Subgroups — #38745661

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×#20263285	C₂	1	2	yes	maximal, complemented
2	#27160982	C₂	1	2	yes	maximal
3	#27160972	C₂	1	4	yes	maximal, complemented
4	#27160949	C₂	1	1	yes	maximal, characteristic, complemented
5	#27160939	C₂	1	2	yes	maximal, complemented
6	C₄⋊C₄×Dic₁₉₇	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊Dic₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₄×C₁₉₇	C₂	1	1	yes	maximal, characteristic
9	#20263285	—	2	8	no	—
10	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Dic₁₉₇	C₂²	1	4	yes	—
16	C₂×C₂²⋊Dic₁₉₇	C₂²	1	4	yes	—
17	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
18	C₂×C₄×Dic₁₉₇	C₂²	1	4	yes	—
19	#20263233	C₂²	1	2	yes	—
20	#20263233	C₂²	1	4	yes	—
21	C₂²×D₄×C₁₉₇	C₂²	1	1	yes	characteristic
22	C₂³×C₄×C₁₉₇	C₂²	1	2	yes	—
23	C₄×D₄×C₁₉₇	C₄	1	8	yes	complemented
24	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₁₉₇	C₂²	1	2	yes	—
26	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
27	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
28	C₂²×Dic₁₉₇	C₂³	1	4	yes	—
29	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
30	C₄⋊Dic₁₉₇	—	2	4	no	—
31	C₄⋊Dic₁₉₇	—	2	2	no	—
32	C₄⋊Dic₁₉₇	—	2	4	no	—
33	C₄⋊Dic₁₉₇	—	4	2	no	—
34	C₄⋊Dic₁₉₇	D₄	1	4	yes	complemented
35	C₂²⋊Dic₁₉₇	—	2	8	no	—
36	C₂²⋊Dic₁₉₇	—	4	4	no	—
37	C₄×Dic₁₉₇	—	2	8	no	—
38	C₄×Dic₁₉₇	—	2	4	no	—
39	C₂⁴×C₁₉₇	C₂³	1	2	yes	—
40	C₂×D₄×C₁₉₇	—	2	2	no	—
41	C₂×D₄×C₁₉₇	—	2	2	no	—
42	C₂×D₄×C₁₉₇	C₂×C₄	1	4	yes	—
43	C₂²×C₄×C₁₉₇	—	2	4	no	—
44	C₂²×C₄×C₁₉₇	—	2	4	no	—
45	C₂²×C₄×C₁₉₇	C₂×C₄	1	8	yes	—
46	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
48	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₁₉₇	C₂×C₄	1	4	yes	—
51	C₂²⋊C₄×C₁₉₇	C₂×C₄	1	8	yes	—
52	C₄²×C₁₉₇	C₂×C₄	1	4	yes	—
53	C₂×Dic₁₉₇	—	2	2	no	—
54	C₂×Dic₁₉₇	—	2	2	no	—
55	C₂×Dic₁₉₇	—	2	4	no	—
56	C₂×Dic₁₉₇	—	2	2	no	—
57	C₂×Dic₁₉₇	—	2	2	no	—
58	C₂×Dic₁₉₇	—	2	4	no	—
59	C₂×Dic₁₉₇	—	2	2	no	—
60	C₂×Dic₁₉₇	—	2	4	no	—
61	C₂×Dic₁₉₇	C₂×D₄	1	4	yes	—
62	C₂³×C₁₉₇	—	2	2	no	—
63	C₂³×C₁₉₇	—	2	2	no	—
64	C₂³×C₁₉₇	—	2	2	no	—
65	C₂³×C₁₉₇	—	2	2	no	—
66	C₂³×C₁₉₇	—	2	2	no	—
67	C₂³×C₁₉₇	—	2	2	no	—
68	C₂³×C₁₉₇	C₂²×C₄	1	4	yes	—
69	C₂³×C₁₉₇	C₂⁴	1	1	yes	characteristic
70	D₄×C₁₉₇	—	2	8	no	—
71	C₂×C₄×C₁₉₇	—	2	2	no	—
72	C₂×C₄×C₁₉₇	—	2	2	no	—
73	C₂×C₄×C₁₉₇	—	2	1	no	—
74	C₂×C₄×C₁₉₇	—	2	1	no	—
75	C₂×C₄×C₁₉₇	—	2	16	no	—
76	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₁₉₇	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	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	—	2	4	no	—
85	C₂²×C₁₉₇	—	2	4	no	—
86	C₂²×C₁₉₇	—	2	4	no	—
87	C₂²×C₁₉₇	—	4	2	no	—
88	C₂²×C₁₉₇	—	4	2	no	—
89	C₂²×C₁₉₇	—	4	2	no	—
90	C₂²×C₁₉₇	—	4	2	no	—
91	C₂²×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
92	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
93	C₂²×C₁₉₇	C₂×#41	1	1	yes	characteristic
94	C₂²×C₁₉₇	C₂×#41	1	1	yes	characteristic
95	C₂²×C₁₉₇	C₂×#41	1	1	yes	characteristic
96	C₂²×C₁₉₇	H(5,𝔽₂)	1	1	yes	characteristic
97	C₂²×C₁₉₇	2₋¹⁺⁴	1	1	yes	characteristic
98	C₄×C₁₉₇	—	2	4	no	—
99	C₄×C₁₉₇	—	2	2	no	—
100	C₄×C₁₉₇	#117	1	4	yes	complemented
101	C₄×C₁₉₇	C₄×D₄	1	4	yes	—
102	Dic₁₉₇	—	2	4	no	—
103	Dic₁₉₇	—	4	4	no	—
104	Dic₁₉₇	—	4	2	no	—
105	C₂×C₁₉₇	—	4	4	no	—
106	C₂×C₁₉₇	C₂×#117	1	1	yes	characteristic
107	C₂×C₁₉₇	C₂×C₄×D₄	1	1	yes	characteristic
108	C₂×C₁₉₇	#520	1	1	yes	characteristic
109	C₂×C₁₉₇	#547	1	1	yes	characteristic
110	C₂×C₁₉₇	#550	1	1	yes	characteristic
111	C₂×C₁₉₇	#555	1	1	yes	characteristic
112	C₂×C₁₉₇	#556	1	1	yes	characteristic
113	C₁₉₇	#2354	1	1	yes	characteristic, complemented, Sylow
114	#2354	—	197	1	no	maximal, Sylow
115	C₂×C₄×D₄	D₁₉₇	1	1	yes	characteristic
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	—	197	1	no	—
123	C₂²×D₄	C₂×D₁₉₇	1	1	yes	characteristic
124	C₂³×C₄	C₂×D₁₉₇	1	2	yes	—
125	C₄×D₄	Dic₁₉₇	1	8	yes	complemented
126	#117	—	394	8	no	—
127	C₂×C₄⋊C₄	—	197	2	no	—
128	C₂×C₄⋊C₄	—	197	2	no	—
129	C₂×C₄⋊C₄	—	197	1	no	—
130	C₂×C₄⋊C₄	—	197	2	no	—
131	C₂×C₄⋊C₄	—	197	1	no	—
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	C₂×D₁₉₇	1	2	yes	—
136	C₂×C₄²	—	197	4	no	—
137	C₂×C₄²	—	197	2	no	—
138	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
139	#95	—	197	4	no	—
140	#95	—	197	2	no	—
141	C₂⁴	C₂²×D₁₉₇	1	2	yes	—
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	—	2	2	no	—
144	C₂×D₄	C₂×Dic₁₉₇	1	4	yes	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	—	2	4	no	—
147	C₂²×C₄	—	197	2	no	—
148	C₂²×C₄	—	197	4	no	—
149	C₂²×C₄	—	197	2	no	—
150	C₂²×C₄	C₂×Dic₁₉₇	1	8	yes	—
151	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
152	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
153	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
155	C₄⋊C₄	—	788	2	no	—
156	C₄⋊C₄	—	394	4	no	—
157	C₄⋊C₄	—	394	2	no	—
158	C₄⋊C₄	—	394	4	no	—
159	C₄⋊C₄	—	197	4	no	—
160	C₄⋊C₄	C₂×Dic₁₉₇	1	4	yes	—
161	C₂²⋊C₄	—	788	4	no	—
162	C₂²⋊C₄	—	394	8	no	—
163	C₂²⋊C₄	C₂×Dic₁₉₇	1	8	yes	—
164	C₄²	—	394	8	no	—
165	C₄²	—	394	4	no	—
166	C₄²	C₂×Dic₁₉₇	1	4	yes	—
167	C₂³	—	2	2	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	2	2	no	—
170	C₂³	—	2	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	C₂²×Dic₁₉₇	1	4	yes	—
174	C₂³	C₂³×D₁₉₇	1	1	yes	characteristic
175	D₄	—	2	8	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	16	no	—
181	C₂×C₄	—	394	4	no	—
182	C₂×C₄	—	394	2	no	—
183	C₂×C₄	—	394	2	no	—
184	C₂×C₄	—	394	4	no	—
185	C₂×C₄	—	394	2	no	—
186	C₂×C₄	—	394	4	no	—
187	C₂×C₄	—	394	2	no	—
188	C₂×C₄	—	394	2	no	—
189	C₂×C₄	—	197	4	no	—
190	C₂×C₄	C₂²×Dic₁₉₇	1	2	yes	—
191	C₂×C₄	C₂²×Dic₁₉₇	1	4	yes	—
192	C₂×C₄	C₂²×Dic₁₉₇	1	2	yes	—
193	C₂×C₄	C₂²×Dic₁₉₇	1	2	yes	—
194	C₂×C₄	D₄×D₁₉₇	1	2	yes	—
195	C₂×C₄	#15691357	1	2	yes	—
196	C₂×C₄	#15691358	1	2	yes	—
197	C₂×C₄	#15691358	1	2	yes	—
198	C₂²	—	2	4	no	—
199	C₂²	—	2	4	no	—
200	C₂²	—	2	4	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	—	4	2	no	—
204	C₂²	—	4	2	no	—
205	C₂²	C₂³×Dic₁₉₇	1	1	yes	characteristic
206	C₂²	C₂×D₄×D₁₉₇	1	1	yes	characteristic
207	C₂²	C₂×#15691357	1	1	yes	characteristic
208	C₂²	C₂×#15691358	1	1	yes	characteristic
209	C₂²	C₂×#15691358	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	—	788	4	no	—
215	C₄	—	788	2	no	—
216	C₄	—	394	4	no	—
217	C₄	#20263285	1	4	yes	complemented
218	C₄	D₄×Dic₁₉₇	1	4	yes	—
219	C₂	—	4	4	no	—
220	C₂	C₂×#20263285	1	1	yes	characteristic
221	C₂	C₂×D₄×Dic₁₉₇	1	1	yes	characteristic
222	C₂	#27161562	1	1	yes	characteristic
223	C₂	#27161734	1	1	yes	characteristic
224	C₂	#27161750	1	1	yes	characteristic
225	C₂	#27161794	1	1	yes	characteristic
226	C₂	#27161799	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉₇
commutator	C₂²×C₁₉₇	C₂³×C₄
Fitting	C₂×C₄×D₄×C₁₉₇	C₂
socle, cosocle	C₂³×C₁₉₇	C₂⁴
prime radical	C₂²×D₄×C₁₉₇	C₂²
hypercenter	C₂×C₄×D₄	D₁₉₇
hypocenter	C₁₉₇	#2354
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#20263285	C₂	semidirect
3. #27160972	C₂	semidirect
4. #27160949	C₂	semidirect
5. #27160939	C₂	semidirect
6. C₄⋊C₄×Dic₁₉₇	C₂	semidirect
7. C₄×C₄⋊Dic₁₉₇	C₂	semidirect
23. C₄×D₄×C₁₉₇	C₄	semidirect
34. C₄⋊Dic₁₉₇	D₄	semidirect
100. C₄×C₁₉₇	#117	semidirect
113. C₁₉₇	#2354	semidirect
125. C₄×D₄	Dic₁₉₇	semidirect
217. C₄	#20263285	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20263285	2
#27160982	2
#27160972	2
#27160949	2
#27160939	2
C₄⋊C₄×Dic₁₉₇	2
C₄×C₄⋊Dic₁₉₇	2
C₂×C₄×D₄×C₁₉₇	2
#2354	197