Subgroups — #2850×C₂₇₇ #47980239

Order 35456.

696 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#124×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊Q₈×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#121×C₂₇₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄²×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#391×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#390×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#388×C₂₇₇	C₂	1	4	yes	maximal, complemented
8	#386×C₂₇₇	C₂	1	4	yes	maximal, complemented
9	#382×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₈×C₂₇₇	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₂₇₇	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₂₇₇	C₂²	1	2	yes	—
13	C₂³×C₄×C₂₇₇	C₂²	1	2	yes	—
14	C₂³×C₄×C₂₇₇	C₂²	1	1	yes	characteristic
15	#124×C₂₇₇	—	2	4	no	—
16	C₂²⋊Q₈×C₂₇₇	—	2	4	no	—
17	#121×C₂₇₇	—	2	4	no	—
18	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₂₇₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₂₇₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₂₇₇	C₂²	1	4	yes	—
24	C₂×C₂²⋊C₄×C₂₇₇	C₂²	1	4	yes	—
25	C₂×C₄²×C₂₇₇	—	2	4	no	—
26	C₂×C₄²×C₂₇₇	C₂²	1	2	yes	—
27	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
29	#95×C₂₇₇	C₂²	1	4	yes	—
30	#95×C₂₇₇	C₂²	1	4	yes	—
31	C₂⁴×C₂₇₇	C₂³	1	2	yes	—
32	C₂⁴×C₂₇₇	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₂₇₇	—	2	2	no	—
34	C₂×Q₈×C₂₇₇	—	2	2	no	—
35	C₂×Q₈×C₂₇₇	—	2	2	no	—
36	C₂×D₄×C₂₇₇	—	2	4	no	—
37	C₂×D₄×C₂₇₇	—	2	8	no	—
38	C₂×D₄×C₂₇₇	—	2	2	no	—
39	C₂×D₄×C₂₇₇	—	2	2	no	—
40	C₂×D₄×C₂₇₇	—	2	2	no	—
41	C₂²×C₄×C₂₇₇	—	2	4	no	—
42	C₂²×C₄×C₂₇₇	—	2	4	no	—
43	C₂²×C₄×C₂₇₇	—	2	2	no	—
44	C₂²×C₄×C₂₇₇	—	2	4	no	—
45	C₂²×C₄×C₂₇₇	—	2	2	no	—
46	C₂²×C₄×C₂₇₇	D₄	1	4	yes	—
47	C₂²×C₄×C₂₇₇	C₂³	1	2	yes	—
48	C₂²×C₄×C₂₇₇	C₂³	1	2	yes	—
49	C₂²×C₄×C₂₇₇	C₂³	1	2	yes	—
50	C₂²×C₄×C₂₇₇	C₂³	1	2	yes	—
51	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₂₇₇	C₂³	1	2	yes	—
53	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₂₇₇	—	2	2	no	—
55	C₄⋊C₄×C₂₇₇	—	2	2	no	—
56	C₄⋊C₄×C₂₇₇	—	2	4	no	—
57	C₂²⋊C₄×C₂₇₇	—	2	4	no	—
58	C₂²⋊C₄×C₂₇₇	—	2	4	no	—
59	C₂²⋊C₄×C₂₇₇	—	2	8	no	—
60	C₂²⋊C₄×C₂₇₇	—	2	8	no	—
61	C₄²×C₂₇₇	—	2	4	no	—
62	C₄²×C₂₇₇	—	2	2	no	—
63	C₄²×C₂₇₇	D₄	1	4	yes	complemented
64	C₂³×C₂₇₇	—	2	1	no	—
65	C₂³×C₂₇₇	—	2	1	no	—
66	C₂³×C₂₇₇	—	2	2	no	—
67	C₂³×C₂₇₇	—	2	2	no	—
68	C₂³×C₂₇₇	—	2	2	no	—
69	C₂³×C₂₇₇	—	2	4	no	—
70	C₂³×C₂₇₇	—	2	2	no	—
71	C₂³×C₂₇₇	—	2	4	no	—
72	C₂³×C₂₇₇	C₂×D₄	1	2	yes	—
73	C₂³×C₂₇₇	#41	1	4	yes	—
74	C₂³×C₂₇₇	C₂⁴	1	1	yes	characteristic
75	Q₈×C₂₇₇	—	4	4	no	—
76	D₄×C₂₇₇	—	4	8	no	—
77	D₄×C₂₇₇	—	4	4	no	—
78	C₂×C₄×C₂₇₇	—	2	16	no	—
79	C₂×C₄×C₂₇₇	—	2	1	no	—
80	C₂×C₄×C₂₇₇	—	2	2	no	—
81	C₂×C₄×C₂₇₇	—	2	1	no	—
82	C₂×C₄×C₂₇₇	—	2	2	no	—
83	C₂×C₄×C₂₇₇	—	2	2	no	—
84	C₂×C₄×C₂₇₇	—	2	2	no	—
85	C₂×C₄×C₂₇₇	—	2	8	no	—
86	C₂×C₄×C₂₇₇	—	2	2	no	—
87	C₂×C₄×C₂₇₇	—	2	2	no	—
88	C₂×C₄×C₂₇₇	—	2	2	no	—
89	C₂×C₄×C₂₇₇	—	2	2	no	—
90	C₂×C₄×C₂₇₇	—	2	4	no	—
91	C₂×C₄×C₂₇₇	—	2	4	no	—
92	C₂×C₄×C₂₇₇	—	2	2	no	—
93	C₂×C₄×C₂₇₇	C₂×D₄	1	4	yes	—
94	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₂₇₇	#41	1	4	yes	—
98	C₂×C₄×C₂₇₇	#41	1	2	yes	—
99	C₂²×C₂₇₇	—	2	2	no	—
100	C₂²×C₂₇₇	—	2	4	no	—
101	C₂²×C₂₇₇	—	2	4	no	—
102	C₂²×C₂₇₇	—	2	2	no	—
103	C₂²×C₂₇₇	—	4	2	no	—
104	C₂²×C₂₇₇	—	4	4	no	—
105	C₂²×C₂₇₇	—	4	2	no	—
106	C₂²×C₂₇₇	—	4	4	no	—
107	C₂²×C₂₇₇	—	4	2	no	—
108	C₂²×C₂₇₇	#124	1	4	yes	complemented
109	C₂²×C₂₇₇	C₂²×D₄	1	1	yes	characteristic
110	C₂²×C₂₇₇	C₂²×D₄	1	1	yes	characteristic
111	C₂²×C₂₇₇	C₂×#41	1	1	yes	characteristic
112	C₂²×C₂₇₇	C₂×#41	1	2	yes	—
113	C₂²×C₂₇₇	C₂×#41	1	2	yes	—
114	C₄×C₂₇₇	—	2	2	no	—
115	C₄×C₂₇₇	—	2	4	no	—
116	C₄×C₂₇₇	—	2	4	no	—
117	C₄×C₂₇₇	—	4	2	no	—
118	C₄×C₂₇₇	—	4	2	no	—
119	C₄×C₂₇₇	—	4	2	no	—
120	C₄×C₂₇₇	#121	1	4	yes	—
121	C₂×C₂₇₇	—	2	4	no	—
122	C₂×C₂₇₇	—	8	2	no	—
123	C₂×C₂₇₇	C₂×#121	1	1	yes	characteristic
124	C₂×C₂₇₇	#525	1	2	yes	—
125	C₂×C₂₇₇	C₂×#124	1	1	yes	characteristic
126	C₂×C₂₇₇	#529	1	2	yes	—
127	C₂×C₂₇₇	#532	1	1	yes	characteristic
128	C₂₇₇	#2850	1	1	yes	characteristic, direct product factor, Sylow
129	#2850	C₂₇₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂×#124	C₂×C₂₇₇	1	1	yes	characteristic, complemented
131	C₂×C₂²⋊Q₈	C₂×C₂₇₇	1	1	yes	characteristic, complemented
132	C₂×#121	C₂×C₂₇₇	1	1	yes	characteristic
133	C₂²×C₄²	C₂×C₂₇₇	1	1	yes	characteristic, complemented
134	#391	C₂×C₂₇₇	1	1	yes	characteristic, complemented
135	#390	C₂×C₂₇₇	1	1	yes	characteristic, complemented
136	#388	C₂×C₂₇₇	1	4	yes	complemented
137	#386	C₂×C₂₇₇	1	4	yes	complemented
138	#382	C₂×C₂₇₇	1	1	yes	characteristic, complemented
139	C₂²×Q₈	C₂²×C₂₇₇	1	1	yes	characteristic
140	C₂²×D₄	C₂²×C₂₇₇	1	1	yes	characteristic
141	C₂²×D₄	C₂²×C₂₇₇	1	2	yes	—
142	C₂³×C₄	C₂²×C₂₇₇	1	2	yes	—
143	C₂³×C₄	C₂²×C₂₇₇	1	1	yes	characteristic
144	#124	—	2	4	no	—
145	C₂²⋊Q₈	—	2	4	no	—
146	#121	—	2	4	no	—
147	C₂×C₄⋊C₄	C₂²×C₂₇₇	1	1	yes	characteristic
148	C₂×C₄⋊C₄	C₂²×C₂₇₇	1	1	yes	characteristic
149	C₂×C₄⋊C₄	C₂²×C₂₇₇	1	2	yes	—
150	C₂×C₂²⋊C₄	C₂²×C₂₇₇	1	2	yes	—
151	C₂×C₂²⋊C₄	C₂²×C₂₇₇	1	2	yes	—
152	C₂×C₂²⋊C₄	C₂²×C₂₇₇	1	4	yes	—
153	C₂×C₂²⋊C₄	C₂²×C₂₇₇	1	4	yes	—
154	C₂×C₄²	—	2	4	no	—
155	C₂×C₄²	C₂²×C₂₇₇	1	2	yes	—
156	C₂×C₄²	C₂²×C₂₇₇	1	1	yes	characteristic
157	C₂×C₄²	C₂²×C₂₇₇	1	1	yes	characteristic
158	#95	C₂²×C₂₇₇	1	4	yes	—
159	#95	C₂²×C₂₇₇	1	4	yes	—
160	C₂⁴	C₂³×C₂₇₇	1	2	yes	—
161	C₂⁴	C₂³×C₂₇₇	1	1	yes	characteristic
162	C₂×Q₈	—	2	2	no	—
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	—	2	2	no	—
165	C₂×D₄	—	2	4	no	—
166	C₂×D₄	—	2	8	no	—
167	C₂×D₄	—	2	2	no	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	2	2	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	D₄×C₂₇₇	1	4	yes	—
176	C₂²×C₄	C₂³×C₂₇₇	1	2	yes	—
177	C₂²×C₄	C₂³×C₂₇₇	1	2	yes	—
178	C₂²×C₄	C₂³×C₂₇₇	1	2	yes	—
179	C₂²×C₄	C₂³×C₂₇₇	1	2	yes	—
180	C₂²×C₄	C₂³×C₂₇₇	1	1	yes	characteristic
181	C₂²×C₄	C₂³×C₂₇₇	1	2	yes	—
182	C₂²×C₄	C₂³×C₂₇₇	1	1	yes	characteristic
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	4	no	—
187	C₂²⋊C₄	—	2	4	no	—
188	C₂²⋊C₄	—	2	8	no	—
189	C₂²⋊C₄	—	2	8	no	—
190	C₄²	—	2	4	no	—
191	C₄²	—	2	2	no	—
192	C₄²	D₄×C₂₇₇	1	4	yes	complemented
193	C₂³	—	2	1	no	—
194	C₂³	—	2	1	no	—
195	C₂³	—	2	2	no	—
196	C₂³	—	2	2	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	2	4	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	4	no	—
201	C₂³	C₂×D₄×C₂₇₇	1	2	yes	—
202	C₂³	#41×C₂₇₇	1	4	yes	—
203	C₂³	C₂⁴×C₂₇₇	1	1	yes	characteristic
204	Q₈	—	4	4	no	—
205	D₄	—	4	8	no	—
206	D₄	—	4	4	no	—
207	C₂×C₄	—	2	16	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	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	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	C₂×D₄×C₂₇₇	1	4	yes	—
223	C₂×C₄	C₂×D₄×C₂₇₇	1	2	yes	—
224	C₂×C₄	C₂×D₄×C₂₇₇	1	2	yes	—
225	C₂×C₄	C₂×D₄×C₂₇₇	1	2	yes	—
226	C₂×C₄	#41×C₂₇₇	1	4	yes	—
227	C₂×C₄	#41×C₂₇₇	1	2	yes	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	2	4	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	4	2	no	—
233	C₂²	—	4	4	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	4	no	—
236	C₂²	—	4	2	no	—
237	C₂²	#124×C₂₇₇	1	4	yes	complemented
238	C₂²	C₂²×D₄×C₂₇₇	1	1	yes	characteristic
239	C₂²	C₂²×D₄×C₂₇₇	1	1	yes	characteristic
240	C₂²	C₂×#41×C₂₇₇	1	1	yes	characteristic
241	C₂²	C₂×#41×C₂₇₇	1	2	yes	—
242	C₂²	C₂×#41×C₂₇₇	1	2	yes	—
243	C₄	—	2	2	no	—
244	C₄	—	2	4	no	—
245	C₄	—	2	4	no	—
246	C₄	—	4	2	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	2	no	—
249	C₄	#121×C₂₇₇	1	4	yes	—
250	C₂	—	2	4	no	—
251	C₂	—	8	2	no	—
252	C₂	C₂×#121×C₂₇₇	1	1	yes	characteristic
253	C₂	#525×C₂₇₇	1	2	yes	—
254	C₂	C₂×#124×C₂₇₇	1	1	yes	characteristic
255	C₂	#529×C₂₇₇	1	2	yes	—
256	C₂	#532×C₂₇₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₇₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₇₇
prime radical	C₂×#121×C₂₇₇	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₇₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#124×C₂₇₇	C₂	semidirect
2. C₂×C₂²⋊Q₈×C₂₇₇	C₂	semidirect
4. C₂²×C₄²×C₂₇₇	C₂	semidirect
5. #391×C₂₇₇	C₂	semidirect
6. #390×C₂₇₇	C₂	semidirect
7. #388×C₂₇₇	C₂	semidirect
8. #386×C₂₇₇	C₂	semidirect
9. #382×C₂₇₇	C₂	semidirect
63. C₄²×C₂₇₇	D₄	semidirect
108. C₂²×C₂₇₇	#124	semidirect
128. C₂₇₇	#2850	direct (N × H)
129. #2850	C₂₇₇	direct (N × H)
130. C₂×#124	C₂×C₂₇₇	semidirect
131. C₂×C₂²⋊Q₈	C₂×C₂₇₇	semidirect
133. C₂²×C₄²	C₂×C₂₇₇	semidirect
134. #391	C₂×C₂₇₇	semidirect
135. #390	C₂×C₂₇₇	semidirect
136. #388	C₂×C₂₇₇	semidirect
137. #386	C₂×C₂₇₇	semidirect
138. #382	C₂×C₂₇₇	semidirect
192. C₄²	D₄×C₂₇₇	semidirect
237. C₂²	#124×C₂₇₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#124×C₂₇₇	2
C₂×C₂²⋊Q₈×C₂₇₇	2
C₂×#121×C₂₇₇	2
C₂²×C₄²×C₂₇₇	2
#391×C₂₇₇	2
#390×C₂₇₇	2
#388×C₂₇₇	2
#386×C₂₇₇	2
#382×C₂₇₇	2
#2850	277

Subgroups — #2850×C277 #47980239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2850×C₂₇₇ #47980239