Subgroups — #47985774

Order 35456.

484 conjugacy classes of subgroups.

The subgroup lattice has 276 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
2	#31799896	C₂	1	2	yes	maximal
3	#31799890	C₂	1	2	yes	maximal
4	#31799887	C₂	1	2	yes	maximal, complemented
5	#31799885	C₂	1	2	yes	maximal, complemented
6	#31799871	C₂	1	1	yes	maximal, characteristic, complemented
7	#31799867	C₂	1	1	yes	maximal, characteristic, complemented
8	#31799862	C₂	1	1	yes	maximal, characteristic
9	C₄⋊C₄×Dic₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄⋊C₄×Dic₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#380×C₂₇₇	C₂	1	1	yes	maximal, characteristic
12	#22579839	C₄	1	8	yes	complemented
13	C₂×C₄⋊Dic₂₇₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₂₇₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₂₇₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₂₇₇	C₂²	1	1	yes	characteristic
17	C₂×#17984542	C₂²	1	2	yes	—
18	C₂×#17984542	C₂²	1	2	yes	—
19	C₂×C₂²⋊Dic₂₇₇	C₂²	1	2	yes	—
20	C₂×C₂²⋊Dic₂₇₇	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊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	C₂×C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
26	#22579787	C₂²	1	2	yes	—
27	#22579787	C₂²	1	2	yes	—
28	#22579787	C₂²	1	2	yes	—
29	#22579787	C₂²	1	2	yes	—
30	#22579787	C₂²	1	2	yes	—
31	C₂³×C₄×C₂₇₇	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₂₇₇	C₂²	1	2	yes	—
35	#95×C₂₇₇	C₂²	1	2	yes	—
36	C₂²×Dic₂₇₇	C₂³	1	2	yes	—
37	C₂²×Dic₂₇₇	C₂³	1	2	yes	—
38	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
39	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
40	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
41	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
42	C₄⋊Dic₂₇₇	—	2	2	no	—
43	C₄⋊Dic₂₇₇	—	2	2	no	—
44	C₄⋊Dic₂₇₇	C₂×C₄	1	4	yes	—
45	C₄⋊Dic₂₇₇	C₂×C₄	1	4	yes	—
46	#17984542	—	4	2	no	—
47	#17984542	—	4	2	no	—
48	C₂²⋊Dic₂₇₇	—	4	2	no	—
49	C₂²⋊Dic₂₇₇	C₂×C₄	1	4	yes	—
50	C₂²⋊Dic₂₇₇	C₂×C₄	1	4	yes	—
51	C₄×Dic₂₇₇	—	4	2	no	—
52	C₄×Dic₂₇₇	—	4	2	no	—
53	C₄×Dic₂₇₇	C₂×C₄	1	4	yes	—
54	C₄×Dic₂₇₇	C₂×C₄	1	4	yes	—
55	C₂⁴×C₂₇₇	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₂₇₇	—	2	2	no	—
57	C₂²×C₄×C₂₇₇	—	2	2	no	—
58	C₂²×C₄×C₂₇₇	C₂×C₄	1	4	yes	—
59	C₂²×C₄×C₂₇₇	C₂³	1	2	yes	—
60	C₂²×C₄×C₂₇₇	C₂³	1	2	yes	—
61	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₂₇₇	—	2	2	no	—
64	C₄⋊C₄×C₂₇₇	—	2	2	no	—
65	C₂²⋊C₄×C₂₇₇	—	4	2	no	—
66	C₂×Dic₂₇₇	—	2	1	no	—
67	C₂×Dic₂₇₇	—	2	1	no	—
68	C₂×Dic₂₇₇	—	2	1	no	—
69	C₂×Dic₂₇₇	—	2	1	no	—
70	C₂×Dic₂₇₇	—	2	1	no	—
71	C₂×Dic₂₇₇	—	2	1	no	—
72	C₂×Dic₂₇₇	—	2	1	no	—
73	C₂×Dic₂₇₇	—	2	1	no	—
74	C₂×Dic₂₇₇	—	2	2	no	—
75	C₂×Dic₂₇₇	—	2	2	no	—
76	C₂×Dic₂₇₇	—	2	2	no	—
77	C₂×Dic₂₇₇	—	2	2	no	—
78	C₂×Dic₂₇₇	—	2	2	no	—
79	C₂×Dic₂₇₇	—	2	2	no	—
80	C₂×Dic₂₇₇	C₂²×C₄	1	2	yes	—
81	C₂×Dic₂₇₇	C₂²×C₄	1	2	yes	—
82	C₂×Dic₂₇₇	C₂²×C₄	1	2	yes	—
83	C₂×Dic₂₇₇	C₂²×C₄	1	2	yes	—
84	C₂³×C₂₇₇	—	2	1	no	—
85	C₂³×C₂₇₇	—	2	1	no	—
86	C₂³×C₂₇₇	—	2	1	no	—
87	C₂³×C₂₇₇	—	2	1	no	—
88	C₂³×C₂₇₇	—	2	1	no	—
89	C₂³×C₂₇₇	—	2	1	no	—
90	C₂³×C₂₇₇	C₂²×C₄	1	2	yes	—
91	C₂³×C₂₇₇	C₂⁴	1	1	yes	characteristic
92	C₂×C₄×C₂₇₇	—	2	1	no	—
93	C₂×C₄×C₂₇₇	—	2	1	no	—
94	C₂×C₄×C₂₇₇	—	2	8	no	—
95	C₂×C₄×C₂₇₇	—	2	2	no	—
96	C₂×C₄×C₂₇₇	—	2	2	no	—
97	C₂×C₄×C₂₇₇	—	2	2	no	—
98	C₂×C₄×C₂₇₇	—	2	2	no	—
99	C₂×C₄×C₂₇₇	—	2	2	no	—
100	C₂×C₄×C₂₇₇	—	2	2	no	—
101	C₂×C₄×C₂₇₇	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₂₇₇	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₂₇₇	#41	1	2	yes	—
104	C₂×C₄×C₂₇₇	#41	1	2	yes	—
105	C₂²×C₂₇₇	—	2	2	no	—
106	C₂²×C₂₇₇	—	2	2	no	—
107	C₂²×C₂₇₇	—	2	2	no	—
108	C₂²×C₂₇₇	—	4	1	no	—
109	C₂²×C₂₇₇	—	4	1	no	—
110	C₂²×C₂₇₇	—	4	1	no	—
111	C₂²×C₂₇₇	—	4	1	no	—
112	C₂²×C₂₇₇	C₂³×C₄	1	1	yes	characteristic
113	C₂²×C₂₇₇	C₂×#41	1	1	yes	characteristic
114	C₂²×C₂₇₇	C₂×#41	1	1	yes	characteristic
115	C₂²×C₂₇₇	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂²×C₂₇₇	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂²×C₂₇₇	2₋¹⁺⁴	1	1	yes	characteristic
118	C₂²×C₂₇₇	2₋¹⁺⁴	1	1	yes	characteristic
119	C₄×C₂₇₇	—	2	2	no	—
120	C₄×C₂₇₇	—	4	2	no	—
121	C₄×C₂₇₇	—	4	2	no	—
122	C₄×C₂₇₇	#117	1	4	yes	—
123	Dic₂₇₇	—	2	2	no	—
124	Dic₂₇₇	—	2	2	no	—
125	Dic₂₇₇	—	2	2	no	—
126	Dic₂₇₇	—	2	2	no	—
127	Dic₂₇₇	—	4	2	no	—
128	Dic₂₇₇	—	4	2	no	—
129	C₂×C₂₇₇	—	4	2	no	—
130	C₂×C₂₇₇	C₂×#117	1	1	yes	characteristic
131	C₂×C₂₇₇	#520	1	1	yes	characteristic
132	C₂×C₂₇₇	#520	1	1	yes	characteristic
133	C₂×C₂₇₇	#540	1	1	yes	characteristic
134	C₂×C₂₇₇	#541	1	1	yes	characteristic
135	C₂×C₂₇₇	#542	1	1	yes	characteristic
136	C₂×C₂₇₇	#542	1	1	yes	characteristic
137	C₂₇₇	#2314	1	1	yes	characteristic, complemented, Sylow
138	#2314	—	277	1	no	maximal, Sylow
139	C₂×#117	—	277	1	no	—
140	#388	—	277	2	no	—
141	#388	—	277	2	no	—
142	#387	—	277	2	no	—
143	#387	—	277	2	no	—
144	#384	—	277	1	no	—
145	#382	—	277	1	no	—
146	#380	—	277	1	no	—
147	#380	D₂₇₇	1	1	yes	characteristic
148	C₄×C₄⋊C₄	—	277	1	no	—
149	C₄×C₄⋊C₄	—	277	1	no	—
150	C₂³×C₄	C₂×D₂₇₇	1	1	yes	characteristic
151	#117	—	277	8	no	—
152	C₂×C₄⋊C₄	—	277	1	no	—
153	C₂×C₄⋊C₄	—	277	1	no	—
154	C₂×C₄⋊C₄	—	277	1	no	—
155	C₂×C₄⋊C₄	—	277	1	no	—
156	C₂×C₄⋊C₄	—	277	2	no	—
157	C₂×C₄⋊C₄	—	277	2	no	—
158	C₂×C₄⋊C₄	C₂×D₂₇₇	1	1	yes	characteristic
159	C₂×C₄⋊C₄	C₂×D₂₇₇	1	1	yes	characteristic
160	C₂×C₂²⋊C₄	—	277	1	no	—
161	C₂×C₂²⋊C₄	—	277	1	no	—
162	C₂×C₂²⋊C₄	—	277	2	no	—
163	C₂×C₂²⋊C₄	C₂×D₂₇₇	1	2	yes	—
164	C₂×C₄²	—	277	1	no	—
165	C₂×C₄²	—	277	1	no	—
166	C₂×C₄²	—	277	2	no	—
167	C₂×C₄²	—	277	2	no	—
168	#95	—	277	2	no	—
169	#95	—	277	2	no	—
170	#95	—	277	2	no	—
171	#95	—	277	2	no	—
172	#95	—	277	2	no	—
173	#95	C₂×D₂₇₇	1	2	yes	—
174	C₂⁴	C₂²×D₂₇₇	1	1	yes	characteristic
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	277	1	no	—
178	C₂²×C₄	—	277	1	no	—
179	C₂²×C₄	—	277	1	no	—
180	C₂²×C₄	—	277	1	no	—
181	C₂²×C₄	—	277	2	no	—
182	C₂²×C₄	—	277	2	no	—
183	C₂²×C₄	C₄×D₂₇₇	1	4	yes	—
184	C₂²×C₄	C₂²×D₂₇₇	1	2	yes	—
185	C₂²×C₄	C₂²×D₂₇₇	1	2	yes	—
186	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
187	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	277	4	no	—
191	C₄⋊C₄	—	277	4	no	—
192	C₄⋊C₄	—	554	2	no	—
193	C₄⋊C₄	—	554	2	no	—
194	C₄⋊C₄	—	1108	2	no	—
195	C₄⋊C₄	—	1108	2	no	—
196	C₂²⋊C₄	—	4	2	no	—
197	C₂²⋊C₄	—	277	4	no	—
198	C₂²⋊C₄	—	277	4	no	—
199	C₂²⋊C₄	—	1108	2	no	—
200	C₄²	—	277	4	no	—
201	C₄²	—	277	4	no	—
202	C₄²	—	1108	2	no	—
203	C₄²	—	1108	2	no	—
204	C₂³	—	2	1	no	—
205	C₂³	—	2	1	no	—
206	C₂³	—	2	1	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	2	1	no	—
209	C₂³	—	2	1	no	—
210	C₂³	C₂×C₄×D₂₇₇	1	2	yes	—
211	C₂³	C₂³×D₂₇₇	1	1	yes	characteristic
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	277	2	no	—
222	C₂×C₄	—	277	2	no	—
223	C₂×C₄	—	277	2	no	—
224	C₂×C₄	—	277	2	no	—
225	C₂×C₄	—	554	1	no	—
226	C₂×C₄	—	554	1	no	—
227	C₂×C₄	—	554	1	no	—
228	C₂×C₄	—	554	1	no	—
229	C₂×C₄	—	554	1	no	—
230	C₂×C₄	—	554	1	no	—
231	C₂×C₄	—	554	1	no	—
232	C₂×C₄	—	554	1	no	—
233	C₂×C₄	—	554	2	no	—
234	C₂×C₄	—	554	2	no	—
235	C₂×C₄	—	554	2	no	—
236	C₂×C₄	—	554	2	no	—
237	C₂×C₄	—	554	2	no	—
238	C₂×C₄	—	554	2	no	—
239	C₂×C₄	C₂×C₄×D₂₇₇	1	2	yes	—
240	C₂×C₄	C₂×C₄×D₂₇₇	1	2	yes	—
241	C₂×C₄	#17984562	1	2	yes	—
242	C₂×C₄	#17984562	1	2	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	1	no	—
250	C₂²	C₂²×C₄×D₂₇₇	1	1	yes	characteristic
251	C₂²	C₂×#17984562	1	1	yes	characteristic
252	C₂²	C₂×#17984562	1	1	yes	characteristic
253	C₂²	#22579925	1	1	yes	characteristic
254	C₂²	#22579926	1	1	yes	characteristic
255	C₂²	#22579927	1	1	yes	characteristic
256	C₂²	#22579928	1	1	yes	characteristic
257	C₄	—	2	2	no	—
258	C₄	—	4	2	no	—
259	C₄	—	4	2	no	—
260	C₄	—	554	2	no	—
261	C₄	—	554	2	no	—
262	C₄	—	554	2	no	—
263	C₄	—	554	2	no	—
264	C₄	—	1108	2	no	—
265	C₄	—	1108	2	no	—
266	C₄	#22579836	1	4	yes	—
267	C₂	—	4	2	no	—
268	C₂	C₂×#22579836	1	1	yes	characteristic
269	C₂	#31800482	1	1	yes	characteristic
270	C₂	#31800483	1	1	yes	characteristic
271	C₂	#31800605	1	1	yes	characteristic
272	C₂	#31800612	1	1	yes	characteristic
273	C₂	#31800619	1	1	yes	characteristic
274	C₂	#31800625	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₇₇
commutator	C₂²×C₂₇₇	C₂³×C₄
Fitting	#380×C₂₇₇	C₂
socle, cosocle	C₂³×C₂₇₇	C₂⁴
prime radical	C₂⁴×C₂₇₇	C₂³
hypercenter	#380	D₂₇₇
hypocenter	C₂₇₇	#2314
LCS	G ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂³ ◁ #380

Complements

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

N	H ≅ G/N	type
4. #31799887	C₂	semidirect
5. #31799885	C₂	semidirect
6. #31799871	C₂	semidirect
7. #31799867	C₂	semidirect
9. C₄⋊C₄×Dic₂₇₇	C₂	semidirect
10. C₄⋊C₄×Dic₂₇₇	C₂	semidirect
12. #22579839	C₄	semidirect
137. C₂₇₇	#2314	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
#31799896	2
#31799890	2
#31799887	2
#31799885	2
#31799871	2
#31799867	2
#31799862	2
C₄⋊C₄×Dic₂₇₇	2
#380×C₂₇₇	2
#2314	277