161	C₂×C₈	—	97	1	no	—
162	C₂×C₈	—	97	1	no	—
163	C₂×C₈	—	97	2	no	—
164	C₂×C₈	C₂²×D₉₇	1	2	yes	—
165	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
166	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
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	2	no	—
172	C₄⋊C₄	—	97	2	no	—
173	C₄⋊C₄	—	97	1	no	—
174	C₄⋊C₄	C₄×D₉₇	1	2	yes	—
175	C₄⋊C₄	C₄×D₉₇	1	1	yes	characteristic
176	C₂²⋊C₄	—	97	2	no	—
177	C₂²⋊C₄	—	97	2	no	—
178	C₄²	—	97	1	no	—
179	C₄²	—	97	1	no	—
180	C₄²	—	97	1	no	—
181	C₄²	—	97	2	no	—
182	C₄²	C₂²×D₉₇	1	1	yes	characteristic
183	C₄²	C₂²×D₉₇	1	2	yes	—
184	C₂³	—	97	1	no	—
185	Q₈	—	2	2	no	—
186	Q₈	—	194	4	no	—
187	Q₈	—	194	2	no	—
188	C₂×C₄	—	97	1	no	—
189	C₂×C₄	—	97	1	no	—
190	C₂×C₄	—	97	1	no	—
191	C₂×C₄	—	97	2	no	—
192	C₂×C₄	—	97	2	no	—
193	C₂×C₄	—	194	2	no	—
194	C₂×C₄	—	194	2	no	—
195	C₂×C₄	—	194	1	no	—
196	C₂×C₄	—	194	1	no	—
197	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
199	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
200	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
201	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	2	1	no	—
204	C₈	—	2	1	no	—
205	C₈	—	194	1	no	—
206	C₈	—	194	1	no	—
207	C₈	—	194	2	no	—
208	C₂²	—	194	1	no	—
209	C₂²	—	194	1	no	—
210	C₂²	—	194	1	no	—
211	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	97	2	no	—
216	C₄	—	194	1	no	—
217	C₄	—	194	2	no	—
218	C₄	—	194	1	no	—
219	C₄	—	194	2	no	—
220	C₄	#15690816	1	2	yes	—
221	C₄	#15690857	1	1	yes	characteristic
222	C₄	#15690858	1	1	yes	characteristic
223	C₂	—	388	1	no	—
224	C₂	#20261211	1	1	yes	characteristic
225	C₂	#131×D₉₇	1	1	yes	characteristic
226	C₂	C₂×#15690816	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	C₈×Q₈×C₉₇	C₂
prime radical	C₂²×D₉₇	C₂²×C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₈×Q₈	D₉₇
hypocenter	C₉₇	#2986
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. #20260866	C₂	semidirect
3. #20260865	C₂	semidirect
4. #20260864	C₂	semidirect
5. C₈×Dic₁₉₄	C₂	semidirect
6. #20260859	C₂	semidirect
11. C₈×Q₈×C₉₇	C₂	semidirect
114. C₉₇	#2986	semidirect
121. C₈×Q₈	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261289	2
#20260866	2
#20260865	2
#20260864	2
C₈×Dic₁₉₄	2
#20260859	2
#20260795	2
#20260794	2
#20260787	2
#20260786	2
C₈×Q₈×C₉₇	2
#2986	97