Subgroups — #47986168

Order 35456.

620 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22579856	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₄×D₂₇₇	C₂	1	1	yes	maximal, characteristic
3	#31799909	C₂	1	1	yes	maximal, characteristic, complemented
4	#31799900	C₂	1	1	yes	maximal, characteristic, complemented
5	#31799897	C₂	1	2	yes	maximal, complemented
6	#31799886	C₂	1	2	yes	maximal, complemented
7	#31799873	C₂	1	2	yes	maximal
8	C₄⋊C₄×Dic₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#17984540	C₂	1	2	yes	maximal
10	C₄×#17984540	C₂	1	1	yes	maximal, characteristic
11	C₂×C₄×Q₈×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₂₇₇	C₂²	1	2	yes	—
13	C₂²×C₄×D₂₇₇	C₂²	1	1	yes	characteristic
14	#22579856	C₄	1	8	yes	complemented
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×#17984542	C₂²	1	2	yes	—
20	C₄⋊C₄×D₂₇₇	—	2	4	no	—
21	C₂×#17984540	C₂²	1	1	yes	characteristic
22	C₂×#17984540	C₂²	1	2	yes	—
23	C₂×#17984540	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₂₇₇	C₂²	1	2	yes	—
26	#22579787	C₂²	1	1	yes	characteristic
27	#22579787	C₂²	1	2	yes	—
28	#22579787	C₂²	1	1	yes	characteristic
29	#22579787	C₂²	1	2	yes	—
30	#22579787	C₂²	1	2	yes	—
31	#22579787	C₂²	1	2	yes	—
32	C₂²×Q₈×C₂₇₇	C₂²	1	1	yes	characteristic
33	C₄×Q₈×C₂₇₇	—	2	4	no	—
34	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	2	yes	—
35	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
36	C₂×C₄²×C₂₇₇	C₂²	1	2	yes	—
37	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
38	C₂³×D₂₇₇	C₂³	1	1	yes	characteristic
39	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
40	C₂²×Dic₂₇₇	C₂³	1	2	yes	—
41	C₂²×Dic₂₇₇	C₂³	1	2	yes	—
42	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
43	C₂²×Dic₂₇₇	C₂³	1	1	yes	characteristic
44	C₂×C₄×D₂₇₇	—	2	4	no	—
45	C₂×C₄×D₂₇₇	—	2	2	no	—
46	C₂×C₄×D₂₇₇	—	2	4	no	—
47	C₂×C₄×D₂₇₇	—	2	2	no	—
48	C₂×C₄×D₂₇₇	—	4	2	no	—
49	C₂×C₄×D₂₇₇	C₂×C₄	1	4	yes	—
50	C₄⋊Dic₂₇₇	—	2	2	no	—
51	C₄⋊Dic₂₇₇	C₂×C₄	1	4	yes	complemented
52	#17984542	—	2	4	no	—
53	#17984542	—	4	2	no	—
54	#17984542	C₂×C₄	1	8	yes	complemented
55	#17984540	—	2	2	no	—
56	#17984540	C₂×C₄	1	8	yes	—
57	#17984540	Q₈	1	4	yes	complemented
58	C₄×Dic₂₇₇	—	2	2	no	—
59	C₄×Dic₂₇₇	—	4	2	no	—
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	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₂₇₇	C₂³	1	2	yes	—
65	C₂²×C₄×C₂₇₇	C₂³	1	2	yes	—
66	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₂₇₇	—	2	4	no	—
69	C₄⋊C₄×C₂₇₇	D₄	1	4	yes	—
70	C₄²×C₂₇₇	—	2	4	no	—
71	C₄²×C₂₇₇	—	2	2	no	—
72	C₂²×D₂₇₇	—	2	1	no	—
73	C₂²×D₂₇₇	—	2	1	no	—
74	C₂²×D₂₇₇	—	2	1	no	—
75	C₂²×D₂₇₇	—	2	1	no	—
76	C₂²×D₂₇₇	C₂²×C₄	1	2	yes	—
77	C₂²×D₂₇₇	C₂×Q₈	1	2	yes	—
78	C₂²×D₂₇₇	#41	1	2	yes	—
79	C₂×Dic₂₇₇	—	2	1	no	—
80	C₂×Dic₂₇₇	—	2	1	no	—
81	C₂×Dic₂₇₇	—	2	1	no	—
82	C₂×Dic₂₇₇	—	2	1	no	—
83	C₂×Dic₂₇₇	—	2	2	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	—	2	2	no	—
89	C₂×Dic₂₇₇	—	2	2	no	—
90	C₂×Dic₂₇₇	—	2	1	no	—
91	C₂×Dic₂₇₇	C₂²×C₄	1	4	yes	—
92	C₂×Dic₂₇₇	C₂²×C₄	1	2	yes	—
93	C₂×Dic₂₇₇	C₂×Q₈	1	2	yes	—
94	C₄×D₂₇₇	—	2	8	no	—
95	C₄×D₂₇₇	—	4	8	no	—
96	C₂³×C₂₇₇	C₂⁴	1	1	yes	characteristic
97	Q₈×C₂₇₇	—	2	8	no	—
98	C₂×C₄×C₂₇₇	—	2	1	no	—
99	C₂×C₄×C₂₇₇	—	2	2	no	—
100	C₂×C₄×C₂₇₇	—	2	2	no	—
101	C₂×C₄×C₂₇₇	—	2	2	no	—
102	C₂×C₄×C₂₇₇	—	2	2	no	—
103	C₂×C₄×C₂₇₇	—	2	1	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	C₂²×C₄	1	4	yes	—
108	C₂×C₄×C₂₇₇	C₂²×C₄	1	2	yes	—
109	C₂×C₄×C₂₇₇	C₂×D₄	1	4	yes	—
110	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₂₇₇	C₂×Q₈	1	2	yes	—
112	C₂×C₄×C₂₇₇	#41	1	2	yes	—
113	C₂²×C₂₇₇	C₂³×C₄	1	1	yes	characteristic
114	C₂²×C₂₇₇	C₂²×D₄	1	1	yes	characteristic
115	C₂²×C₂₇₇	C₂²×Q₈	1	1	yes	characteristic
116	C₂²×C₂₇₇	C₂×#41	1	1	yes	characteristic
117	C₂²×C₂₇₇	C₂×#41	1	1	yes	characteristic
118	C₂²×C₂₇₇	H(5,𝔽₂)	1	1	yes	characteristic
119	C₂²×C₂₇₇	2₋¹⁺⁴	1	1	yes	characteristic
120	C₂×D₂₇₇	—	2	2	no	—
121	C₂×D₂₇₇	—	2	2	no	—
122	C₂×D₂₇₇	—	2	2	no	—
123	C₂×D₂₇₇	—	2	2	no	—
124	C₂×D₂₇₇	—	2	2	no	—
125	C₂×D₂₇₇	—	2	2	no	—
126	C₂×D₂₇₇	C₄×Q₈	1	4	yes	complemented
127	C₄×C₂₇₇	—	2	4	no	—
128	C₄×C₂₇₇	—	2	4	no	—
129	C₄×C₂₇₇	—	2	2	no	—
130	C₄×C₂₇₇	—	4	1	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	2	no	—
136	Dic₂₇₇	—	4	1	no	—
137	C₂×C₂₇₇	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂×C₂₇₇	C₂×C₄×Q₈	1	1	yes	characteristic
139	C₂×C₂₇₇	#520	1	1	yes	characteristic
140	C₂×C₂₇₇	#546	1	1	yes	characteristic
141	C₂×C₂₇₇	D₄×Q₈	1	1	yes	characteristic
142	C₂×C₂₇₇	#552	1	1	yes	characteristic
143	C₂×C₂₇₇	#554	1	1	yes	characteristic
144	D₂₇₇	—	2	4	no	—
145	C₂₇₇	#2360	1	1	yes	characteristic, complemented, Sylow
146	#2360	—	277	1	no	maximal, Sylow
147	C₂×C₂²⋊Q₈	—	277	1	no	—
148	C₂×C₄×Q₈	D₂₇₇	1	1	yes	characteristic, complemented
149	C₂²×C₄⋊C₄	—	277	1	no	—
150	#391	—	277	1	no	—
151	#389	—	277	2	no	—
152	#389	—	277	1	no	—
153	#387	—	277	2	no	—
154	#385	—	277	2	no	—
155	C₄×C₄⋊C₄	—	277	1	no	—
156	C₄×C₂²⋊C₄	—	277	1	no	—
157	C₄×C₂²⋊C₄	—	277	2	no	—
158	C₂²×Q₈	C₂×D₂₇₇	1	1	yes	characteristic
159	C₂³×C₄	—	277	1	no	—
160	C₂³×C₄	—	277	2	no	—
161	C₂²⋊Q₈	—	277	8	no	—
162	C₄×Q₈	—	2	4	no	—
163	C₂×C₄⋊C₄	—	277	1	no	—
164	C₂×C₄⋊C₄	—	277	2	no	—
165	C₂×C₄⋊C₄	—	277	2	no	—
166	C₂×C₄⋊C₄	—	277	1	no	—
167	C₂×C₄⋊C₄	—	277	2	no	—
168	C₂×C₄⋊C₄	—	554	4	no	—
169	C₂×C₄⋊C₄	C₂×D₂₇₇	1	2	yes	—
170	C₂×C₄⋊C₄	C₂×D₂₇₇	1	1	yes	characteristic
171	C₂×C₂²⋊C₄	—	277	2	no	—
172	C₂×C₂²⋊C₄	—	277	1	no	—
173	C₂×C₂²⋊C₄	—	277	1	no	—
174	C₂×C₄²	—	277	1	no	—
175	C₂×C₄²	—	277	2	no	—
176	C₂×C₄²	C₂×D₂₇₇	1	2	yes	—
177	C₂×C₄²	C₂×D₂₇₇	1	1	yes	characteristic
178	#95	—	277	1	no	—
179	#95	—	277	2	no	—
180	#95	—	277	1	no	—
181	#95	—	277	2	no	—
182	#95	—	277	2	no	—
183	#95	—	277	2	no	—
184	C₂⁴	—	277	1	no	—
185	C₂×Q₈	—	2	2	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	C₄×D₂₇₇	1	4	yes	complemented
188	C₂²×C₄	—	277	1	no	—
189	C₂²×C₄	—	277	2	no	—
190	C₂²×C₄	—	277	1	no	—
191	C₂²×C₄	—	277	4	no	—
192	C₂²×C₄	—	277	1	no	—
193	C₂²×C₄	—	277	2	no	—
194	C₂²×C₄	—	554	4	no	—
195	C₂²×C₄	—	554	2	no	—
196	C₂²×C₄	—	554	2	no	—
197	C₂²×C₄	—	554	4	no	—
198	C₂²×C₄	—	1108	2	no	—
199	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₂₇₇	1	2	yes	—
201	C₂²×C₄	C₂²×D₂₇₇	1	2	yes	—
202	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
203	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	277	8	no	—
206	C₄⋊C₄	—	277	4	no	—
207	C₄⋊C₄	—	554	4	no	—
208	C₄⋊C₄	—	554	2	no	—
209	C₄⋊C₄	—	1108	2	no	—
210	C₄⋊C₄	C₂²⋊D₂₇₇	1	4	yes	—
211	C₂²⋊C₄	—	277	8	no	—
212	C₂²⋊C₄	—	277	4	no	—
213	C₂²⋊C₄	—	554	2	no	—
214	C₄²	—	2	4	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	554	2	no	—
217	C₄²	—	1108	2	no	—
218	C₂³	—	277	2	no	—
219	C₂³	—	277	2	no	—
220	C₂³	—	277	2	no	—
221	C₂³	—	554	1	no	—
222	C₂³	—	554	1	no	—
223	C₂³	—	554	1	no	—
224	C₂³	—	554	1	no	—
225	C₂³	C₂³×D₂₇₇	1	1	yes	characteristic
226	Q₈	—	2	8	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	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	1	no	—
242	C₂×C₄	—	554	1	no	—
243	C₂×C₄	—	554	2	no	—
244	C₂×C₄	—	554	2	no	—
245	C₂×C₄	—	554	8	no	—
246	C₂×C₄	—	554	2	no	—
247	C₂×C₄	—	554	1	no	—
248	C₂×C₄	—	554	2	no	—
249	C₂×C₄	—	554	1	no	—
250	C₂×C₄	—	554	2	no	—
251	C₂×C₄	—	554	1	no	—
252	C₂×C₄	—	1108	8	no	—
253	C₂×C₄	C₂×C₄×D₂₇₇	1	4	yes	—
254	C₂×C₄	C₂×C₄×D₂₇₇	1	2	yes	—
255	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	4	yes	—
256	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	2	yes	—
257	C₂×C₄	Q₈×D₂₇₇	1	2	yes	—
258	C₂×C₄	#17984563	1	2	yes	—
259	C₂²	—	277	4	no	—
260	C₂²	—	554	2	no	—
261	C₂²	—	554	2	no	—
262	C₂²	—	554	2	no	—
263	C₂²	—	554	2	no	—
264	C₂²	—	554	2	no	—
265	C₂²	—	554	2	no	—
266	C₂²	C₂²×C₄×D₂₇₇	1	1	yes	characteristic
267	C₂²	C₂²×C₂²⋊D₂₇₇	1	1	yes	characteristic
268	C₂²	C₂×Q₈×D₂₇₇	1	1	yes	characteristic
269	C₂²	#41×D₂₇₇	1	1	yes	characteristic
270	C₂²	C₂×#17984563	1	1	yes	characteristic
271	C₂²	#22579926	1	1	yes	characteristic
272	C₂²	#22579928	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	4	no	—
276	C₄	—	4	1	no	—
277	C₄	—	554	2	no	—
278	C₄	—	554	2	no	—
279	C₄	—	554	4	no	—
280	C₄	—	1108	2	no	—
281	C₄	—	1108	1	no	—
282	C₄	C₄×C₂²⋊D₂₇₇	1	4	yes	—
283	C₂	—	554	4	no	—
284	C₂	C₂×C₄×C₂²⋊D₂₇₇	1	1	yes	characteristic
285	C₂	C₄×Q₈×D₂₇₇	1	1	yes	characteristic
286	C₂	#31800484	1	1	yes	characteristic
287	C₂	#31800640	1	1	yes	characteristic
288	C₂	Q₈×C₂²⋊D₂₇₇	1	1	yes	characteristic
289	C₂	#31800693	1	1	yes	characteristic
290	C₂	#31800712	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₇₇
commutator	C₂²×C₂₇₇	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₂₇₇	C₂
socle, cosocle	C₂³×C₂₇₇	C₂⁴
prime radical	C₂³×D₂₇₇	C₂³
hypercenter	C₂×C₄×Q₈	D₂₇₇
hypocenter	C₂₇₇	#2360
LCS	G ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
3. #31799909	C₂	semidirect
4. #31799900	C₂	semidirect
5. #31799897	C₂	semidirect
6. #31799886	C₂	semidirect
8. C₄⋊C₄×Dic₂₇₇	C₂	semidirect
11. C₂×C₄×Q₈×C₂₇₇	C₂	semidirect
14. #22579856	C₄	semidirect
51. C₄⋊Dic₂₇₇	C₂×C₄	semidirect
54. #17984542	C₂×C₄	semidirect
57. #17984540	Q₈	semidirect
62. C₂×Q₈×C₂₇₇	C₂×C₄	semidirect
126. C₂×D₂₇₇	C₄×Q₈	semidirect
145. C₂₇₇	#2360	semidirect
148. C₂×C₄×Q₈	D₂₇₇	semidirect
187. C₂×Q₈	C₄×D₂₇₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22579856	2
C₂×C₄⋊C₄×D₂₇₇	2
#31799909	2
#31799900	2
#31799897	2
#31799886	2
#31799873	2
C₄⋊C₄×Dic₂₇₇	2
C₄×#17984540	2
C₂×C₄×Q₈×C₂₇₇	2
#2360	277