Subgroups — #38751420

Order 25216.

500 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20263334	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#20263333	C₂	1	1	yes	maximal, characteristic, complemented
3	#27161625	C₂	1	1	yes	maximal, characteristic
4	#27161154	C₂	1	4	yes	maximal, complemented
5	#27161142	C₂	1	4	yes	maximal, complemented
6	#27161102	C₂	1	1	yes	maximal, characteristic, complemented
7	#27161045	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×Dic₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#408×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15691358	C₂²	1	1	yes	characteristic
11	#20263334	—	2	4	no	—
12	#20263333	—	2	4	no	—
13	C₂×#15691342	C₂²	1	1	yes	characteristic
14	C₂×#15691342	C₂²	1	2	yes	—
15	C₂×#15691341	C₂²	1	2	yes	—
16	C₂²×#13096802	C₂²	1	2	yes	—
17	C₂²×#13096802	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₁₉₇	C₂²	1	2	yes	—
19	#20263324	C₄	1	1	yes	characteristic
20	#20263322	C₄	1	1	yes	characteristic
21	#20263315	C₄	1	1	yes	characteristic
22	#20263313	C₄	1	1	yes	characteristic
23	#20263298	C₄	1	4	yes	—
24	C₄×C₂²⋊D₁₉₇	C₂²	1	4	yes	—
25	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
26	#20263255	C₂²	1	4	yes	—
27	#20263238	C₂²	1	4	yes	—
28	C₈×Dic₁₉₇	C₂²	1	2	yes	—
29	C₈×Dic₁₉₇	C₂²	1	2	yes	—
30	C₂×#41×C₁₉₇	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
32	C₂²×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
33	C₂²⋊C₈×C₁₉₇	C₂²	1	4	yes	—
34	#15691358	—	2	2	no	—
35	#15691358	—	2	2	no	—
36	C₂×Dic₃₉₄	C₂×C₄	1	1	yes	characteristic
37	C₂×D₇₈₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₂²⋊D₁₉₇	C₂×C₄	1	4	yes	—
39	C₂×C₂²⋊D₁₉₇	C₂×C₄	1	2	yes	—
40	C₂²×Dic₁₉₇	C₂×C₄	1	2	yes	—
41	C₂×C₄×D₁₉₇	C₂³	1	2	yes	—
42	#15691342	—	2	4	no	—
43	#15691342	—	2	2	no	—
44	#15691341	—	2	4	no	—
45	C₂×#13096802	—	2	4	no	—
46	C₂×#13096802	D₄	1	4	yes	complemented
47	C₂×#13096802	C₂³	1	2	yes	—
48	C₂×#13096802	C₂³	1	1	yes	characteristic
49	C₂×#13096802	C₂³	1	1	yes	characteristic
50	C₈×D₁₉₇	—	2	4	no	—
51	#15691337	C₂×C₄	1	4	yes	—
52	C₂²⋊Dic₁₉₇	C₂×C₄	1	4	yes	—
53	#15691335	C₂×C₄	1	4	yes	—
54	C₄×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
55	C₄×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
56	C₄×Dic₁₉₇	C₂³	1	2	yes	—
57	#41×C₁₉₇	—	2	2	no	—
58	#41×C₁₉₇	—	2	2	no	—
59	C₂×Q₈×C₁₉₇	C₂×C₄	1	1	yes	characteristic
60	C₂×D₄×C₁₉₇	C₂×C₄	1	2	yes	—
61	C₂×D₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
63	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
64	M₁₆×C₁₉₇	—	2	2	no	—
65	C₂×C₈×C₁₉₇	—	2	2	no	—
66	C₂×C₈×C₁₉₇	C₂³	1	2	yes	—
67	C₂×C₈×C₁₉₇	C₂³	1	2	yes	—
68	C₂²×D₁₉₇	C₂²×C₄	1	2	yes	—
69	Dic₃₉₄	—	2	2	no	—
70	D₇₈₈	—	2	2	no	—
71	C₂²⋊D₁₉₇	—	2	4	no	—
72	C₂²⋊D₁₉₇	—	2	8	no	—
73	C₂×Dic₁₉₇	—	2	4	no	—
74	C₂×Dic₁₉₇	C₂²×C₄	1	2	yes	—
75	C₂×Dic₁₉₇	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₉₇	C₂²×C₄	1	2	yes	—
77	C₄×D₁₉₇	—	2	2	no	—
78	C₄×D₁₉₇	—	2	2	no	—
79	#13096802	—	2	2	no	—
80	#13096802	C₂×D₄	1	2	yes	—
81	#13096802	C₂×D₄	1	2	yes	—
82	C₂³×C₁₉₇	C₂²×C₄	1	2	yes	—
83	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
84	Q₈×C₁₉₇	—	2	2	no	—
85	D₄×C₁₉₇	—	2	4	no	—
86	D₄×C₁₉₇	—	2	2	no	—
87	C₂×C₄×C₁₉₇	—	2	2	no	—
88	C₂×C₄×C₁₉₇	—	2	2	no	—
89	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₁₉₇	#41	1	2	yes	—
93	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
94	C₈×C₁₉₇	—	2	2	no	—
95	C₈×C₁₉₇	—	2	2	no	—
96	C₂²×C₁₉₇	—	2	2	no	—
97	C₂²×C₁₉₇	—	2	2	no	—
98	C₂²×C₁₉₇	—	2	4	no	—
99	C₂²×C₁₉₇	C₄×D₄	1	2	yes	—
100	C₂²×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₁₉₇	—	2	2	no	—
102	C₂×D₁₉₇	—	2	4	no	—
103	C₄×C₁₉₇	—	2	2	no	—
104	C₄×C₁₉₇	C₄×D₄	1	2	yes	—
105	C₄×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
106	C₄×C₁₉₇	C₂×#41	1	1	yes	characteristic
107	Dic₁₉₇	—	2	2	no	—
108	Dic₁₉₇	—	2	2	no	—
109	Dic₁₉₇	#131	1	4	yes	complemented
110	C₂×C₁₉₇	—	2	2	no	—
111	C₂×C₁₉₇	—	4	2	no	—
112	C₂×C₁₉₇	C₂×C₄×D₄	1	1	yes	characteristic
113	C₂×C₁₉₇	C₂×#131	1	2	yes	—
114	D₁₉₇	—	4	2	no	—
115	C₁₉₇	#2931	1	1	yes	characteristic, complemented, Sylow
116	#2931	—	197	1	no	maximal, Sylow
117	C₂×#131	—	197	1	no	—
118	C₂×#131	—	197	1	no	—
119	#532	—	197	1	no	—
120	#436	—	197	4	no	—
121	C₈×D₄	—	197	4	no	—
122	#423	—	197	1	no	—
123	#408	—	197	1	no	—
124	#408	D₁₉₇	1	1	yes	characteristic, complemented
125	C₂×C₄×C₈	—	197	1	no	—
126	C₂×#41	—	197	1	no	—
127	C₂×#41	C₂×D₁₉₇	1	1	yes	characteristic
128	#131	—	394	4	no	—
129	#131	—	394	4	no	—
130	C₂×M₁₆	—	197	1	no	—
131	C₂×M₁₆	—	197	2	no	—
132	C₂×M₁₆	—	197	2	no	—
133	C₂×M₁₆	C₂×D₁₉₇	1	1	yes	characteristic
134	C₂²×C₈	—	197	2	no	—
135	C₂²×C₈	—	197	2	no	—
136	C₂²×C₈	—	197	1	no	—
137	C₂²×C₈	C₂×D₁₉₇	1	1	yes	characteristic
138	C₄⋊Q₈	—	197	1	no	—
139	#127	—	197	1	no	—
140	#124	—	197	1	no	—
141	#124	—	197	1	no	—
142	#121	—	197	4	no	—
143	C₄×D₄	—	197	4	no	—
144	C₂×C₄²	—	197	1	no	—
145	C₄⋊C₈	—	197	4	no	—
146	C₂²⋊C₈	—	197	4	no	—
147	C₂²⋊C₈	C₂×D₁₉₇	1	4	yes	—
148	C₄×C₈	—	197	2	no	—
149	C₄×C₈	—	197	2	no	—
150	#41	—	2	2	no	—
151	#41	—	2	2	no	—
152	#41	—	394	2	no	—
153	#41	—	394	2	no	—
154	C₂×Q₈	—	197	1	no	—
155	C₂×Q₈	C₄×D₁₉₇	1	1	yes	characteristic
156	C₂×D₄	—	197	4	no	—
157	C₂×D₄	—	197	2	no	—
158	C₂×D₄	—	197	1	no	—
159	C₂×D₄	C₄×D₁₉₇	1	2	yes	—
160	C₂×D₄	C₄×D₁₉₇	1	1	yes	characteristic
161	C₂²×C₄	—	197	2	no	—
162	C₂²×C₄	—	197	2	no	—
163	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
164	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
165	M₁₆	—	2	2	no	—
166	M₁₆	—	394	4	no	—
167	M₁₆	—	394	2	no	—
168	M₁₆	—	394	4	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	394	4	no	—
171	C₂×C₈	—	394	4	no	—
172	C₂×C₈	—	197	2	no	—
173	C₂×C₈	—	197	4	no	—
174	C₂×C₈	—	197	1	no	—
175	C₂×C₈	—	197	1	no	—
176	C₂×C₈	C₂²×D₁₉₇	1	2	yes	—
177	C₂×C₈	C₂²×D₁₉₇	1	2	yes	—
178	C₄⋊C₄	—	197	4	no	—
179	C₂²⋊C₄	—	197	4	no	—
180	C₂²⋊C₄	—	197	4	no	—
181	C₄²	—	197	1	no	—
182	C₄²	—	197	1	no	—
183	C₄²	—	197	2	no	—
184	C₂³	—	197	2	no	—
185	C₂³	C₂×C₄×D₁₉₇	1	2	yes	—
186	C₂³	C₂×C₄×D₁₉₇	1	1	yes	characteristic
187	Q₈	—	2	2	no	—
188	Q₈	—	394	2	no	—
189	D₄	—	2	4	no	—
190	D₄	—	2	2	no	—
191	D₄	—	394	4	no	—
192	D₄	—	394	8	no	—
193	D₄	—	394	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	394	4	no	—
197	C₂×C₄	—	394	2	no	—
198	C₂×C₄	—	394	2	no	—
199	C₂×C₄	—	197	2	no	—
200	C₂×C₄	—	197	2	no	—
201	C₂×C₄	—	197	2	no	—
202	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
204	C₂×C₄	D₄×D₁₉₇	1	2	yes	—
205	C₂×C₄	#15691357	1	2	yes	—
206	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	2	2	no	—
209	C₈	—	394	2	no	—
210	C₈	—	197	2	no	—
211	C₈	—	197	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	394	2	no	—
216	C₂²	—	394	4	no	—
217	C₂²	#20263286	1	2	yes	—
218	C₂²	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
219	C₄	—	2	2	no	—
220	C₄	—	394	2	no	—
221	C₄	—	394	2	no	—
222	C₄	—	197	4	no	—
223	C₄	#20263286	1	2	yes	—
224	C₄	C₂×D₄×D₁₉₇	1	1	yes	characteristic
225	C₄	C₂×#15691357	1	1	yes	characteristic
226	C₂	—	2	2	no	—
227	C₂	—	4	2	no	—
228	C₂	—	788	2	no	—
229	C₂	C₂×#20263286	1	1	yes	characteristic
230	C₂	#131×D₁₉₇	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉₇
commutator, socle	C₂²×C₁₉₇	C₂³×C₄
Fitting	#408×C₁₉₇	C₂
prime radical	#27161625	C₂
cosocle	C₂×C₄×C₁₉₇	C₂⁴
hypercenter	#408	D₁₉₇
hypocenter	C₁₉₇	#2931
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
1. C₂×#20263334	C₂	semidirect
2. C₂×#20263333	C₂	semidirect
4. #27161154	C₂	semidirect
5. #27161142	C₂	semidirect
6. #27161102	C₂	semidirect
7. #27161045	C₂	semidirect
8. C₂×C₈×Dic₁₉₇	C₂	semidirect
9. #408×C₁₉₇	C₂	semidirect
46. C₂×#13096802	D₄	semidirect
109. Dic₁₉₇	#131	semidirect
115. C₁₉₇	#2931	semidirect
124. #408	D₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20263334	2
C₂×#20263333	2
#27161625	2
#27161154	2
#27161142	2
#27161102	2
#27161045	2
C₂×C₈×Dic₁₉₇	2
#408×C₁₉₇	2
#2931	197