Subgroups — #3273×C₂₂₇ #44397229

Order 29056.

520 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×AGL(1,ℤ/8)×C₂₂₇	C₂	1	2	yes	maximal, complemented
2	#546×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	#463×C₂₂₇	C₂	1	2	yes	maximal, complemented
6	#452×C₂₂₇	C₂	1	2	yes	maximal, complemented
7	#447×C₂₂₇	C₂	1	2	yes	maximal, complemented
8	C₄×D₈×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×D₄×C₂₂₇	C₂²	1	2	yes	complemented
12	AGL(1,ℤ/8)×C₂₂₇	—	2	8	no	—
13	C₂×Q₁₆×C₂₂₇	C₂²	1	1	yes	characteristic, complemented
14	C₂×SD₁₆×C₂₂₇	C₂²	1	2	yes	complemented
15	C₂×SD₁₆×C₂₂₇	C₂²	1	2	yes	complemented
16	C₂×D₈×C₂₂₇	C₂²	1	2	yes	—
17	C₂×D₈×C₂₂₇	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₂₂₇	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₂₂₇	C₂²	1	2	yes	complemented
20	#124×C₂₂₇	C₂²	1	2	yes	complemented
21	#123×C₂₂₇	—	2	2	no	—
22	C₂²⋊Q₈×C₂₂₇	C₂²	1	2	yes	complemented
23	#121×C₂₂₇	—	2	2	no	—
24	#121×C₂₂₇	C₂²	1	2	yes	—
25	#120×C₂₂₇	—	2	2	no	—
26	C₄×D₄×C₂₂₇	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₂₂₇	C₂²	1	2	yes	complemented
28	C₂×C₂²⋊C₄×C₂₂₇	—	2	2	no	—
29	#107×C₂₂₇	C₂²	1	1	yes	characteristic, complemented
30	C₄⋊C₈×C₂₂₇	C₂²	1	1	yes	characteristic, complemented
31	#103×C₂₂₇	C₂²	1	2	yes	complemented
32	#102×C₂₂₇	C₂²	1	2	yes	complemented
33	#102×C₂₂₇	C₂²	1	2	yes	complemented
34	C₂²⋊C₈×C₂₂₇	C₂²	1	2	yes	complemented
35	C₄×C₈×C₂₂₇	C₂²	1	1	yes	characteristic, complemented
36	C₂⁴×C₂₂₇	—	2	2	no	—
37	#41×C₂₂₇	—	2	4	no	—
38	#41×C₂₂₇	—	4	2	no	—
39	C₂×Q₈×C₂₂₇	C₂³	1	2	yes	—
40	C₂×D₄×C₂₂₇	—	2	4	no	—
41	C₂×D₄×C₂₂₇	—	2	2	no	—
42	C₂×D₄×C₂₂₇	—	2	2	no	—
43	C₂×D₄×C₂₂₇	—	2	2	no	—
44	C₂×D₄×C₂₂₇	—	4	2	no	—
45	C₂×D₄×C₂₂₇	D₄	1	1	yes	characteristic
46	C₂×D₄×C₂₂₇	C₂³	1	2	yes	—
47	C₂×D₄×C₂₂₇	C₂³	1	2	yes	—
48	C₂²×C₄×C₂₂₇	—	2	2	no	—
49	C₂²×C₄×C₂₂₇	—	2	2	no	—
50	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
51	Q₁₆×C₂₂₇	—	4	1	no	—
52	SD₁₆×C₂₂₇	—	2	4	no	—
53	SD₁₆×C₂₂₇	—	2	4	no	—
54	D₈×C₂₂₇	—	2	4	no	—
55	D₈×C₂₂₇	—	4	1	no	—
56	D₈×C₂₂₇	D₄	1	4	yes	complemented
57	M₁₆×C₂₂₇	—	2	4	no	—
58	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₂₂₇	C₂³	1	2	yes	—
60	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
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	—	2	2	no	—
69	C₂²⋊C₄×C₂₂₇	D₄	1	2	yes	—
70	C₄²×C₂₂₇	C₂³	1	1	yes	characteristic
71	C₂³×C₂₂₇	—	2	2	no	—
72	C₂³×C₂₂₇	—	2	2	no	—
73	C₂³×C₂₂₇	—	4	4	no	—
74	C₂³×C₂₂₇	—	4	4	no	—
75	C₂³×C₂₂₇	—	8	2	no	—
76	C₂³×C₂₂₇	C₂×D₄	1	2	yes	—
77	Q₈×C₂₂₇	—	2	2	no	—
78	Q₈×C₂₂₇	—	2	2	no	—
79	D₄×C₂₂₇	—	2	2	no	—
80	D₄×C₂₂₇	—	2	2	no	—
81	D₄×C₂₂₇	—	2	2	no	—
82	D₄×C₂₂₇	—	4	1	no	—
83	D₄×C₂₂₇	—	4	4	no	—
84	D₄×C₂₂₇	—	4	4	no	—
85	D₄×C₂₂₇	—	8	2	no	—
86	D₄×C₂₂₇	C₂×D₄	1	4	yes	—
87	C₂×C₄×C₂₂₇	—	2	2	no	—
88	C₂×C₄×C₂₂₇	—	2	2	no	—
89	C₂×C₄×C₂₂₇	—	2	2	no	—
90	C₂×C₄×C₂₂₇	—	2	2	no	—
91	C₂×C₄×C₂₂₇	—	4	4	no	—
92	C₂×C₄×C₂₂₇	—	8	2	no	—
93	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
97	C₈×C₂₂₇	—	2	2	no	—
98	C₈×C₂₂₇	—	2	1	no	—
99	C₈×C₂₂₇	C₂×D₄	1	2	yes	—
100	C₂²×C₂₂₇	—	2	2	no	—
101	C₂²×C₂₂₇	—	2	4	no	—
102	C₂²×C₂₂₇	—	2	2	no	—
103	C₂²×C₂₂₇	—	2	2	no	—
104	C₂²×C₂₂₇	—	4	2	no	—
105	C₂²×C₂₂₇	—	4	2	no	—
106	C₂²×C₂₂₇	—	4	2	no	—
107	C₂²×C₂₂₇	—	4	2	no	—
108	C₂²×C₂₂₇	—	4	2	no	—
109	C₂²×C₂₂₇	—	8	8	no	—
110	C₂²×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
111	C₄×C₂₂₇	—	2	2	no	—
112	C₄×C₂₂₇	—	2	1	no	—
113	C₄×C₂₂₇	—	2	1	no	—
114	C₄×C₂₂₇	—	4	2	no	—
115	C₄×C₂₂₇	—	4	2	no	—
116	C₄×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
117	C₄×C₂₂₇	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂×C₂₂₇	—	4	2	no	—
119	C₂×C₂₂₇	—	4	4	no	—
120	C₂×C₂₂₇	—	8	2	no	—
121	C₂×C₂₂₇	D₄²	1	1	yes	characteristic
122	C₂×C₂₂₇	#575	1	1	yes	characteristic
123	C₂×C₂₂₇	#576	1	1	yes	characteristic
124	C₂₂₇	#3273	1	1	yes	characteristic, direct product factor, Sylow
125	#3273	C₂₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂×AGL(1,ℤ/8)	C₂×C₂₂₇	1	2	yes	complemented
127	#546	C₂×C₂₂₇	1	2	yes	complemented
128	#495	C₂×C₂₂₇	1	1	yes	characteristic, complemented
129	#468	C₂×C₂₂₇	1	2	yes	complemented
130	#463	C₂×C₂₂₇	1	2	yes	complemented
131	#452	C₂×C₂₂₇	1	2	yes	complemented
132	#447	C₂×C₂₂₇	1	2	yes	complemented
133	C₄×D₈	C₂×C₂₂₇	1	1	yes	characteristic, complemented
134	#436	C₂×C₂₂₇	1	1	yes	characteristic, complemented
135	C₂×#41	C₂²×C₂₂₇	1	2	yes	—
136	C₂²×D₄	C₂²×C₂₂₇	1	2	yes	complemented
137	AGL(1,ℤ/8)	—	2	8	no	—
138	C₂×Q₁₆	C₂²×C₂₂₇	1	1	yes	characteristic, complemented
139	C₂×SD₁₆	C₂²×C₂₂₇	1	2	yes	complemented
140	C₂×SD₁₆	C₂²×C₂₂₇	1	2	yes	complemented
141	C₂×D₈	C₂²×C₂₂₇	1	2	yes	—
142	C₂×D₈	C₂²×C₂₂₇	1	1	yes	characteristic
143	C₂×D₈	C₂²×C₂₂₇	1	1	yes	characteristic
144	C₂×M₁₆	C₂²×C₂₂₇	1	2	yes	complemented
145	#124	C₂²×C₂₂₇	1	2	yes	complemented
146	#123	—	2	2	no	—
147	C₂²⋊Q₈	C₂²×C₂₂₇	1	2	yes	complemented
148	#121	—	2	2	no	—
149	#121	C₂²×C₂₂₇	1	2	yes	—
150	#120	—	2	2	no	—
151	C₄×D₄	C₂²×C₂₂₇	1	1	yes	characteristic
152	C₄×D₄	C₂²×C₂₂₇	1	2	yes	complemented
153	C₂×C₂²⋊C₄	—	2	2	no	—
154	#107	C₂²×C₂₂₇	1	1	yes	characteristic, complemented
155	C₄⋊C₈	C₂²×C₂₂₇	1	1	yes	characteristic, complemented
156	#103	C₂²×C₂₂₇	1	2	yes	complemented
157	#102	C₂²×C₂₂₇	1	2	yes	complemented
158	#102	C₂²×C₂₂₇	1	2	yes	complemented
159	C₂²⋊C₈	C₂²×C₂₂₇	1	2	yes	complemented
160	C₄×C₈	C₂²×C₂₂₇	1	1	yes	characteristic, complemented
161	C₂⁴	—	2	2	no	—
162	#41	—	2	4	no	—
163	#41	—	4	2	no	—
164	C₂×Q₈	C₂³×C₂₂₇	1	2	yes	—
165	C₂×D₄	—	2	4	no	—
166	C₂×D₄	—	2	2	no	—
167	C₂×D₄	—	2	2	no	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	4	2	no	—
170	C₂×D₄	D₄×C₂₂₇	1	1	yes	characteristic
171	C₂×D₄	C₂³×C₂₂₇	1	2	yes	—
172	C₂×D₄	C₂³×C₂₂₇	1	2	yes	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
176	Q₁₆	—	4	1	no	—
177	SD₁₆	—	2	4	no	—
178	SD₁₆	—	2	4	no	—
179	D₈	—	2	4	no	—
180	D₈	—	4	1	no	—
181	D₈	D₄×C₂₂₇	1	4	yes	complemented
182	M₁₆	—	2	4	no	—
183	C₂×C₈	C₂³×C₂₂₇	1	1	yes	characteristic
184	C₂×C₈	C₂³×C₂₂₇	1	2	yes	—
185	C₂×C₈	C₂³×C₂₂₇	1	1	yes	characteristic
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	D₄×C₂₂₇	1	1	yes	characteristic
188	C₄⋊C₄	C₂³×C₂₂₇	1	2	yes	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	D₄×C₂₂₇	1	2	yes	—
195	C₄²	C₂³×C₂₂₇	1	1	yes	characteristic
196	C₂³	—	2	2	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	4	4	no	—
199	C₂³	—	4	4	no	—
200	C₂³	—	8	2	no	—
201	C₂³	C₂×D₄×C₂₂₇	1	2	yes	—
202	Q₈	—	2	2	no	—
203	Q₈	—	2	2	no	—
204	D₄	—	2	2	no	—
205	D₄	—	2	2	no	—
206	D₄	—	2	2	no	—
207	D₄	—	4	1	no	—
208	D₄	—	4	4	no	—
209	D₄	—	4	4	no	—
210	D₄	—	8	2	no	—
211	D₄	C₂×D₄×C₂₂₇	1	4	yes	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	C₂×D₄×C₂₂₇	1	1	yes	characteristic
219	C₂×C₄	C₂×D₄×C₂₂₇	1	1	yes	characteristic
220	C₂×C₄	C₂×D₄×C₂₂₇	1	2	yes	—
221	C₂×C₄	C₂⁴×C₂₂₇	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	2	1	no	—
224	C₈	C₂×D₄×C₂₂₇	1	2	yes	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	4	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	4	2	no	—
230	C₂²	—	4	2	no	—
231	C₂²	—	4	2	no	—
232	C₂²	—	4	2	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	8	8	no	—
235	C₂²	C₂²×D₄×C₂₂₇	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	2	no	—
241	C₄	C₂²×D₄×C₂₂₇	1	1	yes	characteristic
242	C₄	H(5,𝔽₂)×C₂₂₇	1	1	yes	characteristic
243	C₂	—	4	2	no	—
244	C₂	—	4	4	no	—
245	C₂	—	8	2	no	—
246	C₂	D₄²×C₂₂₇	1	1	yes	characteristic
247	C₂	#575×C₂₂₇	1	1	yes	characteristic
248	C₂	#576×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×AGL(1,ℤ/8)×C₂₂₇	C₂	semidirect
2. #546×C₂₂₇	C₂	semidirect
3. #495×C₂₂₇	C₂	semidirect
4. #468×C₂₂₇	C₂	semidirect
5. #463×C₂₂₇	C₂	semidirect
6. #452×C₂₂₇	C₂	semidirect
7. #447×C₂₂₇	C₂	semidirect
8. C₄×D₈×C₂₂₇	C₂	semidirect
9. #436×C₂₂₇	C₂	semidirect
11. C₂²×D₄×C₂₂₇	C₂²	semidirect
13. C₂×Q₁₆×C₂₂₇	C₂²	semidirect
14. C₂×SD₁₆×C₂₂₇	C₂²	semidirect
15. C₂×SD₁₆×C₂₂₇	C₂²	semidirect
19. C₂×M₁₆×C₂₂₇	C₂²	semidirect
20. #124×C₂₂₇	C₂²	semidirect
22. C₂²⋊Q₈×C₂₂₇	C₂²	semidirect
27. C₄×D₄×C₂₂₇	C₂²	semidirect
29. #107×C₂₂₇	C₂²	semidirect
30. C₄⋊C₈×C₂₂₇	C₂²	semidirect
31. #103×C₂₂₇	C₂²	semidirect
32. #102×C₂₂₇	C₂²	semidirect
33. #102×C₂₂₇	C₂²	semidirect
34. C₂²⋊C₈×C₂₂₇	C₂²	semidirect
35. C₄×C₈×C₂₂₇	C₂²	semidirect
56. D₈×C₂₂₇	D₄	semidirect
124. C₂₂₇	#3273	direct (N × H)
125. #3273	C₂₂₇	direct (N × H)
126. C₂×AGL(1,ℤ/8)	C₂×C₂₂₇	semidirect
127. #546	C₂×C₂₂₇	semidirect
128. #495	C₂×C₂₂₇	semidirect
129. #468	C₂×C₂₂₇	semidirect
130. #463	C₂×C₂₂₇	semidirect
131. #452	C₂×C₂₂₇	semidirect
132. #447	C₂×C₂₂₇	semidirect
133. C₄×D₈	C₂×C₂₂₇	semidirect
134. #436	C₂×C₂₂₇	semidirect
136. C₂²×D₄	C₂²×C₂₂₇	semidirect
138. C₂×Q₁₆	C₂²×C₂₂₇	semidirect
139. C₂×SD₁₆	C₂²×C₂₂₇	semidirect
140. C₂×SD₁₆	C₂²×C₂₂₇	semidirect
144. C₂×M₁₆	C₂²×C₂₂₇	semidirect
145. #124	C₂²×C₂₂₇	semidirect
147. C₂²⋊Q₈	C₂²×C₂₂₇	semidirect
152. C₄×D₄	C₂²×C₂₂₇	semidirect
154. #107	C₂²×C₂₂₇	semidirect
155. C₄⋊C₈	C₂²×C₂₂₇	semidirect
156. #103	C₂²×C₂₂₇	semidirect
157. #102	C₂²×C₂₂₇	semidirect
158. #102	C₂²×C₂₂₇	semidirect
159. C₂²⋊C₈	C₂²×C₂₂₇	semidirect
160. C₄×C₈	C₂²×C₂₂₇	semidirect
181. D₈	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×AGL(1,ℤ/8)×C₂₂₇	2
#546×C₂₂₇	2
#495×C₂₂₇	2
#468×C₂₂₇	2
#463×C₂₂₇	2
#452×C₂₂₇	2
#447×C₂₂₇	2
C₄×D₈×C₂₂₇	2
#436×C₂₂₇	2
#3273	227

Subgroups — #3273×C227 #44397229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3273×C₂₂₇ #44397229