Subgroups — #47986253

Order 35456.

496 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22579879	C₂	1	1	yes	maximal, characteristic
2	#31799907	C₂	1	2	yes	maximal
3	#31799906	C₂	1	2	yes	maximal
4	#31799898	C₂	1	1	yes	maximal, characteristic
5	#31799897	C₂	1	2	yes	maximal
6	C₄⋊C₄×Dic₂₇₇	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₄×Dic₂₇₇	C₂	1	1	yes	maximal, characteristic
8	C₄×#17984542	C₂	1	1	yes	maximal, characteristic
9	C₄×C₄⋊Dic₂₇₇	C₂	1	1	yes	maximal, characteristic
10	#31799853	C₂	1	2	yes	maximal
11	#389×C₂₇₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₅₅₄	C₂²	1	2	yes	—
13	#22579879	C₄	1	8	yes	complemented
14	C₂×C₄⋊Dic₂₇₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₂₇₇	C₂²	1	1	yes	characteristic
16	C₂×#17984542	C₂²	1	1	yes	characteristic
17	C₂×#17984542	C₂²	1	2	yes	—
18	C₂×#17984542	C₂²	1	2	yes	—
19	C₂×#17984542	C₂²	1	1	yes	characteristic
20	C₂×C₄×Dic₂₇₇	C₂²	1	2	yes	—
21	C₂×C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
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	2	yes	—
27	#22579787	C₂²	1	1	yes	characteristic
28	#22579787	C₂²	1	1	yes	characteristic
29	#22579787	C₂²	1	2	yes	—
30	#22579787	C₂²	1	2	yes	—
31	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	2	yes	—
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	C₂×Dic₅₅₄	—	4	2	no	—
37	C₂×Dic₅₅₄	—	4	2	no	—
38	C₂×Dic₅₅₄	C₂×C₄	1	8	yes	—
39	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
40	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
41	C₂²×Dic₂₇₇	C₂³	1	2	yes	—
42	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
43	C₂²×Dic₂₇₇	C₂³	1	2	yes	—
44	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
45	C₄⋊Dic₂₇₇	C₂×C₄	1	4	yes	—
46	C₄⋊Dic₂₇₇	Q₈	1	4	yes	complemented
47	#17984542	—	2	2	no	—
48	#17984542	—	4	2	no	—
49	#17984542	C₂×C₄	1	8	yes	—
50	#17984542	Q₈	1	4	yes	complemented
51	C₄×Dic₂₇₇	—	2	4	no	—
52	C₄×Dic₂₇₇	—	2	2	no	—
53	C₄×Dic₂₇₇	—	4	2	no	—
54	C₄×Dic₂₇₇	—	4	2	no	—
55	C₄×Dic₂₇₇	C₂×C₄	1	4	yes	—
56	C₂²×C₄×C₂₇₇	C₂³	1	2	yes	—
57	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₂₇₇	C₂³	1	2	yes	—
59	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₂₇₇	—	2	2	no	—
62	C₄⋊C₄×C₂₇₇	—	4	2	no	—
63	C₄⋊C₄×C₂₇₇	C₂×C₄	1	4	yes	—
64	C₄²×C₂₇₇	—	2	2	no	—
65	Dic₅₅₄	—	4	8	no	—
66	C₂×Dic₂₇₇	—	2	2	no	—
67	C₂×Dic₂₇₇	—	2	1	no	—
68	C₂×Dic₂₇₇	—	2	1	no	—
69	C₂×Dic₂₇₇	—	2	2	no	—
70	C₂×Dic₂₇₇	—	2	2	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	—	2	1	no	—
77	C₂×Dic₂₇₇	C₂²×C₄	1	2	yes	—
78	C₂×Dic₂₇₇	C₂²×C₄	1	4	yes	—
79	C₂×Dic₂₇₇	C₂²×C₄	1	2	yes	—
80	C₂×Dic₂₇₇	C₂×Q₈	1	2	yes	—
81	C₂×Dic₂₇₇	C₂×Q₈	1	4	yes	—
82	C₂×Dic₂₇₇	C₂×Q₈	1	2	yes	—
83	C₂×Dic₂₇₇	#41	1	2	yes	—
84	C₂³×C₂₇₇	C₂⁴	1	1	yes	characteristic
85	C₂×C₄×C₂₇₇	—	2	1	no	—
86	C₂×C₄×C₂₇₇	—	2	2	no	—
87	C₂×C₄×C₂₇₇	—	2	1	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	2	no	—
92	C₂×C₄×C₂₇₇	—	2	1	no	—
93	C₂×C₄×C₂₇₇	—	2	1	no	—
94	C₂×C₄×C₂₇₇	—	2	2	no	—
95	C₂×C₄×C₂₇₇	C₂²×C₄	1	4	yes	—
96	C₂×C₄×C₂₇₇	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₂₇₇	C₂×Q₈	1	2	yes	—
98	C₂×C₄×C₂₇₇	C₂×Q₈	1	2	yes	—
99	C₂×C₄×C₂₇₇	#41	1	2	yes	—
100	C₂²×C₂₇₇	C₂³×C₄	1	1	yes	characteristic
101	C₂²×C₂₇₇	C₂²×Q₈	1	1	yes	characteristic
102	C₂²×C₂₇₇	C₂²×Q₈	1	1	yes	characteristic
103	C₂²×C₂₇₇	C₂×#41	1	1	yes	characteristic
104	C₂²×C₂₇₇	C₂×#41	1	1	yes	characteristic
105	C₂²×C₂₇₇	H(5,𝔽₂)	1	1	yes	characteristic
106	C₂²×C₂₇₇	H(5,𝔽₂)	1	1	yes	characteristic
107	C₄×C₂₇₇	—	2	4	no	—
108	C₄×C₂₇₇	—	2	2	no	—
109	C₄×C₂₇₇	—	4	1	no	—
110	C₄×C₂₇₇	—	4	2	no	—
111	C₄×C₂₇₇	C₄×Q₈	1	4	yes	—
112	Dic₂₇₇	—	2	4	no	—
113	Dic₂₇₇	—	2	2	no	—
114	Dic₂₇₇	—	2	4	no	—
115	Dic₂₇₇	—	2	2	no	—
116	Dic₂₇₇	—	4	1	no	—
117	Dic₂₇₇	C₄×Q₈	1	4	yes	—
118	C₂×C₂₇₇	C₂×C₄×Q₈	1	1	yes	characteristic
119	C₂×C₂₇₇	C₂×C₄×Q₈	1	1	yes	characteristic
120	C₂×C₂₇₇	#518	1	1	yes	characteristic
121	C₂×C₂₇₇	#554	1	1	yes	characteristic
122	C₂×C₂₇₇	#554	1	1	yes	characteristic
123	C₂×C₂₇₇	#555	1	1	yes	characteristic
124	C₂×C₂₇₇	Q₈²	1	1	yes	characteristic
125	C₂₇₇	#2369	1	1	yes	characteristic, complemented, Sylow
126	#2369	—	277	1	no	maximal, Sylow
127	C₂×C₄⋊Q₈	—	277	1	no	—
128	#391	—	277	2	no	—
129	#391	—	277	2	no	—
130	#389	—	277	2	no	—
131	#389	—	277	1	no	—
132	#389	D₂₇₇	1	1	yes	characteristic
133	C₄×C₄⋊C₄	—	277	1	no	—
134	C₄×C₄⋊C₄	—	277	1	no	—
135	C₄×C₄⋊C₄	—	277	1	no	—
136	C₄×C₄⋊C₄	—	277	1	no	—
137	C₄×C₄⋊C₄	—	277	2	no	—
138	C₂²×Q₈	—	277	2	no	—
139	C₄⋊Q₈	—	277	8	no	—
140	C₂×C₄⋊C₄	—	277	2	no	—
141	C₂×C₄⋊C₄	—	277	1	no	—
142	C₂×C₄⋊C₄	—	277	1	no	—
143	C₂×C₄⋊C₄	—	277	1	no	—
144	C₂×C₄⋊C₄	—	277	1	no	—
145	C₂×C₄⋊C₄	—	277	2	no	—
146	C₂×C₄⋊C₄	C₂×D₂₇₇	1	1	yes	characteristic
147	C₂×C₄⋊C₄	C₂×D₂₇₇	1	2	yes	—
148	C₂×C₄⋊C₄	C₂×D₂₇₇	1	1	yes	characteristic
149	C₂×C₄²	—	277	1	no	—
150	C₂×C₄²	—	277	2	no	—
151	C₂×C₄²	—	277	1	no	—
152	C₂×C₄²	—	277	2	no	—
153	C₂×C₄²	—	277	2	no	—
154	C₂×C₄²	C₂×D₂₇₇	1	1	yes	characteristic
155	#95	—	277	2	no	—
156	#95	—	277	2	no	—
157	#95	—	277	2	no	—
158	#95	—	277	1	no	—
159	#95	—	277	2	no	—
160	#95	—	277	1	no	—
161	#95	C₂×D₂₇₇	1	2	yes	—
162	C₂×Q₈	—	277	8	no	—
163	C₂×Q₈	—	1108	2	no	—
164	C₂×Q₈	—	1108	2	no	—
165	C₂²×C₄	—	277	1	no	—
166	C₂²×C₄	—	277	1	no	—
167	C₂²×C₄	—	277	2	no	—
168	C₂²×C₄	—	277	1	no	—
169	C₂²×C₄	—	277	2	no	—
170	C₂²×C₄	—	277	1	no	—
171	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
172	C₂²×C₄	C₂²×D₂₇₇	1	2	yes	—
173	C₂²×C₄	C₂²×D₂₇₇	1	2	yes	—
174	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
175	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	277	8	no	—
179	C₄⋊C₄	—	277	4	no	—
180	C₄⋊C₄	—	277	4	no	—
181	C₄⋊C₄	—	277	4	no	—
182	C₄⋊C₄	—	554	2	no	—
183	C₄⋊C₄	—	1108	2	no	—
184	C₄⋊C₄	C₄×D₂₇₇	1	4	yes	—
185	C₄²	—	2	2	no	—
186	C₄²	—	277	4	no	—
187	C₄²	—	554	4	no	—
188	C₄²	—	554	2	no	—
189	C₄²	—	1108	2	no	—
190	C₄²	—	1108	2	no	—
191	C₂³	C₂³×D₂₇₇	1	1	yes	characteristic
192	Q₈	—	1108	8	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	277	2	no	—
204	C₂×C₄	—	277	2	no	—
205	C₂×C₄	—	277	2	no	—
206	C₂×C₄	—	277	2	no	—
207	C₂×C₄	—	277	4	no	—
208	C₂×C₄	—	277	4	no	—
209	C₂×C₄	—	277	2	no	—
210	C₂×C₄	—	554	2	no	—
211	C₂×C₄	—	554	1	no	—
212	C₂×C₄	—	554	2	no	—
213	C₂×C₄	—	554	1	no	—
214	C₂×C₄	—	554	2	no	—
215	C₂×C₄	—	554	1	no	—
216	C₂×C₄	—	554	2	no	—
217	C₂×C₄	—	554	1	no	—
218	C₂×C₄	—	554	1	no	—
219	C₂×C₄	—	554	1	no	—
220	C₂×C₄	—	554	1	no	—
221	C₂×C₄	C₂×C₄×D₂₇₇	1	4	yes	—
222	C₂×C₄	C₂×C₄×D₂₇₇	1	2	yes	—
223	C₂×C₄	Q₈×D₂₇₇	1	2	yes	—
224	C₂×C₄	Q₈×D₂₇₇	1	2	yes	—
225	C₂×C₄	#17984562	1	2	yes	—
226	C₂²	C₂²×C₄×D₂₇₇	1	1	yes	characteristic
227	C₂²	C₂×Q₈×D₂₇₇	1	1	yes	characteristic
228	C₂²	C₂×Q₈×D₂₇₇	1	1	yes	characteristic
229	C₂²	#41×D₂₇₇	1	1	yes	characteristic
230	C₂²	C₂×#17984562	1	1	yes	characteristic
231	C₂²	#22579925	1	1	yes	characteristic
232	C₂²	#22579926	1	1	yes	characteristic
233	C₄	—	2	2	no	—
234	C₄	—	2	4	no	—
235	C₄	—	4	2	no	—
236	C₄	—	4	1	no	—
237	C₄	—	277	4	no	—
238	C₄	—	554	4	no	—
239	C₄	—	554	2	no	—
240	C₄	—	554	2	no	—
241	C₄	—	554	4	no	—
242	C₄	—	1108	1	no	—
243	C₄	#22579845	1	4	yes	—
244	C₂	C₄×Q₈×D₂₇₇	1	1	yes	characteristic
245	C₂	C₂×#22579845	1	1	yes	characteristic
246	C₂	#31800474	1	1	yes	characteristic
247	C₂	#31800708	1	1	yes	characteristic
248	C₂	#31800710	1	1	yes	characteristic
249	C₂	#31800716	1	1	yes	characteristic
250	C₂	#31800732	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	#2369
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
13. #22579879	C₄	semidirect
46. C₄⋊Dic₂₇₇	Q₈	semidirect
50. #17984542	Q₈	semidirect
125. C₂₇₇	#2369	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22579879	2
#31799907	2
#31799906	2
#31799898	2
#31799897	2
C₄⋊C₄×Dic₂₇₇	2
C₄×#17984542	2
C₄×C₄⋊Dic₂₇₇	2
#31799853	2
#389×C₂₇₇	2
#2369	277