Subgroups — #3277×C₂₂₇ #44397233

Order 29056.

476 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#137×C₂₂₇	C₂	1	2	yes	maximal, complemented
2	#548×C₂₂₇	C₂	1	2	yes	maximal, complemented
3	#495×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#468×C₂₂₇	C₂	1	2	yes	maximal, complemented
5	#464×C₂₂₇	C₂	1	2	yes	maximal, complemented
6	#451×C₂₂₇	C₂	1	2	yes	maximal, complemented
7	#449×C₂₂₇	C₂	1	2	yes	maximal, complemented
8	C₄×Q₁₆×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#436×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₂₂₇	C₂²	1	2	yes	—
11	C₂²×Q₈×C₂₂₇	C₂²	1	2	yes	—
12	#137×C₂₂₇	—	2	8	no	—
13	C₂×Q₁₆×C₂₂₇	C₂²	1	2	yes	—
14	C₂×Q₁₆×C₂₂₇	C₂²	1	1	yes	characteristic
15	C₂×Q₁₆×C₂₂₇	C₂²	1	1	yes	characteristic
16	C₂×SD₁₆×C₂₂₇	C₂²	1	2	yes	—
17	C₂×SD₁₆×C₂₂₇	C₂²	1	2	yes	—
18	C₂×D₈×C₂₂₇	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₂₂₇	C₂²	1	2	yes	—
20	#124×C₂₂₇	—	2	2	no	—
21	#124×C₂₂₇	C₂²	1	2	yes	—
22	C₂²⋊Q₈×C₂₂₇	—	2	2	no	—
23	C₂²⋊Q₈×C₂₂₇	C₂²	1	2	yes	—
24	#121×C₂₂₇	—	2	2	no	—
25	#121×C₂₂₇	C₂²	1	2	yes	—
26	C₄×Q₈×C₂₂₇	C₂²	1	2	yes	—
27	C₄×D₄×C₂₂₇	—	2	2	no	—
28	C₄×D₄×C₂₂₇	C₂²	1	1	yes	characteristic
29	#107×C₂₂₇	C₂²	1	1	yes	characteristic
30	C₄⋊C₈×C₂₂₇	C₂²	1	1	yes	characteristic
31	#103×C₂₂₇	C₂²	1	2	yes	—
32	#103×C₂₂₇	C₂²	1	2	yes	—
33	#102×C₂₂₇	C₂²	1	2	yes	—
34	C₂²⋊C₈×C₂₂₇	C₂²	1	2	yes	—
35	C₄×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
36	#41×C₂₂₇	—	2	4	no	—
37	#41×C₂₂₇	—	4	2	no	—
38	C₂×Q₈×C₂₂₇	—	2	2	no	—
39	C₂×Q₈×C₂₂₇	—	2	4	no	—
40	C₂×Q₈×C₂₂₇	—	4	2	no	—
41	C₂×Q₈×C₂₂₇	C₂³	1	2	yes	—
42	C₂×Q₈×C₂₂₇	C₂³	1	2	yes	—
43	C₂×D₄×C₂₂₇	—	2	2	no	—
44	C₂×D₄×C₂₂₇	—	2	2	no	—
45	C₂×D₄×C₂₂₇	D₄	1	1	yes	characteristic
46	C₂×D₄×C₂₂₇	C₂³	1	2	yes	—
47	C₂²×C₄×C₂₂₇	—	2	2	no	—
48	C₂²×C₄×C₂₂₇	—	2	2	no	—
49	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
50	Q₁₆×C₂₂₇	—	2	4	no	—
51	Q₁₆×C₂₂₇	—	4	1	no	—
52	Q₁₆×C₂₂₇	D₄	1	4	yes	complemented
53	SD₁₆×C₂₂₇	—	2	4	no	—
54	SD₁₆×C₂₂₇	—	2	4	no	—
55	D₈×C₂₂₇	—	4	1	no	—
56	M₁₆×C₂₂₇	—	2	4	no	—
57	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₂₂₇	C₂³	1	2	yes	—
59	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₂₂₇	—	2	2	no	—
61	C₄⋊C₄×C₂₂₇	—	2	2	no	—
62	C₄⋊C₄×C₂₂₇	D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₂₂₇	C₂³	1	2	yes	—
64	C₂²⋊C₄×C₂₂₇	—	2	2	no	—
65	C₂²⋊C₄×C₂₂₇	—	2	2	no	—
66	C₂²⋊C₄×C₂₂₇	—	2	2	no	—
67	C₂²⋊C₄×C₂₂₇	—	2	2	no	—
68	C₂²⋊C₄×C₂₂₇	D₄	1	2	yes	—
69	C₄²×C₂₂₇	—	2	2	no	—
70	C₄²×C₂₂₇	C₂³	1	1	yes	characteristic
71	C₂³×C₂₂₇	—	2	2	no	—
72	C₂³×C₂₂₇	C₂×D₄	1	2	yes	—
73	Q₈×C₂₂₇	—	2	2	no	—
74	Q₈×C₂₂₇	—	2	2	no	—
75	Q₈×C₂₂₇	—	2	2	no	—
76	Q₈×C₂₂₇	—	4	4	no	—
77	Q₈×C₂₂₇	C₂×D₄	1	4	yes	—
78	D₄×C₂₂₇	—	2	2	no	—
79	D₄×C₂₂₇	—	2	2	no	—
80	D₄×C₂₂₇	—	4	1	no	—
81	D₄×C₂₂₇	—	4	4	no	—
82	D₄×C₂₂₇	—	4	4	no	—
83	C₂×C₄×C₂₂₇	—	2	2	no	—
84	C₂×C₄×C₂₂₇	—	2	2	no	—
85	C₂×C₄×C₂₂₇	—	2	2	no	—
86	C₂×C₄×C₂₂₇	—	2	2	no	—
87	C₂×C₄×C₂₂₇	—	2	2	no	—
88	C₂×C₄×C₂₂₇	—	4	4	no	—
89	C₂×C₄×C₂₂₇	—	4	4	no	—
90	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
94	C₈×C₂₂₇	—	2	2	no	—
95	C₈×C₂₂₇	—	2	1	no	—
96	C₈×C₂₂₇	C₂×D₄	1	2	yes	—
97	C₂²×C₂₂₇	—	2	2	no	—
98	C₂²×C₂₂₇	—	2	2	no	—
99	C₂²×C₂₂₇	—	2	2	no	—
100	C₂²×C₂₂₇	—	4	2	no	—
101	C₂²×C₂₂₇	—	4	2	no	—
102	C₂²×C₂₂₇	—	4	2	no	—
103	C₂²×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
104	C₄×C₂₂₇	—	2	2	no	—
105	C₄×C₂₂₇	—	2	4	no	—
106	C₄×C₂₂₇	—	2	1	no	—
107	C₄×C₂₂₇	—	2	1	no	—
108	C₄×C₂₂₇	—	4	2	no	—
109	C₄×C₂₂₇	—	4	2	no	—
110	C₄×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
111	C₄×C₂₂₇	H(5,𝔽₂)	1	1	yes	characteristic
112	C₂×C₂₂₇	—	4	2	no	—
113	C₂×C₂₂₇	—	8	2	no	—
114	C₂×C₂₂₇	D₄²	1	1	yes	characteristic
115	C₂×C₂₂₇	#575	1	1	yes	characteristic
116	C₂×C₂₂₇	#578	1	1	yes	characteristic
117	C₂₂₇	#3277	1	1	yes	characteristic, direct product factor, Sylow
118	#3277	C₂₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂×#137	C₂×C₂₂₇	1	2	yes	complemented
120	#548	C₂×C₂₂₇	1	2	yes	complemented
121	#495	C₂×C₂₂₇	1	1	yes	characteristic, complemented
122	#468	C₂×C₂₂₇	1	2	yes	complemented
123	#464	C₂×C₂₂₇	1	2	yes	complemented
124	#451	C₂×C₂₂₇	1	2	yes	complemented
125	#449	C₂×C₂₂₇	1	2	yes	complemented
126	C₄×Q₁₆	C₂×C₂₂₇	1	1	yes	characteristic, complemented
127	#436	C₂×C₂₂₇	1	1	yes	characteristic, complemented
128	C₂×#41	C₂²×C₂₂₇	1	2	yes	—
129	C₂²×Q₈	C₂²×C₂₂₇	1	2	yes	—
130	#137	—	2	8	no	—
131	C₂×Q₁₆	C₂²×C₂₂₇	1	2	yes	—
132	C₂×Q₁₆	C₂²×C₂₂₇	1	1	yes	characteristic
133	C₂×Q₁₆	C₂²×C₂₂₇	1	1	yes	characteristic
134	C₂×SD₁₆	C₂²×C₂₂₇	1	2	yes	—
135	C₂×SD₁₆	C₂²×C₂₂₇	1	2	yes	—
136	C₂×D₈	C₂²×C₂₂₇	1	1	yes	characteristic
137	C₂×M₁₆	C₂²×C₂₂₇	1	2	yes	—
138	#124	—	2	2	no	—
139	#124	C₂²×C₂₂₇	1	2	yes	—
140	C₂²⋊Q₈	—	2	2	no	—
141	C₂²⋊Q₈	C₂²×C₂₂₇	1	2	yes	—
142	#121	—	2	2	no	—
143	#121	C₂²×C₂₂₇	1	2	yes	—
144	C₄×Q₈	C₂²×C₂₂₇	1	2	yes	—
145	C₄×D₄	—	2	2	no	—
146	C₄×D₄	C₂²×C₂₂₇	1	1	yes	characteristic
147	#107	C₂²×C₂₂₇	1	1	yes	characteristic
148	C₄⋊C₈	C₂²×C₂₂₇	1	1	yes	characteristic
149	#103	C₂²×C₂₂₇	1	2	yes	—
150	#103	C₂²×C₂₂₇	1	2	yes	—
151	#102	C₂²×C₂₂₇	1	2	yes	—
152	C₂²⋊C₈	C₂²×C₂₂₇	1	2	yes	—
153	C₄×C₈	C₂²×C₂₂₇	1	1	yes	characteristic
154	#41	—	2	4	no	—
155	#41	—	4	2	no	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	—	2	4	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂×Q₈	C₂³×C₂₂₇	1	2	yes	—
160	C₂×Q₈	C₂³×C₂₂₇	1	2	yes	—
161	C₂×D₄	—	2	2	no	—
162	C₂×D₄	—	2	2	no	—
163	C₂×D₄	D₄×C₂₂₇	1	1	yes	characteristic
164	C₂×D₄	C₂³×C₂₂₇	1	2	yes	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
168	Q₁₆	—	2	4	no	—
169	Q₁₆	—	4	1	no	—
170	Q₁₆	D₄×C₂₂₇	1	4	yes	complemented
171	SD₁₆	—	2	4	no	—
172	SD₁₆	—	2	4	no	—
173	D₈	—	4	1	no	—
174	M₁₆	—	2	4	no	—
175	C₂×C₈	C₂³×C₂₂₇	1	1	yes	characteristic
176	C₂×C₈	C₂³×C₂₂₇	1	2	yes	—
177	C₂×C₈	C₂³×C₂₂₇	1	1	yes	characteristic
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	D₄×C₂₂₇	1	1	yes	characteristic
181	C₄⋊C₄	C₂³×C₂₂₇	1	2	yes	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₂²⋊C₄	D₄×C₂₂₇	1	2	yes	—
187	C₄²	—	2	2	no	—
188	C₄²	C₂³×C₂₂₇	1	1	yes	characteristic
189	C₂³	—	2	2	no	—
190	C₂³	C₂×D₄×C₂₂₇	1	2	yes	—
191	Q₈	—	2	2	no	—
192	Q₈	—	2	2	no	—
193	Q₈	—	2	2	no	—
194	Q₈	—	4	4	no	—
195	Q₈	C₂×D₄×C₂₂₇	1	4	yes	—
196	D₄	—	2	2	no	—
197	D₄	—	2	2	no	—
198	D₄	—	4	1	no	—
199	D₄	—	4	4	no	—
200	D₄	—	4	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	C₂×D₄×C₂₂₇	1	1	yes	characteristic
209	C₂×C₄	C₂×D₄×C₂₂₇	1	1	yes	characteristic
210	C₂×C₄	C₂×D₄×C₂₂₇	1	2	yes	—
211	C₂×C₄	C₂⁴×C₂₂₇	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	—	2	1	no	—
214	C₈	C₂×D₄×C₂₂₇	1	2	yes	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	C₂²×D₄×C₂₂₇	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	2	4	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	4	2	no	—
227	C₄	—	4	2	no	—
228	C₄	C₂²×D₄×C₂₂₇	1	1	yes	characteristic
229	C₄	H(5,𝔽₂)×C₂₂₇	1	1	yes	characteristic
230	C₂	—	4	2	no	—
231	C₂	—	8	2	no	—
232	C₂	D₄²×C₂₂₇	1	1	yes	characteristic
233	C₂	#575×C₂₂₇	1	1	yes	characteristic
234	C₂	#578×C₂₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₂₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₂₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₇ ◁ C₄×D₄×C₂₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#137×C₂₂₇	C₂	semidirect
2. #548×C₂₂₇	C₂	semidirect
3. #495×C₂₂₇	C₂	semidirect
4. #468×C₂₂₇	C₂	semidirect
5. #464×C₂₂₇	C₂	semidirect
6. #451×C₂₂₇	C₂	semidirect
7. #449×C₂₂₇	C₂	semidirect
8. C₄×Q₁₆×C₂₂₇	C₂	semidirect
9. #436×C₂₂₇	C₂	semidirect
52. Q₁₆×C₂₂₇	D₄	semidirect
117. C₂₂₇	#3277	direct (N × H)
118. #3277	C₂₂₇	direct (N × H)
119. C₂×#137	C₂×C₂₂₇	semidirect
120. #548	C₂×C₂₂₇	semidirect
121. #495	C₂×C₂₂₇	semidirect
122. #468	C₂×C₂₂₇	semidirect
123. #464	C₂×C₂₂₇	semidirect
124. #451	C₂×C₂₂₇	semidirect
125. #449	C₂×C₂₂₇	semidirect
126. C₄×Q₁₆	C₂×C₂₂₇	semidirect
127. #436	C₂×C₂₂₇	semidirect
170. Q₁₆	D₄×C₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#137×C₂₂₇	2
#548×C₂₂₇	2
#495×C₂₂₇	2
#468×C₂₂₇	2
#464×C₂₂₇	2
#451×C₂₂₇	2
#449×C₂₂₇	2
C₄×Q₁₆×C₂₂₇	2
#436×C₂₂₇	2
#3277	227

Subgroups — #3277×C227 #44397233

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3277×C₂₂₇ #44397233