Subgroups — #47985658

Order 35456.

936 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22579839	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×Dic₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#31799909	C₂	1	2	yes	maximal, complemented
4	#31799905	C₂	1	2	yes	maximal, complemented
5	#31799882	C₂	1	4	yes	maximal, complemented
6	C₄×C₂²⋊Dic₂₇₇	C₂	1	4	yes	maximal, complemented
7	C₂²×#41×C₂₇₇	C₂	1	1	yes	maximal, characteristic
8	C₂³×Dic₂₇₇	C₂²	1	2	yes	—
9	#22579839	—	2	4	no	—
10	C₂×C₄⋊Dic₂₇₇	C₂²	1	2	yes	—
11	C₂×C₂²⋊Dic₂₇₇	C₂²	1	8	yes	—
12	C₂×C₂²⋊Dic₂₇₇	C₂²	1	2	yes	—
13	C₂×C₄×Dic₂₇₇	—	2	4	no	—
14	C₂×C₄×Dic₂₇₇	C₂²	1	2	yes	—
15	C₂×C₄×Dic₂₇₇	C₂²	1	2	yes	—
16	C₂×C₄×Dic₂₇₇	C₂²	1	2	yes	—
17	#22579787	C₂²	1	8	yes	—
18	C₂×#41×C₂₇₇	—	2	4	no	—
19	C₂×#41×C₂₇₇	—	2	4	no	—
20	C₂×#41×C₂₇₇	C₄	1	8	yes	complemented
21	C₂²×Q₈×C₂₇₇	C₂²	1	1	yes	characteristic
22	C₂²×D₄×C₂₇₇	C₂²	1	2	yes	—
23	C₂²×D₄×C₂₇₇	C₂²	1	1	yes	characteristic
24	C₂³×C₄×C₂₇₇	C₂²	1	2	yes	—
25	C₂³×C₄×C₂₇₇	C₂²	1	1	yes	characteristic
26	C₂²×Dic₂₇₇	—	2	8	no	—
27	C₂²×Dic₂₇₇	—	2	4	no	—
28	C₂²×Dic₂₇₇	C₂³	1	4	yes	—
29	C₂²×Dic₂₇₇	C₂³	1	4	yes	—
30	C₄⋊Dic₂₇₇	—	2	4	no	—
31	C₂²⋊Dic₂₇₇	—	2	16	no	—
32	C₂²⋊Dic₂₇₇	—	2	4	no	—
33	C₄×Dic₂₇₇	—	2	4	no	—
34	C₄×Dic₂₇₇	—	2	4	no	—
35	C₄×Dic₂₇₇	—	2	4	no	—
36	C₂⁴×C₂₇₇	C₂³	1	2	yes	—
37	C₂⁴×C₂₇₇	C₂³	1	1	yes	characteristic
38	#41×C₂₇₇	—	2	32	no	—
39	C₂×Q₈×C₂₇₇	—	2	2	no	—
40	C₂×Q₈×C₂₇₇	—	2	2	no	—
41	C₂×Q₈×C₂₇₇	C₂×C₄	1	4	yes	complemented
42	C₂×D₄×C₂₇₇	—	2	4	no	—
43	C₂×D₄×C₂₇₇	—	2	4	no	—
44	C₂×D₄×C₂₇₇	—	2	2	no	—
45	C₂×D₄×C₂₇₇	—	2	2	no	—
46	C₂×D₄×C₂₇₇	C₂×C₄	1	8	yes	—
47	C₂×D₄×C₂₇₇	C₂×C₄	1	4	yes	complemented
48	C₂²×C₄×C₂₇₇	—	2	4	no	—
49	C₂²×C₄×C₂₇₇	—	2	4	no	—
50	C₂²×C₄×C₂₇₇	C₂×C₄	1	4	yes	—
51	C₂²×C₄×C₂₇₇	C₂×C₄	1	8	yes	complemented
52	C₂²×C₄×C₂₇₇	D₄	1	4	yes	—
53	C₂²×C₄×C₂₇₇	D₄	1	4	yes	—
54	C₂²×C₄×C₂₇₇	C₂³	1	2	yes	—
55	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
57	C₂×Dic₂₇₇	—	2	16	no	—
58	C₂×Dic₂₇₇	—	2	4	no	—
59	C₂×Dic₂₇₇	—	2	4	no	—
60	C₂×Dic₂₇₇	—	2	4	no	—
61	C₂×Dic₂₇₇	—	2	8	no	—
62	C₂×Dic₂₇₇	#41	1	8	yes	complemented
63	C₂³×C₂₇₇	—	2	4	no	—
64	C₂³×C₂₇₇	—	2	2	no	—
65	C₂³×C₂₇₇	—	2	2	no	—
66	C₂³×C₂₇₇	—	2	8	no	—
67	C₂³×C₂₇₇	C₂²×C₄	1	4	yes	—
68	C₂³×C₂₇₇	C₂²×C₄	1	2	yes	—
69	C₂³×C₂₇₇	C₂×D₄	1	2	yes	—
70	C₂³×C₂₇₇	C₂×D₄	1	2	yes	—
71	C₂³×C₂₇₇	C₂⁴	1	1	yes	characteristic
72	Q₈×C₂₇₇	—	2	8	no	—
73	D₄×C₂₇₇	—	2	16	no	—
74	D₄×C₂₇₇	—	2	8	no	—
75	C₂×C₄×C₂₇₇	—	2	2	no	—
76	C₂×C₄×C₂₇₇	—	2	2	no	—
77	C₂×C₄×C₂₇₇	—	2	16	no	—
78	C₂×C₄×C₂₇₇	C₂²⋊C₄	1	16	yes	complemented
79	C₂×C₄×C₂₇₇	C₂²×C₄	1	4	yes	—
80	C₂×C₄×C₂₇₇	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₂₇₇	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
83	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
86	C₂²×C₂₇₇	—	2	4	no	—
87	C₂²×C₂₇₇	—	2	4	no	—
88	C₂²×C₂₇₇	—	2	4	no	—
89	C₂²×C₂₇₇	—	2	4	no	—
90	C₂²×C₂₇₇	—	2	8	no	—
91	C₂²×C₂₇₇	—	4	4	no	—
92	C₂²×C₂₇₇	—	4	4	no	—
93	C₂²×C₂₇₇	C₂×C₂²⋊C₄	1	4	yes	—
94	C₂²×C₂₇₇	C₂³×C₄	1	1	yes	characteristic
95	C₂²×C₂₇₇	C₂²×D₄	1	1	yes	characteristic
96	C₂²×C₂₇₇	C₂²×D₄	1	1	yes	characteristic
97	C₂²×C₂₇₇	C₂×#41	1	4	yes	—
98	C₄×C₂₇₇	—	2	4	no	—
99	C₄×C₂₇₇	C₂×C₂²⋊C₄	1	4	yes	—
100	C₄×C₂₇₇	C₂×C₂²⋊C₄	1	4	yes	—
101	Dic₂₇₇	—	2	8	no	—
102	Dic₂₇₇	—	4	4	no	—
103	C₂×C₂₇₇	—	2	4	no	—
104	C₂×C₂₇₇	—	4	4	no	—
105	C₂×C₂₇₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₂₇₇	C₄×#41	1	2	yes	—
107	C₂×C₂₇₇	#525	1	2	yes	—
108	C₂×C₂₇₇	#532	1	2	yes	—
109	C₂₇₇	#2297	1	1	yes	characteristic, complemented, Sylow
110	#2297	—	277	1	no	maximal, Sylow
111	C₂²×#41	D₂₇₇	1	1	yes	characteristic
112	C₂×#117	—	277	1	no	—
113	C₂²×C₄²	—	277	1	no	—
114	#391	—	277	2	no	—
115	#390	—	277	2	no	—
116	#386	—	277	4	no	—
117	C₄×C₂²⋊C₄	—	277	4	no	—
118	C₂×#41	—	2	4	no	—
119	C₂×#41	—	2	4	no	—
120	C₂×#41	Dic₂₇₇	1	8	yes	complemented
121	C₂²×Q₈	C₂×D₂₇₇	1	1	yes	characteristic
122	C₂²×D₄	C₂×D₂₇₇	1	2	yes	—
123	C₂²×D₄	C₂×D₂₇₇	1	1	yes	characteristic
124	C₂³×C₄	—	277	2	no	—
125	C₂³×C₄	C₂×D₂₇₇	1	2	yes	—
126	C₂³×C₄	C₂×D₂₇₇	1	1	yes	characteristic
127	#117	—	554	4	no	—
128	C₂×C₄⋊C₄	—	277	2	no	—
129	C₂×C₂²⋊C₄	—	277	2	no	—
130	C₂×C₂²⋊C₄	—	277	8	no	—
131	C₂×C₄²	—	277	2	no	—
132	C₂×C₄²	—	277	2	no	—
133	C₂×C₄²	—	277	2	no	—
134	C₂×C₄²	—	554	4	no	—
135	#95	—	277	8	no	—
136	C₂⁴	C₂²×D₂₇₇	1	2	yes	—
137	C₂⁴	C₂²×D₂₇₇	1	1	yes	characteristic
138	#41	—	2	32	no	—
139	C₂×Q₈	—	2	2	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	C₂×Dic₂₇₇	1	4	yes	complemented
142	C₂×D₄	—	2	4	no	—
143	C₂×D₄	—	2	4	no	—
144	C₂×D₄	—	2	2	no	—
145	C₂×D₄	—	2	2	no	—
146	C₂×D₄	C₂×Dic₂₇₇	1	8	yes	—
147	C₂×D₄	C₂×Dic₂₇₇	1	4	yes	complemented
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	—	277	4	no	—
151	C₂²×C₄	—	277	4	no	—
152	C₂²×C₄	—	554	8	no	—
153	C₂²×C₄	—	554	4	no	—
154	C₂²×C₄	C₂×Dic₂₇₇	1	4	yes	—
155	C₂²×C₄	C₂×Dic₂₇₇	1	8	yes	complemented
156	C₂²×C₄	C₂²⋊D₂₇₇	1	4	yes	—
157	C₂²×C₄	C₂²⋊D₂₇₇	1	4	yes	—
158	C₂²×C₄	C₂²×D₂₇₇	1	2	yes	—
159	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
160	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
161	C₄⋊C₄	—	554	4	no	—
162	C₂²⋊C₄	—	554	16	no	—
163	C₂²⋊C₄	—	554	4	no	—
164	C₄²	—	554	4	no	—
165	C₄²	—	554	4	no	—
166	C₄²	—	554	4	no	—
167	C₂³	—	2	4	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	2	2	no	—
170	C₂³	—	2	8	no	—
171	C₂³	C₂²×Dic₂₇₇	1	4	yes	—
172	C₂³	C₂²×Dic₂₇₇	1	2	yes	—
173	C₂³	C₂×C₂²⋊D₂₇₇	1	2	yes	—
174	C₂³	C₂×C₂²⋊D₂₇₇	1	2	yes	—
175	C₂³	C₂³×D₂₇₇	1	1	yes	characteristic
176	Q₈	—	2	8	no	—
177	D₄	—	2	16	no	—
178	D₄	—	2	8	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	16	no	—
182	C₂×C₄	—	277	8	no	—
183	C₂×C₄	—	554	16	no	—
184	C₂×C₄	—	554	4	no	—
185	C₂×C₄	—	554	4	no	—
186	C₂×C₄	—	554	4	no	—
187	C₂×C₄	—	554	8	no	—
188	C₂×C₄	C₂²⋊Dic₂₇₇	1	16	yes	complemented
189	C₂×C₄	C₂²×Dic₂₇₇	1	4	yes	—
190	C₂×C₄	C₂²×Dic₂₇₇	1	2	yes	—
191	C₂×C₄	C₂²×Dic₂₇₇	1	2	yes	—
192	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	2	yes	—
193	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	2	yes	—
194	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	2	yes	—
195	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	2	yes	—
196	C₂²	—	2	4	no	—
197	C₂²	—	2	4	no	—
198	C₂²	—	2	4	no	—
199	C₂²	—	2	4	no	—
200	C₂²	—	2	8	no	—
201	C₂²	—	4	4	no	—
202	C₂²	—	4	4	no	—
203	C₂²	C₂×C₂²⋊Dic₂₇₇	1	4	yes	—
204	C₂²	C₂³×Dic₂₇₇	1	1	yes	characteristic
205	C₂²	C₂²×C₂²⋊D₂₇₇	1	1	yes	characteristic
206	C₂²	C₂²×C₂²⋊D₂₇₇	1	1	yes	characteristic
207	C₂²	#41×D₂₇₇	1	4	yes	—
208	C₄	—	2	4	no	—
209	C₄	—	554	8	no	—
210	C₄	—	1108	4	no	—
211	C₄	C₂×C₂²⋊Dic₂₇₇	1	4	yes	—
212	C₄	C₂×C₂²⋊Dic₂₇₇	1	4	yes	—
213	C₂	—	2	4	no	—
214	C₂	—	4	4	no	—
215	C₂	C₂²×C₂²⋊Dic₂₇₇	1	1	yes	characteristic
216	C₂	#41×Dic₂₇₇	1	2	yes	—
217	C₂	#31800509	1	2	yes	—
218	C₂	#31800548	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₇₇
commutator	C₂²×C₂₇₇	C₂³×C₄
Frattini	C₂³	C₂³×D₂₇₇
Fitting, prime radical	C₂²×#41×C₂₇₇	C₂
socle, cosocle	C₂³×C₂₇₇	C₂⁴
hypercenter	C₂²×#41	D₂₇₇
hypocenter	C₂₇₇	#2297
LCS	G ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#22579839	C₂	semidirect
2. C₂²×C₄×Dic₂₇₇	C₂	semidirect
3. #31799909	C₂	semidirect
4. #31799905	C₂	semidirect
5. #31799882	C₂	semidirect
6. C₄×C₂²⋊Dic₂₇₇	C₂	semidirect
20. C₂×#41×C₂₇₇	C₄	semidirect
41. C₂×Q₈×C₂₇₇	C₂×C₄	semidirect
47. C₂×D₄×C₂₇₇	C₂×C₄	semidirect
51. C₂²×C₄×C₂₇₇	C₂×C₄	semidirect
62. C₂×Dic₂₇₇	#41	semidirect
78. C₂×C₄×C₂₇₇	C₂²⋊C₄	semidirect
109. C₂₇₇	#2297	semidirect
120. C₂×#41	Dic₂₇₇	semidirect
141. C₂×Q₈	C₂×Dic₂₇₇	semidirect
147. C₂×D₄	C₂×Dic₂₇₇	semidirect
155. C₂²×C₄	C₂×Dic₂₇₇	semidirect
188. C₂×C₄	C₂²⋊Dic₂₇₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22579839	2
C₂²×C₄×Dic₂₇₇	2
#31799909	2
#31799905	2
#31799882	2
C₄×C₂²⋊Dic₂₇₇	2
C₂²×#41×C₂₇₇	2
#2297	277