Subgroups — #2362×C₂₇₇ #47979751

Order 35456.

576 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊Q₈×C₂₇₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#117×C₂₇₇	C₂	1	1	yes	maximal, characteristic
4	#391×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#389×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#388×C₂₇₇	C₂	1	2	yes	maximal
7	#388×C₂₇₇	C₂	1	2	yes	maximal
8	#387×C₂₇₇	C₂	1	2	yes	maximal, complemented
9	C₄×C₄⋊C₄×C₂₇₇	C₂	1	2	yes	maximal, complemented
10	C₄×C₄⋊C₄×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊C₄×C₂₇₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₂₇₇	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₂₇₇	C₂²	1	1	yes	characteristic
14	C₂²⋊Q₈×C₂₇₇	C₄	1	8	yes	complemented
15	C₄×Q₈×C₂₇₇	—	2	4	no	—
16	#117×C₂₇₇	—	2	4	no	—
17	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₂₇₇	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₂₇₇	C₂²	1	2	yes	—
27	C₂×C₄²×C₂₇₇	C₂²	1	2	yes	—
28	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
29	C₂×C₄²×C₂₇₇	C₂²	1	2	yes	—
30	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₂₇₇	C₂²	1	2	yes	—
32	#95×C₂₇₇	C₂²	1	1	yes	characteristic
33	#95×C₂₇₇	C₂²	1	1	yes	characteristic
34	#95×C₂₇₇	C₂²	1	2	yes	—
35	#95×C₂₇₇	C₂²	1	2	yes	—
36	#95×C₂₇₇	C₂²	1	2	yes	—
37	#95×C₂₇₇	C₂²	1	2	yes	—
38	C₂⁴×C₂₇₇	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₂₇₇	—	2	2	no	—
40	C₂×Q₈×C₂₇₇	—	2	2	no	—
41	C₂×Q₈×C₂₇₇	C₂×C₄	1	4	yes	—
42	C₂²×C₄×C₂₇₇	—	2	2	no	—
43	C₂²×C₄×C₂₇₇	—	2	2	no	—
44	C₂²×C₄×C₂₇₇	C₂×C₄	1	4	yes	—
45	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₂₇₇	C₂³	1	2	yes	—
47	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₂₇₇	C₂³	1	2	yes	—
50	C₂²×C₄×C₂₇₇	C₂³	1	2	yes	—
51	C₂²×C₄×C₂₇₇	C₂³	1	2	yes	—
52	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₂₇₇	—	2	2	no	—
56	C₄⋊C₄×C₂₇₇	—	2	4	no	—
57	C₄⋊C₄×C₂₇₇	—	2	4	no	—
58	C₄⋊C₄×C₂₇₇	C₂×C₄	1	4	yes	—
59	C₄⋊C₄×C₂₇₇	C₂×C₄	1	8	yes	—
60	C₄⋊C₄×C₂₇₇	D₄	1	4	yes	—
61	C₂²⋊C₄×C₂₇₇	—	2	4	no	—
62	C₂²⋊C₄×C₂₇₇	—	2	2	no	—
63	C₂²⋊C₄×C₂₇₇	—	2	2	no	—
64	C₂²⋊C₄×C₂₇₇	C₂×C₄	1	8	yes	—
65	C₄²×C₂₇₇	—	2	4	no	—
66	C₄²×C₂₇₇	—	2	4	no	—
67	C₄²×C₂₇₇	—	2	2	no	—
68	C₄²×C₂₇₇	—	2	4	no	—
69	C₄²×C₂₇₇	—	2	2	no	—
70	C₂³×C₂₇₇	—	2	1	no	—
71	C₂³×C₂₇₇	—	2	1	no	—
72	C₂³×C₂₇₇	—	2	1	no	—
73	C₂³×C₂₇₇	—	2	1	no	—
74	C₂³×C₂₇₇	—	2	1	no	—
75	C₂³×C₂₇₇	—	2	1	no	—
76	C₂³×C₂₇₇	C₂²×C₄	1	2	yes	—
77	C₂³×C₂₇₇	C₂⁴	1	1	yes	characteristic
78	Q₈×C₂₇₇	—	2	8	no	—
79	C₂×C₄×C₂₇₇	—	2	1	no	—
80	C₂×C₄×C₂₇₇	—	2	1	no	—
81	C₂×C₄×C₂₇₇	—	2	2	no	—
82	C₂×C₄×C₂₇₇	—	2	2	no	—
83	C₂×C₄×C₂₇₇	—	2	2	no	—
84	C₂×C₄×C₂₇₇	—	2	1	no	—
85	C₂×C₄×C₂₇₇	—	2	2	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	1	no	—
91	C₂×C₄×C₂₇₇	—	2	8	no	—
92	C₂×C₄×C₂₇₇	—	2	2	no	—
93	C₂×C₄×C₂₇₇	—	2	1	no	—
94	C₂×C₄×C₂₇₇	—	2	1	no	—
95	C₂×C₄×C₂₇₇	—	2	1	no	—
96	C₂×C₄×C₂₇₇	—	2	1	no	—
97	C₂×C₄×C₂₇₇	—	2	2	no	—
98	C₂×C₄×C₂₇₇	—	2	1	no	—
99	C₂×C₄×C₂₇₇	—	2	2	no	—
100	C₂×C₄×C₂₇₇	—	2	1	no	—
101	C₂×C₄×C₂₇₇	C₂²×C₄	1	4	yes	—
102	C₂×C₄×C₂₇₇	C₂²×C₄	1	4	yes	—
103	C₂×C₄×C₂₇₇	C₂²×C₄	1	2	yes	—
104	C₂×C₄×C₂₇₇	C₂²×C₄	1	2	yes	—
105	C₂×C₄×C₂₇₇	C₂×D₄	1	4	yes	—
106	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₂₇₇	#41	1	2	yes	—
108	C₂×C₄×C₂₇₇	#41	1	2	yes	—
109	C₂×C₄×C₂₇₇	#41	1	2	yes	—
110	C₂²×C₂₇₇	—	2	2	no	—
111	C₂²×C₂₇₇	—	2	2	no	—
112	C₂²×C₂₇₇	—	2	2	no	—
113	C₂²×C₂₇₇	—	4	1	no	—
114	C₂²×C₂₇₇	—	4	1	no	—
115	C₂²×C₂₇₇	—	4	1	no	—
116	C₂²×C₂₇₇	—	4	1	no	—
117	C₂²×C₂₇₇	C₂³×C₄	1	1	yes	characteristic
118	C₂²×C₂₇₇	C₂²×D₄	1	1	yes	characteristic
119	C₂²×C₂₇₇	C₂×#41	1	1	yes	characteristic
120	C₂²×C₂₇₇	C₂×#41	1	1	yes	characteristic
121	C₂²×C₂₇₇	C₂×#41	1	1	yes	characteristic
122	C₂²×C₂₇₇	2₋¹⁺⁴	1	1	yes	characteristic
123	C₂²×C₂₇₇	2₋¹⁺⁴	1	1	yes	characteristic
124	C₄×C₂₇₇	—	2	2	no	—
125	C₄×C₂₇₇	—	2	4	no	—
126	C₄×C₂₇₇	—	2	4	no	—
127	C₄×C₂₇₇	—	2	4	no	—
128	C₄×C₂₇₇	—	2	2	no	—
129	C₄×C₂₇₇	—	2	2	no	—
130	C₄×C₂₇₇	—	4	1	no	—
131	C₄×C₂₇₇	—	4	2	no	—
132	C₄×C₂₇₇	—	4	1	no	—
133	C₄×C₂₇₇	C₄×D₄	1	4	yes	—
134	C₂×C₂₇₇	—	4	2	no	—
135	C₂×C₂₇₇	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂×C₂₇₇	C₄×#41	1	1	yes	characteristic
137	C₂×C₂₇₇	#519	1	1	yes	characteristic
138	C₂×C₂₇₇	#547	1	1	yes	characteristic
139	C₂×C₂₇₇	#548	1	1	yes	characteristic
140	C₂×C₂₇₇	#552	1	1	yes	characteristic
141	C₂×C₂₇₇	#556	1	1	yes	characteristic
142	C₂₇₇	#2362	1	1	yes	characteristic, direct product factor, Sylow
143	#2362	C₂₇₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂×C₂²⋊Q₈	C₂×C₂₇₇	1	1	yes	characteristic
145	C₂×C₄×Q₈	C₂×C₂₇₇	1	1	yes	characteristic, complemented
146	C₂×#117	C₂×C₂₇₇	1	1	yes	characteristic
147	#391	C₂×C₂₇₇	1	1	yes	characteristic, complemented
148	#389	C₂×C₂₇₇	1	1	yes	characteristic, complemented
149	#388	C₂×C₂₇₇	1	2	yes	—
150	#388	C₂×C₂₇₇	1	2	yes	—
151	#387	C₂×C₂₇₇	1	2	yes	complemented
152	C₄×C₄⋊C₄	C₂×C₂₇₇	1	2	yes	complemented
153	C₄×C₄⋊C₄	C₂×C₂₇₇	1	1	yes	characteristic, complemented
154	C₄×C₂²⋊C₄	C₂×C₂₇₇	1	1	yes	characteristic
155	C₂²×Q₈	C₂²×C₂₇₇	1	1	yes	characteristic
156	C₂³×C₄	C₂²×C₂₇₇	1	1	yes	characteristic
157	C₂²⋊Q₈	C₄×C₂₇₇	1	8	yes	complemented
158	C₄×Q₈	—	2	4	no	—
159	#117	—	2	4	no	—
160	C₂×C₄⋊C₄	C₂²×C₂₇₇	1	1	yes	characteristic
161	C₂×C₄⋊C₄	C₂²×C₂₇₇	1	2	yes	—
162	C₂×C₄⋊C₄	C₂²×C₂₇₇	1	2	yes	—
163	C₂×C₄⋊C₄	C₂²×C₂₇₇	1	2	yes	—
164	C₂×C₄⋊C₄	C₂²×C₂₇₇	1	1	yes	characteristic
165	C₂×C₄⋊C₄	C₂²×C₂₇₇	1	1	yes	characteristic
166	C₂×C₂²⋊C₄	C₂²×C₂₇₇	1	2	yes	—
167	C₂×C₂²⋊C₄	C₂²×C₂₇₇	1	1	yes	characteristic
168	C₂×C₂²⋊C₄	C₂²×C₂₇₇	1	1	yes	characteristic
169	C₂×C₂²⋊C₄	C₂²×C₂₇₇	1	2	yes	—
170	C₂×C₄²	C₂²×C₂₇₇	1	2	yes	—
171	C₂×C₄²	C₂²×C₂₇₇	1	1	yes	characteristic
172	C₂×C₄²	C₂²×C₂₇₇	1	2	yes	—
173	C₂×C₄²	C₂²×C₂₇₇	1	1	yes	characteristic
174	C₂×C₄²	C₂²×C₂₇₇	1	2	yes	—
175	#95	C₂²×C₂₇₇	1	1	yes	characteristic
176	#95	C₂²×C₂₇₇	1	1	yes	characteristic
177	#95	C₂²×C₂₇₇	1	2	yes	—
178	#95	C₂²×C₂₇₇	1	2	yes	—
179	#95	C₂²×C₂₇₇	1	2	yes	—
180	#95	C₂²×C₂₇₇	1	2	yes	—
181	C₂⁴	C₂³×C₂₇₇	1	1	yes	characteristic
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	C₂×C₄×C₂₇₇	1	4	yes	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	C₂×C₄×C₂₇₇	1	4	yes	—
188	C₂²×C₄	C₂³×C₂₇₇	1	1	yes	characteristic
189	C₂²×C₄	C₂³×C₂₇₇	1	2	yes	—
190	C₂²×C₄	C₂³×C₂₇₇	1	1	yes	characteristic
191	C₂²×C₄	C₂³×C₂₇₇	1	1	yes	characteristic
192	C₂²×C₄	C₂³×C₂₇₇	1	2	yes	—
193	C₂²×C₄	C₂³×C₂₇₇	1	2	yes	—
194	C₂²×C₄	C₂³×C₂₇₇	1	2	yes	—
195	C₂²×C₄	C₂³×C₂₇₇	1	1	yes	characteristic
196	C₂²×C₄	C₂³×C₂₇₇	1	1	yes	characteristic
197	C₂²×C₄	C₂³×C₂₇₇	1	1	yes	characteristic
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	C₂×C₄×C₂₇₇	1	4	yes	—
202	C₄⋊C₄	C₂×C₄×C₂₇₇	1	8	yes	—
203	C₄⋊C₄	D₄×C₂₇₇	1	4	yes	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₂²⋊C₄	C₂×C₄×C₂₇₇	1	8	yes	—
208	C₄²	—	2	4	no	—
209	C₄²	—	2	4	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	4	no	—
212	C₄²	—	2	2	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	2	1	no	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	C₂²×C₄×C₂₇₇	1	2	yes	—
220	C₂³	C₂⁴×C₂₇₇	1	1	yes	characteristic
221	Q₈	—	2	8	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	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	8	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	C₂²×C₄×C₂₇₇	1	4	yes	—
245	C₂×C₄	C₂²×C₄×C₂₇₇	1	4	yes	—
246	C₂×C₄	C₂²×C₄×C₂₇₇	1	2	yes	—
247	C₂×C₄	C₂²×C₄×C₂₇₇	1	2	yes	—
248	C₂×C₄	C₂×D₄×C₂₇₇	1	4	yes	—
249	C₂×C₄	C₂×D₄×C₂₇₇	1	2	yes	—
250	C₂×C₄	#41×C₂₇₇	1	2	yes	—
251	C₂×C₄	#41×C₂₇₇	1	2	yes	—
252	C₂×C₄	#41×C₂₇₇	1	2	yes	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	4	1	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	C₂³×C₄×C₂₇₇	1	1	yes	characteristic
261	C₂²	C₂²×D₄×C₂₇₇	1	1	yes	characteristic
262	C₂²	C₂×#41×C₂₇₇	1	1	yes	characteristic
263	C₂²	C₂×#41×C₂₇₇	1	1	yes	characteristic
264	C₂²	C₂×#41×C₂₇₇	1	1	yes	characteristic
265	C₂²	2₋¹⁺⁴×C₂₇₇	1	1	yes	characteristic
266	C₂²	2₋¹⁺⁴×C₂₇₇	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	2	4	no	—
269	C₄	—	2	4	no	—
270	C₄	—	2	4	no	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	2	no	—
275	C₄	—	4	1	no	—
276	C₄	C₄×D₄×C₂₇₇	1	4	yes	—
277	C₂	—	4	2	no	—
278	C₂	C₂×C₄×D₄×C₂₇₇	1	1	yes	characteristic
279	C₂	C₄×#41×C₂₇₇	1	1	yes	characteristic
280	C₂	#519×C₂₇₇	1	1	yes	characteristic
281	C₂	#547×C₂₇₇	1	1	yes	characteristic
282	C₂	#548×C₂₇₇	1	1	yes	characteristic
283	C₂	#552×C₂₇₇	1	1	yes	characteristic
284	C₂	#556×C₂₇₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₇₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₇₇
Frattini, cosocle	C₂³	C₂⁴×C₂₇₇
prime radical	C₂⁴×C₂₇₇	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₇₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₈×C₂₇₇	C₂	semidirect
4. #391×C₂₇₇	C₂	semidirect
5. #389×C₂₇₇	C₂	semidirect
8. #387×C₂₇₇	C₂	semidirect
9. C₄×C₄⋊C₄×C₂₇₇	C₂	semidirect
10. C₄×C₄⋊C₄×C₂₇₇	C₂	semidirect
14. C₂²⋊Q₈×C₂₇₇	C₄	semidirect
142. C₂₇₇	#2362	direct (N × H)
143. #2362	C₂₇₇	direct (N × H)
145. C₂×C₄×Q₈	C₂×C₂₇₇	semidirect
147. #391	C₂×C₂₇₇	semidirect
148. #389	C₂×C₂₇₇	semidirect
151. #387	C₂×C₂₇₇	semidirect
152. C₄×C₄⋊C₄	C₂×C₂₇₇	semidirect
153. C₄×C₄⋊C₄	C₂×C₂₇₇	semidirect
157. C₂²⋊Q₈	C₄×C₂₇₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊Q₈×C₂₇₇	2
C₂×C₄×Q₈×C₂₇₇	2
C₂×#117×C₂₇₇	2
#391×C₂₇₇	2
#389×C₂₇₇	2
#388×C₂₇₇	2
#387×C₂₇₇	2
C₄×C₄⋊C₄×C₂₇₇	2
C₄×C₂²⋊C₄×C₂₇₇	2
#2362	277

Subgroups — #2362×C277 #47979751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2362×C₂₇₇ #47979751