Subgroups — #47986262

Order 35456.

464 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22579871	C₂	1	1	yes	maximal, characteristic
2	#31799898	C₂	1	1	yes	maximal, characteristic
3	#31799897	C₂	1	2	yes	maximal
4	#31799887	C₂	1	2	yes	maximal
5	#31799884	C₂	1	2	yes	maximal
6	C₄⋊C₄×Dic₂₇₇	C₂	1	2	yes	maximal
7	C₄⋊C₄×Dic₂₇₇	C₂	1	1	yes	maximal, characteristic
8	C₄⋊C₄×Dic₂₇₇	C₂	1	1	yes	maximal, characteristic
9	C₄×#17984542	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊Dic₂₇₇	C₂	1	1	yes	maximal, characteristic
11	#389×C₂₇₇	C₂	1	1	yes	maximal, characteristic
12	#22579871	C₄	1	8	yes	complemented
13	C₂×C₄⋊Dic₂₇₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₂₇₇	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₂₇₇	C₂²	1	1	yes	characteristic
16	C₂×#17984542	C₂²	1	1	yes	characteristic
17	C₂×#17984542	C₂²	1	1	yes	characteristic
18	C₂×#17984542	C₂²	1	2	yes	—
19	C₂×#17984542	C₂²	1	2	yes	—
20	C₂×C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₂₇₇	C₂²	1	2	yes	—
22	C₂×C₄×Dic₂₇₇	C₂²	1	2	yes	—
23	C₂×C₄×Dic₂₇₇	C₂²	1	2	yes	—
24	C₂×C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
25	#22579787	C₂²	1	2	yes	—
26	#22579787	C₂²	1	1	yes	characteristic
27	#22579787	C₂²	1	1	yes	characteristic
28	#22579787	C₂²	1	2	yes	—
29	#22579787	C₂²	1	2	yes	—
30	#22579787	C₂²	1	2	yes	—
31	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	2	yes	—
32	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
35	#95×C₂₇₇	C₂²	1	2	yes	—
36	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
37	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
38	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
39	C₂²×Dic₂₇₇	C₂³	1	2	yes	—
40	C₂²×Dic₂₇₇	C₂³	1	2	yes	—
41	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
42	C₄⋊Dic₂₇₇	C₂×C₄	1	8	yes	—
43	C₄⋊Dic₂₇₇	C₂×C₄	1	4	yes	—
44	C₄⋊Dic₂₇₇	Q₈	1	4	yes	—
45	#17984542	—	2	2	no	—
46	#17984542	—	2	2	no	—
47	#17984542	—	4	2	no	—
48	#17984542	C₂×C₄	1	8	yes	—
49	C₄×Dic₂₇₇	—	2	4	no	—
50	C₄×Dic₂₇₇	—	2	2	no	—
51	C₄×Dic₂₇₇	—	4	2	no	—
52	C₄×Dic₂₇₇	—	4	2	no	—
53	C₄×Dic₂₇₇	C₂×C₄	1	4	yes	—
54	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₂₇₇	C₂³	1	2	yes	—
56	C₂²×C₄×C₂₇₇	C₂³	1	2	yes	—
57	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×C₂₇₇	—	2	4	no	—
60	C₄⋊C₄×C₂₇₇	—	2	2	no	—
61	C₄⋊C₄×C₂₇₇	C₂×C₄	1	4	yes	—
62	C₄²×C₂₇₇	—	2	2	no	—
63	C₂×Dic₂₇₇	—	2	2	no	—
64	C₂×Dic₂₇₇	—	2	1	no	—
65	C₂×Dic₂₇₇	—	2	1	no	—
66	C₂×Dic₂₇₇	—	2	1	no	—
67	C₂×Dic₂₇₇	—	2	1	no	—
68	C₂×Dic₂₇₇	—	2	2	no	—
69	C₂×Dic₂₇₇	—	2	2	no	—
70	C₂×Dic₂₇₇	—	2	1	no	—
71	C₂×Dic₂₇₇	—	2	2	no	—
72	C₂×Dic₂₇₇	—	2	1	no	—
73	C₂×Dic₂₇₇	—	2	1	no	—
74	C₂×Dic₂₇₇	—	2	1	no	—
75	C₂×Dic₂₇₇	—	2	1	no	—
76	C₂×Dic₂₇₇	C₂²×C₄	1	2	yes	—
77	C₂×Dic₂₇₇	C₂²×C₄	1	2	yes	—
78	C₂×Dic₂₇₇	C₂²×C₄	1	4	yes	—
79	C₂×Dic₂₇₇	C₂×Q₈	1	4	yes	—
80	C₂×Dic₂₇₇	#41	1	2	yes	—
81	C₂³×C₂₇₇	C₂⁴	1	1	yes	characteristic
82	C₂×C₄×C₂₇₇	—	2	1	no	—
83	C₂×C₄×C₂₇₇	—	2	2	no	—
84	C₂×C₄×C₂₇₇	—	2	2	no	—
85	C₂×C₄×C₂₇₇	—	2	1	no	—
86	C₂×C₄×C₂₇₇	—	2	1	no	—
87	C₂×C₄×C₂₇₇	—	2	2	no	—
88	C₂×C₄×C₂₇₇	—	2	2	no	—
89	C₂×C₄×C₂₇₇	—	2	1	no	—
90	C₂×C₄×C₂₇₇	—	2	2	no	—
91	C₂×C₄×C₂₇₇	—	2	1	no	—
92	C₂×C₄×C₂₇₇	C₂²×C₄	1	4	yes	—
93	C₂×C₄×C₂₇₇	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₂₇₇	C₂×Q₈	1	2	yes	—
95	C₂×C₄×C₂₇₇	#41	1	2	yes	—
96	C₂×C₄×C₂₇₇	#41	1	2	yes	—
97	C₂²×C₂₇₇	C₂³×C₄	1	1	yes	characteristic
98	C₂²×C₂₇₇	C₂²×Q₈	1	1	yes	characteristic
99	C₂²×C₂₇₇	C₂×#41	1	1	yes	characteristic
100	C₂²×C₂₇₇	C₂×#41	1	1	yes	characteristic
101	C₂²×C₂₇₇	C₂×#41	1	1	yes	characteristic
102	C₂²×C₂₇₇	H(5,𝔽₂)	1	1	yes	characteristic
103	C₂²×C₂₇₇	2₋¹⁺⁴	1	1	yes	characteristic
104	C₄×C₂₇₇	—	2	4	no	—
105	C₄×C₂₇₇	—	2	2	no	—
106	C₄×C₂₇₇	—	4	1	no	—
107	C₄×C₂₇₇	—	4	2	no	—
108	C₄×C₂₇₇	C₄×Q₈	1	4	yes	—
109	Dic₂₇₇	—	2	2	no	—
110	Dic₂₇₇	—	2	2	no	—
111	Dic₂₇₇	—	2	4	no	—
112	Dic₂₇₇	—	2	2	no	—
113	Dic₂₇₇	—	2	4	no	—
114	Dic₂₇₇	—	4	1	no	—
115	C₂×C₂₇₇	C₂×C₄×Q₈	1	1	yes	characteristic
116	C₂×C₂₇₇	C₄×#41	1	1	yes	characteristic
117	C₂×C₂₇₇	#520	1	1	yes	characteristic
118	C₂×C₂₇₇	#552	1	1	yes	characteristic
119	C₂×C₂₇₇	#553	1	1	yes	characteristic
120	C₂×C₂₇₇	#554	1	1	yes	characteristic
121	C₂×C₂₇₇	#557	1	1	yes	characteristic
122	C₂₇₇	#2370	1	1	yes	characteristic, complemented, Sylow
123	#2370	—	277	1	no	maximal, Sylow
124	C₂×#125	—	277	1	no	—
125	#389	—	277	2	no	—
126	#389	—	277	1	no	—
127	#389	D₂₇₇	1	1	yes	characteristic
128	#387	—	277	2	no	—
129	#387	—	277	2	no	—
130	C₄×C₄⋊C₄	—	277	2	no	—
131	C₄×C₄⋊C₄	—	277	1	no	—
132	C₄×C₄⋊C₄	—	277	1	no	—
133	C₄×C₄⋊C₄	—	277	1	no	—
134	C₄×C₄⋊C₄	—	277	1	no	—
135	#125	—	277	8	no	—
136	C₂×C₄⋊C₄	—	277	2	no	—
137	C₂×C₄⋊C₄	—	277	1	no	—
138	C₂×C₄⋊C₄	—	277	2	no	—
139	C₂×C₄⋊C₄	—	277	1	no	—
140	C₂×C₄⋊C₄	—	277	1	no	—
141	C₂×C₄⋊C₄	—	277	1	no	—
142	C₂×C₄⋊C₄	—	277	2	no	—
143	C₂×C₄⋊C₄	C₂×D₂₇₇	1	1	yes	characteristic
144	C₂×C₄⋊C₄	C₂×D₂₇₇	1	2	yes	—
145	C₂×C₄⋊C₄	C₂×D₂₇₇	1	1	yes	characteristic
146	C₂×C₄²	—	277	1	no	—
147	C₂×C₄²	—	277	2	no	—
148	C₂×C₄²	—	277	1	no	—
149	C₂×C₄²	—	277	2	no	—
150	C₂×C₄²	—	277	2	no	—
151	C₂×C₄²	C₂×D₂₇₇	1	1	yes	characteristic
152	#95	—	277	1	no	—
153	#95	—	277	2	no	—
154	#95	—	277	1	no	—
155	#95	—	277	2	no	—
156	#95	—	277	2	no	—
157	#95	—	277	2	no	—
158	#95	C₂×D₂₇₇	1	2	yes	—
159	C₂²×C₄	—	277	2	no	—
160	C₂²×C₄	—	277	1	no	—
161	C₂²×C₄	—	277	1	no	—
162	C₂²×C₄	—	277	1	no	—
163	C₂²×C₄	—	277	2	no	—
164	C₂²×C₄	—	277	1	no	—
165	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
166	C₂²×C₄	C₂²×D₂₇₇	1	2	yes	—
167	C₂²×C₄	C₂²×D₂₇₇	1	2	yes	—
168	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	277	8	no	—
173	C₄⋊C₄	—	277	8	no	—
174	C₄⋊C₄	—	277	4	no	—
175	C₄⋊C₄	—	277	4	no	—
176	C₄⋊C₄	—	554	2	no	—
177	C₄⋊C₄	—	554	2	no	—
178	C₄⋊C₄	—	1108	2	no	—
179	C₄⋊C₄	C₄×D₂₇₇	1	4	yes	—
180	C₄²	—	2	2	no	—
181	C₄²	—	277	4	no	—
182	C₄²	—	554	4	no	—
183	C₄²	—	554	2	no	—
184	C₄²	—	1108	2	no	—
185	C₄²	—	1108	2	no	—
186	C₂³	C₂³×D₂₇₇	1	1	yes	characteristic
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	277	2	no	—
198	C₂×C₄	—	277	2	no	—
199	C₂×C₄	—	277	4	no	—
200	C₂×C₄	—	277	4	no	—
201	C₂×C₄	—	277	2	no	—
202	C₂×C₄	—	554	2	no	—
203	C₂×C₄	—	554	1	no	—
204	C₂×C₄	—	554	1	no	—
205	C₂×C₄	—	554	1	no	—
206	C₂×C₄	—	554	2	no	—
207	C₂×C₄	—	554	1	no	—
208	C₂×C₄	—	554	2	no	—
209	C₂×C₄	—	554	1	no	—
210	C₂×C₄	—	554	1	no	—
211	C₂×C₄	—	554	2	no	—
212	C₂×C₄	—	554	1	no	—
213	C₂×C₄	—	554	1	no	—
214	C₂×C₄	—	554	1	no	—
215	C₂×C₄	C₂×C₄×D₂₇₇	1	4	yes	—
216	C₂×C₄	C₂×C₄×D₂₇₇	1	2	yes	—
217	C₂×C₄	Q₈×D₂₇₇	1	2	yes	—
218	C₂×C₄	#17984562	1	2	yes	—
219	C₂×C₄	#17984562	1	2	yes	—
220	C₂²	C₂²×C₄×D₂₇₇	1	1	yes	characteristic
221	C₂²	C₂×Q₈×D₂₇₇	1	1	yes	characteristic
222	C₂²	#41×D₂₇₇	1	1	yes	characteristic
223	C₂²	C₂×#17984562	1	1	yes	characteristic
224	C₂²	C₂×#17984562	1	1	yes	characteristic
225	C₂²	#22579926	1	1	yes	characteristic
226	C₂²	#22579928	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	4	no	—
229	C₄	—	4	1	no	—
230	C₄	—	4	2	no	—
231	C₄	—	554	2	no	—
232	C₄	—	554	4	no	—
233	C₄	—	554	2	no	—
234	C₄	—	554	2	no	—
235	C₄	—	554	4	no	—
236	C₄	—	1108	1	no	—
237	C₄	#22579845	1	4	yes	—
238	C₂	C₂×#22579845	1	1	yes	characteristic
239	C₂	C₄×#17984562	1	1	yes	characteristic
240	C₂	#31800484	1	1	yes	characteristic
241	C₂	#31800686	1	1	yes	characteristic
242	C₂	#31800704	1	1	yes	characteristic
243	C₂	#31800708	1	1	yes	characteristic
244	C₂	#31800727	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₇₇
commutator	C₂²×C₂₇₇	C₂³×C₄
Fitting	#389×C₂₇₇	C₂
socle, prime radical, cosocle	C₂³×C₂₇₇	C₂⁴
hypercenter	#389	D₂₇₇
hypocenter	C₂₇₇	#2370
LCS	G ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
12. #22579871	C₄	semidirect
122. C₂₇₇	#2370	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22579871	2
#31799898	2
#31799897	2
#31799887	2
#31799884	2
C₄⋊C₄×Dic₂₇₇	2
C₄×#17984542	2
C₄×C₄⋊Dic₂₇₇	2
#389×C₂₇₇	2
#2370	277