Subgroups — #3303×C₂₂₇ #44397259

Order 29056.

398 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#555×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#547×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#487×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#485×C₂₂₇	C₂	1	2	yes	maximal, complemented
5	#476×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#465×C₂₂₇	C₂	1	2	yes	maximal, complemented
7	#461×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#449×C₂₂₇	C₂	1	2	yes	maximal, complemented
9	C₄×D₈×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×D₄×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#416×C₂₂₇	C₂	1	2	yes	maximal, complemented
12	C₂×#41×C₂₂₇	C₂²	1	2	yes	—
13	C₂×SD₁₆×C₂₂₇	C₂²	1	2	yes	—
14	C₂×D₈×C₂₂₇	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₂₂₇	C₂²	1	2	yes	—
16	C₄⋊Q₈×C₂₂₇	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₂₂₇	C₂²	1	1	yes	characteristic
18	#124×C₂₂₇	—	2	2	no	—
19	#123×C₂₂₇	—	2	2	no	—
20	C₂²⋊Q₈×C₂₂₇	C₂²	1	2	yes	—
21	#121×C₂₂₇	—	2	1	no	—
22	#121×C₂₂₇	—	2	1	no	—
23	#121×C₂₂₇	C₂²	1	2	yes	—
24	C₄×D₄×C₂₂₇	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₂₂₇	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₂₂₇	C₂²	1	1	yes	characteristic
27	#117×C₂₂₇	—	2	1	no	—
28	#117×C₂₂₇	C₂²	1	2	yes	—
29	C₂×C₄⋊C₄×C₂₂₇	—	2	1	no	—
30	#107×C₂₂₇	C₂²	1	1	yes	characteristic
31	#106×C₂₂₇	C₂²	1	2	yes	—
32	C₄⋊C₈×C₂₂₇	C₂²	1	1	yes	characteristic
33	#103×C₂₂₇	C₂²	1	2	yes	—
34	#103×C₂₂₇	C₂²	1	2	yes	—
35	#103×C₂₂₇	C₂²	1	2	yes	—
36	#102×C₂₂₇	C₂²	1	1	yes	characteristic
37	#102×C₂₂₇	C₂²	1	2	yes	—
38	#102×C₂₂₇	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₂₂₇	C₂²	1	2	yes	—
40	C₄×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
41	#41×C₂₂₇	—	2	4	no	—
42	#41×C₂₂₇	—	4	2	no	—
43	C₂×Q₈×C₂₂₇	—	2	1	no	—
44	C₂×Q₈×C₂₂₇	C₂³	1	2	yes	—
45	C₂×D₄×C₂₂₇	—	2	2	no	—
46	C₂×D₄×C₂₂₇	—	2	2	no	—
47	C₂×D₄×C₂₂₇	D₄	1	1	yes	characteristic
48	C₂×D₄×C₂₂₇	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₂₂₇	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₂₂₇	—	2	1	no	—
51	C₂²×C₄×C₂₂₇	—	2	2	no	—
52	C₂²×C₄×C₂₂₇	—	2	1	no	—
53	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
54	SD₁₆×C₂₂₇	—	4	2	no	—
55	D₈×C₂₂₇	—	2	2	no	—
56	C₂×C₈×C₂₂₇	—	4	2	no	—
57	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₂₂₇	C₂³	1	2	yes	—
60	C₄⋊C₄×C₂₂₇	—	2	1	no	—
61	C₄⋊C₄×C₂₂₇	—	2	2	no	—
62	C₄⋊C₄×C₂₂₇	—	2	1	no	—
63	C₄⋊C₄×C₂₂₇	—	2	1	no	—
64	C₄⋊C₄×C₂₂₇	D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₂₂₇	C₂³	1	2	yes	—
66	C₄⋊C₄×C₂₂₇	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₂₂₇	C₂³	1	1	yes	characteristic
68	C₂²⋊C₄×C₂₂₇	—	2	2	no	—
69	C₂²⋊C₄×C₂₂₇	—	2	2	no	—
70	C₂²⋊C₄×C₂₂₇	—	2	2	no	—
71	C₂²⋊C₄×C₂₂₇	—	2	1	no	—
72	C₂²⋊C₄×C₂₂₇	—	2	1	no	—
73	C₂²⋊C₄×C₂₂₇	D₄	1	2	yes	—
74	C₄²×C₂₂₇	—	2	2	no	—
75	C₄²×C₂₂₇	C₂³	1	1	yes	characteristic
76	C₂³×C₂₂₇	—	2	1	no	—
77	C₂³×C₂₂₇	—	2	1	no	—
78	C₂³×C₂₂₇	C₂×D₄	1	2	yes	—
79	Q₈×C₂₂₇	—	2	2	no	—
80	Q₈×C₂₂₇	—	2	2	no	—
81	Q₈×C₂₂₇	—	8	1	no	—
82	D₄×C₂₂₇	—	2	2	no	—
83	D₄×C₂₂₇	—	2	1	no	—
84	D₄×C₂₂₇	—	2	1	no	—
85	D₄×C₂₂₇	—	2	1	no	—
86	D₄×C₂₂₇	—	4	4	no	—
87	D₄×C₂₂₇	—	8	2	no	—
88	D₄×C₂₂₇	#41	1	2	yes	—
89	C₂×C₄×C₂₂₇	—	2	2	no	—
90	C₂×C₄×C₂₂₇	—	2	2	no	—
91	C₂×C₄×C₂₂₇	—	2	1	no	—
92	C₂×C₄×C₂₂₇	—	2	2	no	—
93	C₂×C₄×C₂₂₇	—	2	1	no	—
94	C₂×C₄×C₂₂₇	—	2	2	no	—
95	C₂×C₄×C₂₂₇	—	4	4	no	—
96	C₂×C₄×C₂₂₇	—	4	2	no	—
97	C₂×C₄×C₂₂₇	—	4	2	no	—
98	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
102	C₈×C₂₂₇	—	2	1	no	—
103	C₈×C₂₂₇	—	2	1	no	—
104	C₈×C₂₂₇	—	2	2	no	—
105	C₂²×C₂₂₇	—	2	2	no	—
106	C₂²×C₂₂₇	—	2	2	no	—
107	C₂²×C₂₂₇	—	2	2	no	—
108	C₂²×C₂₂₇	—	2	2	no	—
109	C₂²×C₂₂₇	—	4	1	no	—
110	C₂²×C₂₂₇	—	4	1	no	—
111	C₂²×C₂₂₇	—	4	1	no	—
112	C₂²×C₂₂₇	—	4	1	no	—
113	C₂²×C₂₂₇	—	4	1	no	—
114	C₂²×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
115	C₄×C₂₂₇	—	2	2	no	—
116	C₄×C₂₂₇	—	2	1	no	—
117	C₄×C₂₂₇	—	4	2	no	—
118	C₄×C₂₂₇	—	4	1	no	—
119	C₄×C₂₂₇	—	4	1	no	—
120	C₄×C₂₂₇	—	4	2	no	—
121	C₄×C₂₂₇	#135	1	2	yes	—
122	C₄×C₂₂₇	C₂×#41	1	1	yes	characteristic
123	C₄×C₂₂₇	H(5,𝔽₂)	1	1	yes	characteristic
124	C₂×C₂₂₇	—	4	2	no	—
125	C₂×C₂₂₇	—	4	2	no	—
126	C₂×C₂₂₇	—	8	1	no	—
127	C₂×C₂₂₇	#546	1	1	yes	characteristic
128	C₂×C₂₂₇	C₂×#135	1	1	yes	characteristic
129	C₂×C₂₂₇	#577	1	1	yes	characteristic
130	C₂₂₇	#3303	1	1	yes	characteristic, direct product factor, Sylow
131	#3303	C₂₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	#555	C₂×C₂₂₇	1	1	yes	characteristic, complemented
133	#547	C₂×C₂₂₇	1	1	yes	characteristic, complemented
134	#487	C₂×C₂₂₇	1	1	yes	characteristic, complemented
135	#485	C₂×C₂₂₇	1	2	yes	complemented
136	#476	C₂×C₂₂₇	1	1	yes	characteristic, complemented
137	#465	C₂×C₂₂₇	1	2	yes	complemented
138	#461	C₂×C₂₂₇	1	1	yes	characteristic, complemented
139	#449	C₂×C₂₂₇	1	2	yes	complemented
140	C₄×D₈	C₂×C₂₂₇	1	1	yes	characteristic, complemented
141	C₈×D₄	C₂×C₂₂₇	1	1	yes	characteristic, complemented
142	#416	C₂×C₂₂₇	1	2	yes	complemented
143	C₂×#41	C₂²×C₂₂₇	1	2	yes	—
144	C₂×SD₁₆	C₂²×C₂₂₇	1	2	yes	—
145	C₂×D₈	C₂²×C₂₂₇	1	1	yes	characteristic
146	C₂²×C₈	C₂²×C₂₂₇	1	2	yes	—
147	C₄⋊Q₈	C₂²×C₂₂₇	1	1	yes	characteristic
148	C₄⋊Q₈	C₂²×C₂₂₇	1	1	yes	characteristic
149	#124	—	2	2	no	—
150	#123	—	2	2	no	—
151	C₂²⋊Q₈	C₂²×C₂₂₇	1	2	yes	—
152	#121	—	2	1	no	—
153	#121	—	2	1	no	—
154	#121	C₂²×C₂₂₇	1	2	yes	—
155	C₄×D₄	C₂²×C₂₂₇	1	1	yes	characteristic
156	C₄×D₄	C₂²×C₂₂₇	1	1	yes	characteristic
157	C₄×D₄	C₂²×C₂₂₇	1	1	yes	characteristic
158	#117	—	2	1	no	—
159	#117	C₂²×C₂₂₇	1	2	yes	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	#107	C₂²×C₂₂₇	1	1	yes	characteristic
162	#106	C₂²×C₂₂₇	1	2	yes	—
163	C₄⋊C₈	C₂²×C₂₂₇	1	1	yes	characteristic
164	#103	C₂²×C₂₂₇	1	2	yes	—
165	#103	C₂²×C₂₂₇	1	2	yes	—
166	#103	C₂²×C₂₂₇	1	2	yes	—
167	#102	C₂²×C₂₂₇	1	1	yes	characteristic
168	#102	C₂²×C₂₂₇	1	2	yes	—
169	#102	C₂²×C₂₂₇	1	1	yes	characteristic
170	C₂²⋊C₈	C₂²×C₂₂₇	1	2	yes	—
171	C₄×C₈	C₂²×C₂₂₇	1	1	yes	characteristic
172	#41	—	2	4	no	—
173	#41	—	4	2	no	—
174	C₂×Q₈	—	2	1	no	—
175	C₂×Q₈	C₂³×C₂₂₇	1	2	yes	—
176	C₂×D₄	—	2	2	no	—
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	D₄×C₂₂₇	1	1	yes	characteristic
179	C₂×D₄	C₂³×C₂₂₇	1	1	yes	characteristic
180	C₂×D₄	C₂³×C₂₂₇	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
185	SD₁₆	—	4	2	no	—
186	D₈	—	2	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	C₂³×C₂₂₇	1	1	yes	characteristic
189	C₂×C₈	C₂³×C₂₂₇	1	1	yes	characteristic
190	C₂×C₈	C₂³×C₂₂₇	1	2	yes	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	D₄×C₂₂₇	1	1	yes	characteristic
196	C₄⋊C₄	C₂³×C₂₂₇	1	2	yes	—
197	C₄⋊C₄	C₂³×C₂₂₇	1	1	yes	characteristic
198	C₄⋊C₄	C₂³×C₂₂₇	1	1	yes	characteristic
199	C₂²⋊C₄	—	2	2	no	—
200	C₂²⋊C₄	—	2	2	no	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	D₄×C₂₂₇	1	2	yes	—
205	C₄²	—	2	2	no	—
206	C₄²	C₂³×C₂₂₇	1	1	yes	characteristic
207	C₂³	—	2	1	no	—
208	C₂³	—	2	1	no	—
209	C₂³	C₂×D₄×C₂₂₇	1	2	yes	—
210	Q₈	—	2	2	no	—
211	Q₈	—	2	2	no	—
212	Q₈	—	8	1	no	—
213	D₄	—	2	2	no	—
214	D₄	—	2	1	no	—
215	D₄	—	2	1	no	—
216	D₄	—	2	1	no	—
217	D₄	—	4	4	no	—
218	D₄	—	8	2	no	—
219	D₄	#41×C₂₂₇	1	2	yes	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	C₂×D₄×C₂₂₇	1	2	yes	—
230	C₂×C₄	C₂×D₄×C₂₂₇	1	1	yes	characteristic
231	C₂×C₄	C₂×D₄×C₂₂₇	1	1	yes	characteristic
232	C₂×C₄	C₂⁴×C₂₂₇	1	1	yes	characteristic
233	C₈	—	2	1	no	—
234	C₈	—	2	1	no	—
235	C₈	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	4	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	4	1	no	—
245	C₂²	C₂²×D₄×C₂₂₇	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	1	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	1	no	—
251	C₄	—	4	2	no	—
252	C₄	#135×C₂₂₇	1	2	yes	—
253	C₄	C₂×#41×C₂₂₇	1	1	yes	characteristic
254	C₄	H(5,𝔽₂)×C₂₂₇	1	1	yes	characteristic
255	C₂	—	4	2	no	—
256	C₂	—	4	2	no	—
257	C₂	—	8	1	no	—
258	C₂	#546×C₂₂₇	1	1	yes	characteristic
259	C₂	C₂×#135×C₂₂₇	1	1	yes	characteristic
260	C₂	#577×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. #555×C₂₂₇	C₂	semidirect
2. #547×C₂₂₇	C₂	semidirect
3. #487×C₂₂₇	C₂	semidirect
4. #485×C₂₂₇	C₂	semidirect
5. #476×C₂₂₇	C₂	semidirect
6. #465×C₂₂₇	C₂	semidirect
7. #461×C₂₂₇	C₂	semidirect
8. #449×C₂₂₇	C₂	semidirect
9. C₄×D₈×C₂₂₇	C₂	semidirect
10. C₈×D₄×C₂₂₇	C₂	semidirect
11. #416×C₂₂₇	C₂	semidirect
130. C₂₂₇	#3303	direct (N × H)
131. #3303	C₂₂₇	direct (N × H)
132. #555	C₂×C₂₂₇	semidirect
133. #547	C₂×C₂₂₇	semidirect
134. #487	C₂×C₂₂₇	semidirect
135. #485	C₂×C₂₂₇	semidirect
136. #476	C₂×C₂₂₇	semidirect
137. #465	C₂×C₂₂₇	semidirect
138. #461	C₂×C₂₂₇	semidirect
139. #449	C₂×C₂₂₇	semidirect
140. C₄×D₈	C₂×C₂₂₇	semidirect
141. C₈×D₄	C₂×C₂₂₇	semidirect
142. #416	C₂×C₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
#555×C₂₂₇	2
#547×C₂₂₇	2
#487×C₂₂₇	2
#485×C₂₂₇	2
#476×C₂₂₇	2
#465×C₂₂₇	2
#461×C₂₂₇	2
#449×C₂₂₇	2
C₄×D₈×C₂₂₇	2
C₈×D₄×C₂₂₇	2
#416×C₂₂₇	2
#3303	227

Subgroups — #3303×C227 #44397259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3303×C₂₂₇ #44397259