Subgroups — #27150377

Order 12416.

400 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	#20261203	C₂	1	1	yes	maximal, characteristic
2	#20260797	C₂	1	1	yes	maximal, characteristic, complemented
3	#20260796	C₂	1	1	yes	maximal, characteristic, complemented
4	#20260794	C₂	1	1	yes	maximal, characteristic, complemented
5	#20260787	C₂	1	2	yes	maximal, complemented
6	#20260785	C₂	1	2	yes	maximal, complemented
7	#20260760	C₂	1	2	yes	maximal, complemented
8	#20260700	C₂	1	2	yes	maximal
9	#20260687	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₈×D₉₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#433×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13095511	C₂²	1	1	yes	characteristic
13	C₂×#13095494	C₂²	1	2	yes	—
14	C₂×C₈×D₉₇	C₂²	1	2	yes	—
15	#15690775	C₄	1	2	yes	—
16	#15690772	C₄	1	2	yes	—
17	#15690770	C₄	1	2	yes	—
18	#15690770	C₄	1	2	yes	—
19	#15690767	C₂²	1	1	yes	characteristic
20	#15690767	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
22	C₄²×D₉₇	C₂²	1	1	yes	characteristic
23	C₄²×D₉₇	C₂²	1	1	yes	characteristic
24	C₄⋊#11715895	C₂²	1	2	yes	—
25	#15690739	C₂²	1	2	yes	—
26	#15690739	C₂²	1	2	yes	—
27	C₂²⋊#11715895	C₂²	1	2	yes	—
28	#15690722	C₂²	1	2	yes	—
29	#15690722	C₂²	1	2	yes	—
30	#15690721	C₂²	1	1	yes	characteristic
31	C₄×#11715895	C₂²	1	1	yes	characteristic
32	C₈×Dic₉₇	C₂²	1	2	yes	—
33	C₈×Dic₉₇	C₂²	1	2	yes	—
34	#117×C₉₇	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₉₇	C₂²	1	2	yes	—
36	C₂²⋊C₈×C₉₇	C₂²	1	2	yes	—
37	#97×C₉₇	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
39	#13095511	—	2	4	no	—
40	C₂×Dic₁₉₄	C₂×C₄	1	1	yes	characteristic
41	C₂×D₃₈₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊D₉₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₉₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
48	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
49	#13095494	—	2	4	no	—
50	C₂×#11715895	C₂³	1	2	yes	—
51	C₂×#11715895	C₂³	1	2	yes	—
52	C₈×D₉₇	—	2	4	no	—
53	#13095490	C₂×C₄	1	2	yes	—
54	#13095490	C₂×C₄	1	2	yes	—
55	#13095488	C₂×C₄	1	2	yes	—
56	#13095488	C₂×C₄	1	2	yes	—
57	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
58	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
59	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
60	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
61	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₉₇	C₂³	1	2	yes	—
64	C₂×C₈×C₉₇	C₂³	1	2	yes	—
65	C₄⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
66	C₂²⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
67	C₄²×C₉₇	C₂³	1	1	yes	characteristic
68	C₄²×C₉₇	C₂³	1	1	yes	characteristic
69	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×D₉₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×D₉₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₉₇	—	2	2	no	—
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	C₂²⋊D₉₇	—	2	2	no	—
78	C₂²⋊D₉₇	—	2	2	no	—
79	C₂×Dic₉₇	—	2	4	no	—
80	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
82	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
84	D₃₈₈	—	2	2	no	—
85	C₄×D₉₇	—	2	2	no	—
86	C₄×D₉₇	—	2	4	no	—
87	C₄×D₉₇	—	2	4	no	—
88	C₄×D₉₇	#41	1	4	yes	—
89	Dic₁₉₄	—	2	2	no	—
90	C₈×C₉₇	—	2	2	no	—
91	C₈×C₉₇	—	2	2	no	—
92	#11715895	—	2	2	no	—
93	#11715895	—	2	2	no	—
94	C₂²×C₉₇	—	2	1	no	—
95	C₂²×C₉₇	—	2	2	no	—
96	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
97	C₂×D₉₇	—	2	2	no	—
98	C₂×D₉₇	—	2	1	no	—
99	C₂×D₉₇	—	2	2	no	—
100	C₂×D₉₇	#117	1	2	yes	—
101	C₄×C₉₇	—	2	1	no	—
102	C₄×C₉₇	—	2	2	no	—
103	C₄×C₉₇	—	2	2	no	—
104	C₄×C₉₇	C₂×#41	1	2	yes	—
105	Dic₉₇	—	2	1	no	—
106	Dic₉₇	—	2	2	no	—
107	Dic₉₇	#117	1	2	yes	—
108	Dic₉₇	#131	1	4	yes	—
109	C₂×C₉₇	—	4	1	no	—
110	C₂×C₉₇	C₂×#117	1	1	yes	characteristic
111	C₂×C₉₇	C₂×#131	1	2	yes	—
112	D₉₇	—	2	2	no	—
113	D₉₇	—	4	1	no	—
114	#2922	—	97	1	no	maximal, Sylow
115	C₉₇	#2922	1	1	yes	characteristic, complemented, Sylow
116	C₄×#41	—	97	1	no	—
117	#433	—	97	1	no	—
118	#433	—	97	1	no	—
119	#433	—	97	1	no	—
120	#433	D₉₇	1	1	yes	characteristic, complemented
121	#431	—	97	2	no	—
122	#431	—	97	2	no	—
123	#424	—	97	2	no	—
124	#408	—	97	2	no	—
125	C₄×M₁₆	—	97	1	no	—
126	C₂×C₄×C₈	—	97	1	no	—
127	C₂×#41	—	97	1	no	—
128	C₂×M₁₆	—	97	2	no	—
129	C₂²×C₈	—	97	2	no	—
130	C₄×Q₈	—	97	2	no	—
131	C₄×D₄	—	97	2	no	—
132	C₄×D₄	—	97	2	no	—
133	C₄×D₄	—	97	2	no	—
134	#117	—	97	1	no	—
135	#117	—	97	1	no	—
136	#117	C₂×D₉₇	1	1	yes	characteristic
137	C₂×C₄²	—	97	1	no	—
138	C₂×C₄²	—	97	1	no	—
139	C₂×C₄²	—	97	1	no	—
140	C₄⋊C₈	—	97	2	no	—
141	C₄⋊C₈	—	97	2	no	—
142	C₄⋊C₈	—	97	2	no	—
143	C₄⋊C₈	C₂×D₉₇	1	2	yes	—
144	C₂²⋊C₈	—	97	2	no	—
145	C₂²⋊C₈	—	97	2	no	—
146	C₂²⋊C₈	—	97	2	no	—
147	C₂²⋊C₈	C₂×D₉₇	1	2	yes	—
148	#97	—	97	1	no	—
149	#97	C₂×D₉₇	1	1	yes	characteristic
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	#41	—	194	4	no	—
155	C₂×Q₈	—	97	1	no	—
156	C₂×D₄	—	97	1	no	—
157	C₂×D₄	—	97	1	no	—
158	C₂×D₄	—	97	1	no	—
159	C₂²×C₄	—	97	1	no	—
160	C₂²×C₄	—	97	2	no	—
161	C₂²×C₄	—	97	2	no	—
162	C₂²×C₄	—	97	2	no	—
163	C₂²×C₄	—	97	1	no	—
164	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
165	M₁₆	—	194	4	no	—
166	C₂×C₈	—	97	2	no	—
167	C₂×C₈	—	97	2	no	—
168	C₂×C₈	—	194	4	no	—
169	C₂×C₈	C₂²×D₉₇	1	2	yes	—
170	C₂×C₈	C₂²×D₉₇	1	2	yes	—
171	C₄⋊C₄	—	97	2	no	—
172	C₄⋊C₄	—	97	2	no	—
173	C₄⋊C₄	C₄×D₉₇	1	2	yes	—
174	C₂²⋊C₄	—	97	2	no	—
175	C₂²⋊C₄	—	97	2	no	—
176	C₂²⋊C₄	C₄×D₉₇	1	2	yes	—
177	C₄²	—	97	2	no	—
178	C₄²	—	97	2	no	—
179	C₄²	—	97	2	no	—
180	C₄²	—	97	1	no	—
181	C₄²	—	97	1	no	—
182	C₄²	C₂²×D₉₇	1	1	yes	characteristic
183	C₄²	C₂²×D₉₇	1	1	yes	characteristic
184	C₂³	—	97	1	no	—
185	C₂³	—	97	1	no	—
186	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
187	Q₈	—	194	2	no	—
188	D₄	—	194	2	no	—
189	D₄	—	194	2	no	—
190	D₄	—	194	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	97	2	no	—
193	C₂×C₄	—	97	1	no	—
194	C₂×C₄	—	97	4	no	—
195	C₂×C₄	—	97	2	no	—
196	C₂×C₄	—	97	1	no	—
197	C₂×C₄	—	194	4	no	—
198	C₂×C₄	—	194	4	no	—
199	C₂×C₄	—	194	4	no	—
200	C₂×C₄	—	194	2	no	—
201	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
202	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
204	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	—	2	2	no	—
207	C₈	—	194	2	no	—
208	C₈	—	194	2	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	97	2	no	—
212	C₂²	—	194	1	no	—
213	C₂²	—	194	2	no	—
214	C₂²	—	194	2	no	—
215	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	97	2	no	—
220	C₄	—	97	4	no	—
221	C₄	—	194	1	no	—
222	C₄	—	194	2	no	—
223	C₄	#41×D₉₇	1	2	yes	—
224	C₂	—	4	1	no	—
225	C₂	—	388	1	no	—
226	C₂	—	194	2	no	—
227	C₂	#117×D₉₇	1	1	yes	characteristic
228	C₂	#131×D₉₇	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	#433×C₉₇	C₂
prime radical	C₂×#13095511	C₂²
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#433	D₉₇
hypocenter	C₉₇	#2922
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
2. #20260797	C₂	semidirect
3. #20260796	C₂	semidirect
4. #20260794	C₂	semidirect
5. #20260787	C₂	semidirect
6. #20260785	C₂	semidirect
7. #20260760	C₂	semidirect
9. #20260687	C₂	semidirect
10. C₄×C₈×D₉₇	C₂	semidirect
11. #433×C₉₇	C₂	semidirect
115. C₉₇	#2922	semidirect
120. #433	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261203	2
#20260797	2
#20260796	2
#20260794	2
#20260787	2
#20260785	2
#20260760	2
#20260700	2
#20260687	2
C₄×C₈×D₉₇	2
#433×C₉₇	2
#2922	97