Subgroups — #38745776

Order 25216.

832 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20263323	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₇₈₈	C₂	1	2	yes	maximal, complemented
3	C₂×#20263288	C₂	1	2	yes	maximal, complemented
4	#27160981	C₂	1	2	yes	maximal, complemented
5	#27160980	C₂	1	2	yes	maximal, complemented
6	#27160957	C₂	1	4	yes	maximal, complemented
7	C₄×C₄⋊Dic₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₄⋊C₄×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₇₈₈	C₂²	1	2	yes	—
10	C₂²×D₇₈₈	C₂²	1	2	yes	—
11	C₂²×D₇₈₈	C₂²	1	2	yes	—
12	C₂²×C₄×D₁₉₇	C₂²	1	4	yes	—
13	#20263323	C₄	1	8	yes	complemented
14	C₄×D₇₈₈	—	2	8	no	—
15	#20263288	—	2	8	no	—
16	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
18	C₂×#15691335	C₂²	1	4	yes	—
19	C₂×#15691335	C₂²	1	4	yes	—
20	C₂×#15691335	C₂²	1	4	yes	—
21	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
22	#20263233	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₁₉₇	C₂²	1	2	yes	—
26	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
27	#95×C₁₉₇	C₂²	1	2	yes	—
28	C₂³×D₁₉₇	C₂³	1	4	yes	—
29	C₂×D₇₈₈	—	2	4	no	—
30	C₂×D₇₈₈	—	2	4	no	—
31	C₂×D₇₈₈	—	2	4	no	—
32	C₂×D₇₈₈	—	2	4	no	—
33	C₂×D₇₈₈	—	4	2	no	—
34	C₂×D₇₈₈	—	4	2	no	—
35	C₂×D₇₈₈	C₂×C₄	1	8	yes	complemented
36	C₂×D₇₈₈	C₂×C₄	1	8	yes	complemented
37	C₂×D₇₈₈	C₂×C₄	1	8	yes	complemented
38	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
39	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
40	C₂×C₄×D₁₉₇	—	2	8	no	—
41	C₂×C₄×D₁₉₇	—	2	8	no	—
42	C₂×C₄×D₁₉₇	—	4	4	no	—
43	C₄⋊Dic₁₉₇	—	2	2	no	—
44	C₄⋊Dic₁₉₇	D₄	1	4	yes	complemented
45	#15691335	—	2	8	no	—
46	#15691335	—	2	8	no	—
47	#15691335	—	4	4	no	—
48	C₄×Dic₁₉₇	—	2	4	no	—
49	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₁₉₇	—	2	2	no	—
55	C₄⋊C₄×C₁₉₇	D₄	1	4	yes	complemented
56	C₄²×C₁₉₇	—	2	4	no	—
57	C₄²×C₁₉₇	C₂×C₄	1	4	yes	complemented
58	C₂²×D₁₉₇	—	2	4	no	—
59	C₂²×D₁₉₇	—	2	4	no	—
60	C₂²×D₁₉₇	—	2	4	no	—
61	C₂²×D₁₉₇	—	2	4	no	—
62	C₂²×D₁₉₇	—	2	4	no	—
63	C₂²×D₁₉₇	—	2	4	no	—
64	C₂²×D₁₉₇	C₂²×C₄	1	8	yes	—
65	D₇₈₈	—	2	16	no	—
66	D₇₈₈	—	2	16	no	—
67	D₇₈₈	—	4	8	no	—
68	C₂×Dic₁₉₇	—	2	2	no	—
69	C₂×Dic₁₉₇	—	2	2	no	—
70	C₂×Dic₁₉₇	—	2	2	no	—
71	C₂×Dic₁₉₇	—	2	2	no	—
72	C₂×Dic₁₉₇	—	2	2	no	—
73	C₂×Dic₁₉₇	C₂×D₄	1	4	yes	—
74	C₄×D₁₉₇	—	4	16	no	—
75	C₂³×C₁₉₇	C₂⁴	1	1	yes	characteristic
76	C₂×C₄×C₁₉₇	—	2	2	no	—
77	C₂×C₄×C₁₉₇	—	2	2	no	—
78	C₂×C₄×C₁₉₇	—	2	1	no	—
79	C₂×C₄×C₁₉₇	—	2	1	no	—
80	C₂×C₄×C₁₉₇	—	2	2	no	—
81	C₂×C₄×C₁₉₇	—	2	2	no	—
82	C₂×C₄×C₁₉₇	—	2	2	no	—
83	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₁₉₇	C₂×D₄	1	4	yes	—
88	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₁₉₇	#41	1	2	yes	—
90	C₂×C₄×C₁₉₇	#41	1	2	yes	—
91	C₂²×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
92	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
93	C₂²×C₁₉₇	C₂²×D₄	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	H(5,𝔽₂)	1	1	yes	characteristic
98	C₂×D₁₉₇	—	2	8	no	—
99	C₂×D₁₉₇	—	2	8	no	—
100	C₂×D₁₉₇	—	2	8	no	—
101	C₂×D₁₉₇	—	4	4	no	—
102	C₂×D₁₉₇	—	4	4	no	—
103	C₂×D₁₉₇	—	4	4	no	—
104	C₂×D₁₉₇	—	4	4	no	—
105	C₄×C₁₉₇	—	2	2	no	—
106	C₄×C₁₉₇	—	2	4	no	—
107	C₄×C₁₉₇	—	4	2	no	—
108	C₄×C₁₉₇	C₄×D₄	1	4	yes	complemented
109	C₄×C₁₉₇	C₄×D₄	1	4	yes	complemented
110	Dic₁₉₇	—	2	4	no	—
111	Dic₁₉₇	—	4	2	no	—
112	C₂×C₁₉₇	C₂×C₄×D₄	1	1	yes	characteristic
113	C₂×C₁₉₇	C₂×C₄×D₄	1	1	yes	characteristic
114	C₂×C₁₉₇	#518	1	1	yes	characteristic
115	C₂×C₁₉₇	D₄²	1	1	yes	characteristic
116	C₂×C₁₉₇	#550	1	1	yes	characteristic
117	C₂×C₁₉₇	#550	1	1	yes	characteristic
118	C₂×C₁₉₇	#559	1	1	yes	characteristic
119	D₁₉₇	—	4	8	no	—
120	C₁₉₇	#2366	1	1	yes	characteristic, complemented, Sylow
121	#2366	—	197	1	no	maximal, Sylow
122	C₂×#127	—	197	1	no	—
123	C₂×C₄×D₄	—	197	2	no	—
124	C₂×C₄×D₄	—	197	2	no	—
125	#390	—	197	2	no	—
126	#390	—	197	2	no	—
127	#386	—	197	4	no	—
128	C₄×C₄⋊C₄	—	197	1	no	—
129	C₄×C₄⋊C₄	D₁₉₇	1	1	yes	characteristic, complemented
130	C₂²×D₄	—	197	2	no	—
131	C₂²×D₄	—	197	2	no	—
132	C₂²×D₄	—	197	2	no	—
133	C₂³×C₄	—	197	4	no	—
134	#127	—	197	8	no	—
135	C₄×D₄	—	394	8	no	—
136	C₄×D₄	—	394	8	no	—
137	C₂×C₄⋊C₄	—	197	1	no	—
138	C₂×C₄⋊C₄	—	197	1	no	—
139	C₂×C₄⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
140	C₂×C₄⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
141	C₂×C₂²⋊C₄	—	197	4	no	—
142	C₂×C₂²⋊C₄	—	197	4	no	—
143	C₂×C₂²⋊C₄	—	197	4	no	—
144	C₂×C₄²	—	197	2	no	—
145	C₂×C₄²	C₂×D₁₉₇	1	2	yes	—
146	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
147	#95	—	197	2	no	—
148	#95	C₂×D₁₉₇	1	2	yes	—
149	C₂⁴	—	197	4	no	—
150	C₂×D₄	—	788	2	no	—
151	C₂×D₄	—	788	2	no	—
152	C₂×D₄	—	394	4	no	—
153	C₂×D₄	—	394	4	no	—
154	C₂×D₄	—	394	4	no	—
155	C₂×D₄	—	394	4	no	—
156	C₂×D₄	—	197	8	no	—
157	C₂×D₄	—	197	8	no	—
158	C₂×D₄	—	197	8	no	—
159	C₂²×C₄	—	788	4	no	—
160	C₂²×C₄	—	394	8	no	—
161	C₂²×C₄	—	394	8	no	—
162	C₂²×C₄	—	197	2	no	—
163	C₂²×C₄	—	197	2	no	—
164	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
165	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
166	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
167	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	394	2	no	—
171	C₄⋊C₄	—	197	4	no	—
172	C₄⋊C₄	D₇₈₈	1	4	yes	complemented
173	C₂²⋊C₄	—	788	4	no	—
174	C₂²⋊C₄	—	394	8	no	—
175	C₂²⋊C₄	—	394	8	no	—
176	C₄²	—	2	4	no	—
177	C₄²	—	394	4	no	—
178	C₄²	C₄×D₁₉₇	1	4	yes	complemented
179	C₂³	—	394	4	no	—
180	C₂³	—	394	4	no	—
181	C₂³	—	394	4	no	—
182	C₂³	—	394	4	no	—
183	C₂³	—	394	4	no	—
184	C₂³	—	394	4	no	—
185	C₂³	—	197	8	no	—
186	C₂³	C₂³×D₁₉₇	1	1	yes	characteristic
187	D₄	—	788	8	no	—
188	D₄	—	394	16	no	—
189	D₄	—	394	16	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	788	16	no	—
198	C₂×C₄	—	394	2	no	—
199	C₂×C₄	—	394	2	no	—
200	C₂×C₄	—	394	2	no	—
201	C₂×C₄	—	394	2	no	—
202	C₂×C₄	—	394	2	no	—
203	C₂×C₄	—	197	4	no	—
204	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
207	C₂×C₄	D₄×D₁₉₇	1	2	yes	—
208	C₂×C₄	C₂×D₇₈₈	1	4	yes	—
209	C₂×C₄	C₂×D₇₈₈	1	2	yes	—
210	C₂×C₄	#15691358	1	2	yes	—
211	C₂×C₄	#15691359	1	2	yes	—
212	C₂²	—	788	4	no	—
213	C₂²	—	788	4	no	—
214	C₂²	—	788	4	no	—
215	C₂²	—	788	4	no	—
216	C₂²	—	394	8	no	—
217	C₂²	—	394	8	no	—
218	C₂²	—	394	8	no	—
219	C₂²	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
220	C₂²	C₂×D₄×D₁₉₇	1	1	yes	characteristic
221	C₂²	C₂²×D₇₈₈	1	1	yes	characteristic
222	C₂²	C₂×#15691358	1	1	yes	characteristic
223	C₂²	C₂×#15691359	1	1	yes	characteristic
224	C₂²	#20263372	1	1	yes	characteristic
225	C₂²	#20263372	1	1	yes	characteristic
226	C₄	—	2	4	no	—
227	C₄	—	2	2	no	—
228	C₄	—	4	2	no	—
229	C₄	—	788	2	no	—
230	C₄	—	394	4	no	—
231	C₄	#20263288	1	4	yes	complemented
232	C₄	C₄×D₇₈₈	1	4	yes	complemented
233	C₂	—	788	8	no	—
234	C₂	C₂×#20263288	1	1	yes	characteristic
235	C₂	C₂×C₄×D₇₈₈	1	1	yes	characteristic
236	C₂	#27161553	1	1	yes	characteristic
237	C₂	D₄×D₇₈₈	1	1	yes	characteristic
238	C₂	#27161750	1	1	yes	characteristic
239	C₂	#27161751	1	1	yes	characteristic
240	C₂	#27161814	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	C₂×#20263323	C₂
hypercenter	C₄×C₄⋊C₄	D₁₉₇
hypocenter	C₁₉₇	#2366
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×C₄×D₇₈₈	C₂	semidirect
3. C₂×#20263288	C₂	semidirect
4. #27160981	C₂	semidirect
5. #27160980	C₂	semidirect
6. #27160957	C₂	semidirect
7. C₄×C₄⋊Dic₁₉₇	C₂	semidirect
8. C₄×C₄⋊C₄×C₁₉₇	C₂	semidirect
13. #20263323	C₄	semidirect
35. C₂×D₇₈₈	C₂×C₄	semidirect
36. C₂×D₇₈₈	C₂×C₄	semidirect
37. C₂×D₇₈₈	C₂×C₄	semidirect
44. C₄⋊Dic₁₉₇	D₄	semidirect
55. C₄⋊C₄×C₁₉₇	D₄	semidirect
57. C₄²×C₁₉₇	C₂×C₄	semidirect
108. C₄×C₁₉₇	C₄×D₄	semidirect
109. C₄×C₁₉₇	C₄×D₄	semidirect
120. C₁₉₇	#2366	semidirect
129. C₄×C₄⋊C₄	D₁₉₇	semidirect
172. C₄⋊C₄	D₇₈₈	semidirect
178. C₄²	C₄×D₁₉₇	semidirect
231. C₄	#20263288	semidirect
232. C₄	C₄×D₇₈₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20263323	2
C₂×C₄×D₇₈₈	2
C₂×#20263288	2
#27160981	2
#27160980	2
#27160957	2
C₄×C₄⋊Dic₁₉₇	2
C₄×C₄⋊C₄×C₁₉₇	2
#2366	197