Subgroups — #47986242

Order 35456.

632 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×Dic₂₇₇	C₂	1	1	yes	maximal, characteristic
3	#31799909	C₂	1	1	yes	maximal, characteristic, complemented
4	#31799905	C₂	1	1	yes	maximal, characteristic
5	#31799895	C₂	1	2	yes	maximal, complemented
6	#31799895	C₂	1	2	yes	maximal, complemented
7	#31799877	C₂	1	2	yes	maximal, complemented
8	C₄×C₄⋊Dic₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₄⋊Dic₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²⋊C₄×Dic₂₇₇	C₂	1	2	yes	maximal, complemented
11	C₂×#124×C₂₇₇	C₂	1	1	yes	maximal, characteristic
12	C₂³×Dic₂₇₇	C₂²	1	2	yes	—
13	Q₈×Dic₂₇₇	—	2	4	no	—
14	D₄×Dic₂₇₇	—	2	4	no	—
15	C₂×C₄⋊Dic₂₇₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₂₇₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₂₇₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₂₇₇	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₂₇₇	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊Dic₂₇₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊Dic₂₇₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊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	C₂×C₄×Dic₂₇₇	C₂²	1	2	yes	—
27	#22579787	C₂²	1	1	yes	characteristic
28	#22579787	C₂²	1	2	yes	—
29	#22579787	C₂²	1	2	yes	—
30	#22579787	C₂²	1	1	yes	characteristic
31	#22579787	C₂²	1	2	yes	—
32	C₂²×Q₈×C₂₇₇	C₂²	1	1	yes	characteristic
33	C₂²×D₄×C₂₇₇	C₂²	1	1	yes	characteristic
34	#124×C₂₇₇	C₄	1	8	yes	complemented
35	C₂×C₂²⋊C₄×C₂₇₇	C₂²	1	2	yes	—
36	C₂×C₂²⋊C₄×C₂₇₇	C₂²	1	2	yes	—
37	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
38	C₂²×Dic₂₇₇	—	2	4	no	—
39	C₂²×Dic₂₇₇	—	2	4	no	—
40	C₂²×Dic₂₇₇	—	4	2	no	—
41	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
42	C₂²×Dic₂₇₇	C₂³	1	2	yes	—
43	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
44	C₂²×Dic₂₇₇	C₂³	1	2	yes	—
45	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
46	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
47	C₄⋊Dic₂₇₇	—	2	4	no	—
48	C₄⋊Dic₂₇₇	—	2	2	no	—
49	C₄⋊Dic₂₇₇	—	2	2	no	—
50	C₄⋊Dic₂₇₇	—	2	2	no	—
51	C₄⋊Dic₂₇₇	D₄	1	4	yes	—
52	C₂²⋊Dic₂₇₇	—	2	4	no	—
53	C₂²⋊Dic₂₇₇	—	2	4	no	—
54	C₂²⋊Dic₂₇₇	—	4	2	no	—
55	C₄×Dic₂₇₇	—	2	4	no	—
56	C₄×Dic₂₇₇	—	2	2	no	—
57	C₄×Dic₂₇₇	—	2	2	no	—
58	C₄×Dic₂₇₇	—	4	2	no	—
59	C₂⁴×C₂₇₇	C₂³	1	2	yes	—
60	C₂×Q₈×C₂₇₇	—	2	2	no	—
61	C₂×Q₈×C₂₇₇	—	2	2	no	—
62	C₂×Q₈×C₂₇₇	C₂×C₄	1	4	yes	complemented
63	C₂×D₄×C₂₇₇	—	2	2	no	—
64	C₂×D₄×C₂₇₇	—	2	2	no	—
65	C₂×D₄×C₂₇₇	C₂×C₄	1	4	yes	—
66	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₂₇₇	C₂³	1	2	yes	—
69	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
70	C₂²⋊C₄×C₂₇₇	C₂×C₄	1	8	yes	complemented
71	C₂²⋊C₄×C₂₇₇	C₂×C₄	1	8	yes	complemented
72	C₄²×C₂₇₇	C₂×C₄	1	4	yes	complemented
73	C₂×Dic₂₇₇	—	2	1	no	—
74	C₂×Dic₂₇₇	—	2	1	no	—
75	C₂×Dic₂₇₇	—	2	2	no	—
76	C₂×Dic₂₇₇	—	2	2	no	—
77	C₂×Dic₂₇₇	—	2	1	no	—
78	C₂×Dic₂₇₇	—	2	1	no	—
79	C₂×Dic₂₇₇	—	2	1	no	—
80	C₂×Dic₂₇₇	—	2	1	no	—
81	C₂×Dic₂₇₇	—	2	2	no	—
82	C₂×Dic₂₇₇	—	2	1	no	—
83	C₂×Dic₂₇₇	—	2	1	no	—
84	C₂×Dic₂₇₇	—	2	2	no	—
85	C₂×Dic₂₇₇	—	2	2	no	—
86	C₂×Dic₂₇₇	—	2	1	no	—
87	C₂×Dic₂₇₇	—	2	1	no	—
88	C₂×Dic₂₇₇	—	4	8	no	—
89	C₂×Dic₂₇₇	C₂×D₄	1	4	yes	—
90	C₂×Dic₂₇₇	#41	1	2	yes	—
91	C₂×Dic₂₇₇	#41	1	2	yes	—
92	C₂³×C₂₇₇	—	2	2	no	—
93	C₂³×C₂₇₇	—	2	2	no	—
94	C₂³×C₂₇₇	—	2	2	no	—
95	C₂³×C₂₇₇	—	2	2	no	—
96	C₂³×C₂₇₇	—	2	2	no	—
97	C₂³×C₂₇₇	—	2	2	no	—
98	C₂³×C₂₇₇	C₂²×C₄	1	4	yes	—
99	C₂³×C₂₇₇	C₂⁴	1	1	yes	characteristic
100	Q₈×C₂₇₇	—	2	8	no	—
101	D₄×C₂₇₇	—	2	8	no	—
102	C₂×C₄×C₂₇₇	—	2	2	no	—
103	C₂×C₄×C₂₇₇	—	2	2	no	—
104	C₂×C₄×C₂₇₇	—	2	1	no	—
105	C₂×C₄×C₂₇₇	—	2	1	no	—
106	C₂×C₄×C₂₇₇	—	2	1	no	—
107	C₂×C₄×C₂₇₇	—	2	1	no	—
108	C₂×C₄×C₂₇₇	C₂²×C₄	1	2	yes	—
109	C₂×C₄×C₂₇₇	C₂²×C₄	1	2	yes	—
110	C₂×C₄×C₂₇₇	C₂²×C₄	1	4	yes	—
111	C₂×C₄×C₂₇₇	C₂²×C₄	1	2	yes	—
112	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
113	C₂×C₄×C₂₇₇	#41	1	2	yes	—
114	C₂²×C₂₇₇	—	2	4	no	—
115	C₂²×C₂₇₇	—	2	4	no	—
116	C₂²×C₂₇₇	—	2	4	no	—
117	C₂²×C₂₇₇	—	4	2	no	—
118	C₂²×C₂₇₇	—	4	2	no	—
119	C₂²×C₂₇₇	—	4	2	no	—
120	C₂²×C₂₇₇	—	4	2	no	—
121	C₂²×C₂₇₇	C₂³×C₄	1	1	yes	characteristic
122	C₂²×C₂₇₇	C₂²×D₄	1	1	yes	characteristic
123	C₂²×C₂₇₇	C₂×#41	1	1	yes	characteristic
124	C₂²×C₂₇₇	C₂×#41	1	1	yes	characteristic
125	C₂²×C₂₇₇	C₂×#41	1	1	yes	characteristic
126	C₂²×C₂₇₇	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂²×C₂₇₇	2₋¹⁺⁴	1	1	yes	characteristic
128	C₄×C₂₇₇	—	2	4	no	—
129	C₄×C₂₇₇	—	2	2	no	—
130	C₄×C₂₇₇	—	2	2	no	—
131	C₄×C₂₇₇	C₄×D₄	1	4	yes	—
132	Dic₂₇₇	—	2	2	no	—
133	Dic₂₇₇	—	2	4	no	—
134	Dic₂₇₇	—	2	2	no	—
135	Dic₂₇₇	—	4	1	no	—
136	Dic₂₇₇	—	4	1	no	—
137	Dic₂₇₇	—	4	2	no	—
138	C₂×C₂₇₇	—	4	4	no	—
139	C₂×C₂₇₇	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂×C₂₇₇	C₄×#41	1	1	yes	characteristic
141	C₂×C₂₇₇	#520	1	1	yes	characteristic
142	C₂×C₂₇₇	#546	1	1	yes	characteristic
143	C₂×C₂₇₇	#548	1	1	yes	characteristic
144	C₂×C₂₇₇	#556	1	1	yes	characteristic
145	C₂×C₂₇₇	#559	1	1	yes	characteristic
146	C₂₇₇	#2367	1	1	yes	characteristic, complemented, Sylow
147	#2367	—	277	1	no	maximal, Sylow
148	C₂×#124	D₂₇₇	1	1	yes	characteristic
149	C₂×C₄×Q₈	—	277	1	no	—
150	C₂×C₄×D₄	—	277	1	no	—
151	#391	—	277	1	no	—
152	#390	—	277	1	no	—
153	#388	—	277	2	no	—
154	#388	—	277	2	no	—
155	#385	—	277	2	no	—
156	C₄×C₄⋊C₄	—	277	1	no	—
157	C₄×C₄⋊C₄	—	277	1	no	—
158	C₄×C₂²⋊C₄	—	277	2	no	—
159	C₂²×Q₈	C₂×D₂₇₇	1	1	yes	characteristic
160	C₂²×D₄	C₂×D₂₇₇	1	1	yes	characteristic
161	C₂³×C₄	—	277	2	no	—
162	#124	Dic₂₇₇	1	8	yes	complemented
163	C₄×Q₈	—	554	4	no	—
164	C₄×D₄	—	554	4	no	—
165	C₂×C₄⋊C₄	—	277	1	no	—
166	C₂×C₄⋊C₄	—	277	2	no	—
167	C₂×C₄⋊C₄	—	277	1	no	—
168	C₂×C₄⋊C₄	—	277	1	no	—
169	C₂×C₄⋊C₄	—	277	1	no	—
170	C₂×C₂²⋊C₄	—	277	2	no	—
171	C₂×C₂²⋊C₄	—	277	2	no	—
172	C₂×C₂²⋊C₄	—	277	2	no	—
173	C₂×C₂²⋊C₄	C₂×D₂₇₇	1	2	yes	—
174	C₂×C₂²⋊C₄	C₂×D₂₇₇	1	2	yes	—
175	C₂×C₄²	—	277	1	no	—
176	C₂×C₄²	—	277	2	no	—
177	C₂×C₄²	—	277	1	no	—
178	C₂×C₄²	—	277	2	no	—
179	C₂×C₄²	C₂×D₂₇₇	1	1	yes	characteristic
180	#95	—	277	1	no	—
181	#95	—	277	2	no	—
182	#95	—	277	2	no	—
183	#95	—	277	1	no	—
184	#95	—	277	2	no	—
185	C₂⁴	C₂²×D₂₇₇	1	2	yes	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	C₂×Dic₂₇₇	1	4	yes	complemented
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	C₂×Dic₂₇₇	1	4	yes	—
192	C₂²×C₄	—	277	1	no	—
193	C₂²×C₄	—	277	1	no	—
194	C₂²×C₄	—	277	2	no	—
195	C₂²×C₄	—	277	1	no	—
196	C₂²×C₄	—	277	2	no	—
197	C₂²×C₄	—	277	1	no	—
198	C₂²×C₄	—	554	4	no	—
199	C₂²×C₄	—	554	4	no	—
200	C₂²×C₄	—	1108	2	no	—
201	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
202	C₂²×C₄	C₂²×D₂₇₇	1	2	yes	—
203	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
205	C₄⋊C₄	—	277	4	no	—
206	C₄⋊C₄	—	554	2	no	—
207	C₄⋊C₄	—	554	2	no	—
208	C₄⋊C₄	—	554	4	no	—
209	C₄⋊C₄	—	554	2	no	—
210	C₂²⋊C₄	—	554	4	no	—
211	C₂²⋊C₄	—	554	4	no	—
212	C₂²⋊C₄	—	1108	2	no	—
213	C₂²⋊C₄	C₂×Dic₂₇₇	1	8	yes	complemented
214	C₂²⋊C₄	C₂×Dic₂₇₇	1	8	yes	complemented
215	C₄²	—	554	4	no	—
216	C₄²	—	554	2	no	—
217	C₄²	—	554	2	no	—
218	C₄²	—	1108	2	no	—
219	C₄²	C₂×Dic₂₇₇	1	4	yes	complemented
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	C₂²×Dic₂₇₇	1	4	yes	—
227	C₂³	C₂³×D₂₇₇	1	1	yes	characteristic
228	Q₈	—	2	8	no	—
229	D₄	—	2	8	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	277	4	no	—
237	C₂×C₄	—	277	2	no	—
238	C₂×C₄	—	277	2	no	—
239	C₂×C₄	—	554	1	no	—
240	C₂×C₄	—	554	1	no	—
241	C₂×C₄	—	554	2	no	—
242	C₂×C₄	—	554	2	no	—
243	C₂×C₄	—	554	1	no	—
244	C₂×C₄	—	554	1	no	—
245	C₂×C₄	—	554	1	no	—
246	C₂×C₄	—	554	1	no	—
247	C₂×C₄	—	554	2	no	—
248	C₂×C₄	—	554	1	no	—
249	C₂×C₄	—	554	1	no	—
250	C₂×C₄	—	554	2	no	—
251	C₂×C₄	—	554	2	no	—
252	C₂×C₄	—	554	1	no	—
253	C₂×C₄	—	554	1	no	—
254	C₂×C₄	—	1108	8	no	—
255	C₂×C₄	C₂²×Dic₂₇₇	1	2	yes	—
256	C₂×C₄	C₂²×Dic₂₇₇	1	2	yes	—
257	C₂×C₄	C₂²×Dic₂₇₇	1	4	yes	—
258	C₂×C₄	C₂²×Dic₂₇₇	1	2	yes	—
259	C₂×C₄	D₄×D₂₇₇	1	2	yes	—
260	C₂×C₄	#17984562	1	2	yes	—
261	C₂²	—	2	4	no	—
262	C₂²	—	2	4	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	C₂³×Dic₂₇₇	1	1	yes	characteristic
269	C₂²	C₂×D₄×D₂₇₇	1	1	yes	characteristic
270	C₂²	#41×D₂₇₇	1	1	yes	characteristic
271	C₂²	#41×D₂₇₇	1	1	yes	characteristic
272	C₂²	C₂×#17984562	1	1	yes	characteristic
273	C₂²	#22579926	1	1	yes	characteristic
274	C₂²	#22579927	1	1	yes	characteristic
275	C₄	—	2	4	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	554	2	no	—
279	C₄	—	554	4	no	—
280	C₄	—	554	2	no	—
281	C₄	—	1108	1	no	—
282	C₄	—	1108	1	no	—
283	C₄	—	1108	2	no	—
284	C₄	D₄×Dic₂₇₇	1	4	yes	—
285	C₂	—	4	4	no	—
286	C₂	C₂×D₄×Dic₂₇₇	1	1	yes	characteristic
287	C₂	#41×Dic₂₇₇	1	1	yes	characteristic
288	C₂	#31800485	1	1	yes	characteristic
289	C₂	#31800645	1	1	yes	characteristic
290	C₂	#31800660	1	1	yes	characteristic
291	C₂	#31800721	1	1	yes	characteristic
292	C₂	#31800734	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₇₇
commutator	C₂²×C₂₇₇	C₂³×C₄
Fitting	C₂×#124×C₂₇₇	C₂
socle, cosocle	C₂³×C₂₇₇	C₂⁴
prime radical	C₂²×D₄×C₂₇₇	C₂²
hypercenter	C₂×#124	D₂₇₇
hypocenter	C₂₇₇	#2367
LCS	G ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂³ ◁ C₂×#124

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×Dic₂₇₇	C₂	semidirect
3. #31799909	C₂	semidirect
5. #31799895	C₂	semidirect
6. #31799895	C₂	semidirect
7. #31799877	C₂	semidirect
8. C₄×C₄⋊Dic₂₇₇	C₂	semidirect
9. C₄×C₄⋊Dic₂₇₇	C₂	semidirect
10. C₂²⋊C₄×Dic₂₇₇	C₂	semidirect
34. #124×C₂₇₇	C₄	semidirect
62. C₂×Q₈×C₂₇₇	C₂×C₄	semidirect
70. C₂²⋊C₄×C₂₇₇	C₂×C₄	semidirect
71. C₂²⋊C₄×C₂₇₇	C₂×C₄	semidirect
72. C₄²×C₂₇₇	C₂×C₄	semidirect
146. C₂₇₇	#2367	semidirect
162. #124	Dic₂₇₇	semidirect
188. C₂×Q₈	C₂×Dic₂₇₇	semidirect
213. C₂²⋊C₄	C₂×Dic₂₇₇	semidirect
214. C₂²⋊C₄	C₂×Dic₂₇₇	semidirect
219. C₄²	C₂×Dic₂₇₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×Dic₂₇₇	2
C₂×D₄×Dic₂₇₇	2
#31799909	2
#31799905	2
#31799895	2
#31799877	2
C₄×C₄⋊Dic₂₇₇	2
C₂²⋊C₄×Dic₂₇₇	2
C₂×#124×C₂₇₇	2
#2367	277