Subgroups — #38745659

Order 25216.

632 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×C₄×D₇₈₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#20263284	C₂	1	2	yes	maximal, complemented
3	#27160981	C₂	1	2	yes	maximal
4	#27160971	C₂	1	4	yes	maximal, complemented
5	#27160949	C₂	1	1	yes	maximal, characteristic, complemented
6	#27160936	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊Dic₁₉₇	C₂	1	2	yes	maximal, complemented
8	C₄×C₄⋊C₄×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₇₈₈	C₂²	1	1	yes	characteristic
10	C₂²×C₄×D₁₉₇	C₂²	1	2	yes	—
11	C₄×D₇₈₈	C₄	1	8	yes	complemented
12	#20263284	—	2	8	no	—
13	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
17	C₂×#15691335	C₂²	1	4	yes	—
18	C₂×#15691335	C₂²	1	4	yes	—
19	C₂×#15691335	C₂²	1	2	yes	—
20	C₂×C₄×Dic₁₉₇	C₂²	1	4	yes	—
21	#20263233	C₂²	1	4	yes	—
22	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₁₉₇	C₂²	1	2	yes	—
25	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
26	#95×C₁₉₇	C₂²	1	2	yes	—
27	C₂³×D₁₉₇	C₂³	1	2	yes	—
28	C₂×D₇₈₈	—	2	2	no	—
29	C₂×D₇₈₈	—	2	2	no	—
30	C₂×D₇₈₈	C₂×C₄	1	4	yes	—
31	C₂²×Dic₁₉₇	C₂³	1	4	yes	—
32	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
33	C₂×C₄×D₁₉₇	—	2	4	no	—
34	C₂×C₄×D₁₉₇	—	2	4	no	—
35	C₂×C₄×D₁₉₇	C₂×C₄	1	8	yes	—
36	C₄⋊Dic₁₉₇	—	2	4	no	—
37	C₄⋊Dic₁₉₇	—	2	4	no	—
38	C₄⋊Dic₁₉₇	—	4	2	no	—
39	C₄⋊Dic₁₉₇	C₂×C₄	1	4	yes	—
40	#15691335	—	2	8	no	—
41	#15691335	—	4	4	no	—
42	#15691335	C₂×C₄	1	8	yes	—
43	C₄×Dic₁₉₇	—	2	8	no	—
44	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
47	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
49	C₄⋊C₄×C₁₉₇	—	2	2	no	—
50	C₄⋊C₄×C₁₉₇	D₄	1	4	yes	complemented
51	C₄²×C₁₉₇	—	2	4	no	—
52	C₄²×C₁₉₇	C₂×C₄	1	4	yes	—
53	C₂²×D₁₉₇	—	2	2	no	—
54	C₂²×D₁₉₇	—	2	2	no	—
55	C₂²×D₁₉₇	—	2	2	no	—
56	C₂²×D₁₉₇	—	2	2	no	—
57	C₂²×D₁₉₇	—	2	2	no	—
58	C₂²×D₁₉₇	—	2	2	no	—
59	C₂²×D₁₉₇	C₂²×C₄	1	4	yes	—
60	D₇₈₈	—	2	8	no	—
61	C₂×Dic₁₉₇	—	2	2	no	—
62	C₂×Dic₁₉₇	—	2	2	no	—
63	C₂×Dic₁₉₇	—	2	4	no	—
64	C₂×Dic₁₉₇	—	2	4	no	—
65	C₂×Dic₁₉₇	—	2	4	no	—
66	C₂×Dic₁₉₇	C₂²×C₄	1	4	yes	—
67	C₄×D₁₉₇	—	2	16	no	—
68	C₂³×C₁₉₇	C₂⁴	1	1	yes	characteristic
69	C₂×C₄×C₁₉₇	—	2	2	no	—
70	C₂×C₄×C₁₉₇	—	2	1	no	—
71	C₂×C₄×C₁₉₇	—	2	1	no	—
72	C₂×C₄×C₁₉₇	—	2	2	no	—
73	C₂×C₄×C₁₉₇	—	2	2	no	—
74	C₂×C₄×C₁₉₇	—	2	2	no	—
75	C₂×C₄×C₁₉₇	—	2	2	no	—
76	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₁₉₇	C₂×D₄	1	4	yes	—
80	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₁₉₇	#41	1	2	yes	—
82	C₂×C₄×C₁₉₇	#41	1	2	yes	—
83	C₂×C₄×C₁₉₇	#41	1	2	yes	—
84	C₂²×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
85	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
86	C₂²×C₁₉₇	C₂×#41	1	1	yes	characteristic
87	C₂²×C₁₉₇	C₂×#41	1	1	yes	characteristic
88	C₂²×C₁₉₇	C₂×#41	1	1	yes	characteristic
89	C₂²×C₁₉₇	H(5,𝔽₂)	1	1	yes	characteristic
90	C₂²×C₁₉₇	2₋¹⁺⁴	1	1	yes	characteristic
91	C₂×D₁₉₇	—	2	4	no	—
92	C₂×D₁₉₇	—	2	4	no	—
93	C₂×D₁₉₇	—	2	4	no	—
94	C₂×D₁₉₇	—	4	2	no	—
95	C₂×D₁₉₇	—	4	2	no	—
96	C₂×D₁₉₇	—	4	2	no	—
97	C₂×D₁₉₇	—	4	2	no	—
98	C₄×C₁₉₇	—	2	2	no	—
99	C₄×C₁₉₇	—	2	4	no	—
100	C₄×C₁₉₇	—	4	2	no	—
101	C₄×C₁₉₇	#117	1	4	yes	complemented
102	C₄×C₁₉₇	C₄×D₄	1	4	yes	—
103	Dic₁₉₇	—	2	4	no	—
104	Dic₁₉₇	—	4	4	no	—
105	C₂×C₁₉₇	C₂×#117	1	1	yes	characteristic
106	C₂×C₁₉₇	C₂×C₄×D₄	1	1	yes	characteristic
107	C₂×C₁₉₇	#520	1	1	yes	characteristic
108	C₂×C₁₉₇	#547	1	1	yes	characteristic
109	C₂×C₁₉₇	#550	1	1	yes	characteristic
110	C₂×C₁₉₇	#555	1	1	yes	characteristic
111	C₂×C₁₉₇	#556	1	1	yes	characteristic
112	D₁₉₇	—	4	4	no	—
113	C₁₉₇	#2354	1	1	yes	characteristic, complemented, Sylow
114	#2354	—	197	1	no	maximal, Sylow
115	C₂×C₄×D₄	—	197	1	no	—
116	C₂×#117	—	197	2	no	—
117	#390	—	197	2	no	—
118	#388	—	197	4	no	—
119	#384	—	197	1	no	—
120	#380	—	197	2	no	—
121	C₄×C₄⋊C₄	—	197	2	no	—
122	C₄×C₄⋊C₄	D₁₉₇	1	1	yes	characteristic, complemented
123	C₂²×D₄	—	197	1	no	—
124	C₂³×C₄	—	197	2	no	—
125	C₄×D₄	—	197	8	no	—
126	#117	—	394	8	no	—
127	C₂×C₄⋊C₄	—	197	2	no	—
128	C₂×C₄⋊C₄	—	197	2	no	—
129	C₂×C₄⋊C₄	—	197	2	no	—
130	C₂×C₄⋊C₄	—	197	1	no	—
131	C₂×C₄⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
132	C₂×C₄⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
133	C₂×C₂²⋊C₄	—	197	4	no	—
134	C₂×C₂²⋊C₄	—	197	4	no	—
135	C₂×C₂²⋊C₄	—	197	2	no	—
136	C₂×C₄²	—	197	4	no	—
137	C₂×C₄²	C₂×D₁₉₇	1	2	yes	—
138	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
139	#95	—	197	4	no	—
140	#95	C₂×D₁₉₇	1	2	yes	—
141	C₂⁴	—	197	2	no	—
142	C₂×D₄	—	394	2	no	—
143	C₂×D₄	—	394	2	no	—
144	C₂×D₄	—	197	4	no	—
145	C₂²×C₄	—	394	4	no	—
146	C₂²×C₄	—	394	4	no	—
147	C₂²×C₄	—	197	4	no	—
148	C₂²×C₄	—	197	2	no	—
149	C₂²×C₄	—	197	8	no	—
150	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
151	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
152	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
153	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	788	2	no	—
157	C₄⋊C₄	—	394	4	no	—
158	C₄⋊C₄	—	394	4	no	—
159	C₄⋊C₄	—	197	4	no	—
160	C₄⋊C₄	D₇₈₈	1	4	yes	complemented
161	C₂²⋊C₄	—	788	4	no	—
162	C₂²⋊C₄	—	394	8	no	—
163	C₂²⋊C₄	—	197	8	no	—
164	C₄²	—	2	4	no	—
165	C₄²	—	394	8	no	—
166	C₄²	C₄×D₁₉₇	1	4	yes	—
167	C₂³	—	394	2	no	—
168	C₂³	—	394	2	no	—
169	C₂³	—	394	2	no	—
170	C₂³	—	394	2	no	—
171	C₂³	—	394	2	no	—
172	C₂³	—	394	2	no	—
173	C₂³	—	197	4	no	—
174	C₂³	C₂³×D₁₉₇	1	1	yes	characteristic
175	D₄	—	394	8	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	394	4	no	—
184	C₂×C₄	—	394	4	no	—
185	C₂×C₄	—	394	4	no	—
186	C₂×C₄	—	394	2	no	—
187	C₂×C₄	—	394	2	no	—
188	C₂×C₄	—	394	16	no	—
189	C₂×C₄	—	197	4	no	—
190	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
191	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
192	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
193	C₂×C₄	C₂×D₇₈₈	1	4	yes	—
194	C₂×C₄	C₂×D₇₈₈	1	2	yes	—
195	C₂×C₄	#15691357	1	2	yes	—
196	C₂×C₄	#15691358	1	2	yes	—
197	C₂×C₄	#15691359	1	2	yes	—
198	C₂²	—	788	2	no	—
199	C₂²	—	788	2	no	—
200	C₂²	—	788	2	no	—
201	C₂²	—	788	2	no	—
202	C₂²	—	394	4	no	—
203	C₂²	—	394	4	no	—
204	C₂²	—	394	4	no	—
205	C₂²	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
206	C₂²	C₂²×D₇₈₈	1	1	yes	characteristic
207	C₂²	C₂×#15691357	1	1	yes	characteristic
208	C₂²	C₂×#15691358	1	1	yes	characteristic
209	C₂²	C₂×#15691359	1	1	yes	characteristic
210	C₂²	#20263372	1	1	yes	characteristic
211	C₂²	#20263373	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	2	2	no	—
214	C₄	—	4	2	no	—
215	C₄	—	788	4	no	—
216	C₄	—	394	4	no	—
217	C₄	#20263284	1	4	yes	complemented
218	C₄	C₄×D₇₈₈	1	4	yes	—
219	C₂	—	788	4	no	—
220	C₂	C₂×#20263284	1	1	yes	characteristic
221	C₂	C₂×C₄×D₇₈₈	1	1	yes	characteristic
222	C₂	#27161559	1	1	yes	characteristic
223	C₂	#27161733	1	1	yes	characteristic
224	C₂	#27161751	1	1	yes	characteristic
225	C₂	#27161794	1	1	yes	characteristic
226	C₂	#27161802	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉₇
commutator	C₂²×C₁₉₇	C₂³×C₄
Fitting	C₄×C₄⋊C₄×C₁₉₇	C₂
socle, cosocle	C₂³×C₁₉₇	C₂⁴
prime radical	C₂²×D₇₈₈	C₂²
hypercenter	C₄×C₄⋊C₄	D₁₉₇
hypocenter	C₁₉₇	#2354
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#20263284	C₂	semidirect
4. #27160971	C₂	semidirect
5. #27160949	C₂	semidirect
6. #27160936	C₂	semidirect
7. C₄×C₄⋊Dic₁₉₇	C₂	semidirect
8. C₄×C₄⋊C₄×C₁₉₇	C₂	semidirect
11. C₄×D₇₈₈	C₄	semidirect
50. C₄⋊C₄×C₁₉₇	D₄	semidirect
101. C₄×C₁₉₇	#117	semidirect
113. C₁₉₇	#2354	semidirect
122. C₄×C₄⋊C₄	D₁₉₇	semidirect
160. C₄⋊C₄	D₇₈₈	semidirect
217. C₄	#20263284	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₇₈₈	2
C₂×#20263284	2
#27160981	2
#27160971	2
#27160949	2
#27160936	2
C₄×C₄⋊Dic₁₉₇	2
C₄×C₄⋊C₄×C₁₉₇	2
#2354	197