Subgroups — #3057×C₂₂₇ #44397013

Order 29056.

492 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×SD₁₆×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#131×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#536×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#535×C₂₂₇	C₂	1	1	yes	maximal, characteristic
5	#468×C₂₂₇	C₂	1	4	yes	maximal, complemented
6	#465×C₂₂₇	C₂	1	4	yes	maximal, complemented
7	#427×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#419×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#418×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₂₂₇	C₂²	1	1	yes	characteristic
11	C₂²×Q₈×C₂₂₇	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₂₂₇	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×C₂₂₇	—	2	4	no	—
14	C₂×SD₁₆×C₂₂₇	C₂²	1	2	yes	—
15	C₂×SD₁₆×C₂₂₇	C₂²	1	2	yes	—
16	#131×C₂₂₇	—	2	4	no	—
17	C₂×M₁₆×C₂₂₇	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₂₂₇	C₂²	1	2	yes	complemented
19	C₂²×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₂₂₇	C₂²	1	2	yes	complemented
21	C₄⋊Q₈×C₂₂₇	—	2	1	no	—
22	#127×C₂₂₇	—	2	1	no	—
23	#124×C₂₂₇	—	2	1	no	—
24	#124×C₂₂₇	—	2	1	no	—
25	#123×C₂₂₇	—	2	2	no	—
26	C₂²⋊Q₈×C₂₂₇	C₂²	1	4	yes	complemented
27	#121×C₂₂₇	C₂²	1	4	yes	—
28	#120×C₂₂₇	—	2	2	no	—
29	#117×C₂₂₇	C₂²	1	1	yes	characteristic
30	#117×C₂₂₇	C₂²	1	1	yes	characteristic
31	#107×C₂₂₇	C₂²	1	2	yes	complemented
32	#106×C₂₂₇	C₂²	1	2	yes	complemented
33	#103×C₂₂₇	C₂²	1	4	yes	complemented
34	#102×C₂₂₇	C₂²	1	4	yes	—
35	C₂⁴×C₂₂₇	—	2	1	no	—
36	#41×C₂₂₇	—	2	2	no	—
37	#41×C₂₂₇	—	2	2	no	—
38	C₂×Q₈×C₂₂₇	—	2	1	no	—
39	C₂×Q₈×C₂₂₇	—	2	2	no	—
40	C₂×Q₈×C₂₂₇	—	2	2	no	—
41	C₂×Q₈×C₂₂₇	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₂₂₇	C₂³	1	2	yes	—
43	C₂×D₄×C₂₂₇	—	2	4	no	—
44	C₂×D₄×C₂₂₇	—	2	2	no	—
45	C₂×D₄×C₂₂₇	—	2	1	no	—
46	C₂×D₄×C₂₂₇	—	2	2	no	—
47	C₂×D₄×C₂₂₇	D₄	1	2	yes	—
48	C₂×D₄×C₂₂₇	D₄	1	1	yes	characteristic
49	C₂×D₄×C₂₂₇	C₂³	1	2	yes	—
50	C₂²×C₄×C₂₂₇	—	2	1	no	—
51	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
52	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
53	SD₁₆×C₂₂₇	—	2	4	no	—
54	SD₁₆×C₂₂₇	—	2	4	no	—
55	M₁₆×C₂₂₇	—	2	2	no	—
56	M₁₆×C₂₂₇	—	2	4	no	—
57	C₂×C₈×C₂₂₇	—	2	4	no	—
58	C₂×C₈×C₂₂₇	D₄	1	4	yes	complemented
59	C₂×C₈×C₂₂₇	C₂³	1	2	yes	—
60	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₂₂₇	—	2	4	no	—
63	C₄⋊C₄×C₂₂₇	C₂³	1	2	yes	—
64	C₄⋊C₄×C₂₂₇	C₂³	1	2	yes	—
65	C₂²⋊C₄×C₂₂₇	—	2	1	no	—
66	C₂²⋊C₄×C₂₂₇	—	2	1	no	—
67	C₂²⋊C₄×C₂₂₇	—	2	4	no	—
68	C₂²⋊C₄×C₂₂₇	—	2	4	no	—
69	C₄²×C₂₂₇	—	2	1	no	—
70	C₄²×C₂₂₇	—	2	1	no	—
71	C₂³×C₂₂₇	—	2	2	no	—
72	C₂³×C₂₂₇	—	4	2	no	—
73	C₂³×C₂₂₇	—	8	2	no	—
74	C₂³×C₂₂₇	C₂×D₄	1	2	yes	—
75	C₂³×C₂₂₇	C₂×D₄	1	1	yes	characteristic
76	Q₈×C₂₂₇	—	2	2	no	—
77	Q₈×C₂₂₇	—	2	2	no	—
78	Q₈×C₂₂₇	—	2	2	no	—
79	Q₈×C₂₂₇	—	4	2	no	—
80	D₄×C₂₂₇	—	2	4	no	—
81	D₄×C₂₂₇	—	2	2	no	—
82	D₄×C₂₂₇	—	2	2	no	—
83	D₄×C₂₂₇	—	2	2	no	—
84	D₄×C₂₂₇	—	4	2	no	—
85	D₄×C₂₂₇	—	8	4	no	—
86	C₂×C₄×C₂₂₇	—	2	2	no	—
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	2	no	—
92	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₂₂₇	#41	1	2	yes	—
96	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
97	C₈×C₂₂₇	—	2	2	no	—
98	C₈×C₂₂₇	C₂×D₄	1	2	yes	—
99	C₈×C₂₂₇	C₂×D₄	1	2	yes	—
100	C₂²×C₂₂₇	—	2	2	no	—
101	C₂²×C₂₂₇	—	2	2	no	—
102	C₂²×C₂₂₇	—	2	4	no	—
103	C₂²×C₂₂₇	—	4	2	no	—
104	C₂²×C₂₂₇	—	4	4	no	—
105	C₂²×C₂₂₇	—	8	4	no	—
106	C₂²×C₂₂₇	#121	1	2	yes	—
107	C₂²×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
108	C₄×C₂₂₇	—	2	2	no	—
109	C₄×C₂₂₇	—	4	2	no	—
110	C₄×C₂₂₇	—	4	2	no	—
111	C₄×C₂₂₇	—	4	2	no	—
112	C₄×C₂₂₇	#121	1	2	yes	—
113	C₄×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
114	C₄×C₂₂₇	C₂×#41	1	1	yes	characteristic
115	C₂×C₂₂₇	—	2	2	no	—
116	C₂×C₂₂₇	—	4	2	no	—
117	C₂×C₂₂₇	—	8	2	no	—
118	C₂×C₂₂₇	C₂×#121	1	1	yes	characteristic
119	C₂×C₂₂₇	#577	1	2	yes	—
120	C₂₂₇	#3057	1	1	yes	characteristic, direct product factor, Sylow
121	#3057	C₂₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂²×SD₁₆	C₂×C₂₂₇	1	1	yes	characteristic, complemented
123	C₂×#131	C₂×C₂₂₇	1	1	yes	characteristic, complemented
124	#536	C₂×C₂₂₇	1	1	yes	characteristic, complemented
125	#535	C₂×C₂₂₇	1	1	yes	characteristic
126	#468	C₂×C₂₂₇	1	4	yes	complemented
127	#465	C₂×C₂₂₇	1	4	yes	complemented
128	#427	C₂×C₂₂₇	1	1	yes	characteristic, complemented
129	#419	C₂×C₂₂₇	1	1	yes	characteristic, complemented
130	#418	C₂×C₂₂₇	1	1	yes	characteristic, complemented
131	C₂×#41	C₂²×C₂₂₇	1	1	yes	characteristic
132	C₂²×Q₈	C₂²×C₂₂₇	1	1	yes	characteristic
133	C₂²×D₄	C₂²×C₂₂₇	1	1	yes	characteristic
134	C₂×SD₁₆	—	2	4	no	—
135	C₂×SD₁₆	C₂²×C₂₂₇	1	2	yes	—
136	C₂×SD₁₆	C₂²×C₂₂₇	1	2	yes	—
137	#131	—	2	4	no	—
138	C₂×M₁₆	C₂²×C₂₂₇	1	1	yes	characteristic
139	C₂×M₁₆	C₂²×C₂₂₇	1	2	yes	complemented
140	C₂²×C₈	C₂²×C₂₂₇	1	1	yes	characteristic
141	C₂²×C₈	C₂²×C₂₂₇	1	2	yes	complemented
142	C₄⋊Q₈	—	2	1	no	—
143	#127	—	2	1	no	—
144	#124	—	2	1	no	—
145	#124	—	2	1	no	—
146	#123	—	2	2	no	—
147	C₂²⋊Q₈	C₂²×C₂₂₇	1	4	yes	complemented
148	#121	C₂²×C₂₂₇	1	4	yes	—
149	#120	—	2	2	no	—
150	#117	C₂²×C₂₂₇	1	1	yes	characteristic
151	#117	C₂²×C₂₂₇	1	1	yes	characteristic
152	#107	C₂²×C₂₂₇	1	2	yes	complemented
153	#106	C₂²×C₂₂₇	1	2	yes	complemented
154	#103	C₂²×C₂₂₇	1	4	yes	complemented
155	#102	C₂²×C₂₂₇	1	4	yes	—
156	C₂⁴	—	2	1	no	—
157	#41	—	2	2	no	—
158	#41	—	2	2	no	—
159	C₂×Q₈	—	2	1	no	—
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	D₄×C₂₂₇	1	1	yes	characteristic
163	C₂×Q₈	C₂³×C₂₂₇	1	2	yes	—
164	C₂×D₄	—	2	4	no	—
165	C₂×D₄	—	2	2	no	—
166	C₂×D₄	—	2	1	no	—
167	C₂×D₄	—	2	2	no	—
168	C₂×D₄	D₄×C₂₂₇	1	2	yes	—
169	C₂×D₄	D₄×C₂₂₇	1	1	yes	characteristic
170	C₂×D₄	C₂³×C₂₂₇	1	2	yes	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
173	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
174	SD₁₆	—	2	4	no	—
175	SD₁₆	—	2	4	no	—
176	M₁₆	—	2	2	no	—
177	M₁₆	—	2	4	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	D₄×C₂₂₇	1	4	yes	complemented
180	C₂×C₈	C₂³×C₂₂₇	1	2	yes	—
181	C₂×C₈	C₂³×C₂₂₇	1	1	yes	characteristic
182	C₂×C₈	C₂³×C₂₂₇	1	1	yes	characteristic
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	C₂³×C₂₂₇	1	2	yes	—
185	C₄⋊C₄	C₂³×C₂₂₇	1	2	yes	—
186	C₂²⋊C₄	—	2	1	no	—
187	C₂²⋊C₄	—	2	1	no	—
188	C₂²⋊C₄	—	2	4	no	—
189	C₂²⋊C₄	—	2	4	no	—
190	C₄²	—	2	1	no	—
191	C₄²	—	2	1	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	4	2	no	—
194	C₂³	—	8	2	no	—
195	C₂³	C₂×D₄×C₂₂₇	1	2	yes	—
196	C₂³	C₂×D₄×C₂₂₇	1	1	yes	characteristic
197	Q₈	—	2	2	no	—
198	Q₈	—	2	2	no	—
199	Q₈	—	2	2	no	—
200	Q₈	—	4	2	no	—
201	D₄	—	2	4	no	—
202	D₄	—	2	2	no	—
203	D₄	—	2	2	no	—
204	D₄	—	2	2	no	—
205	D₄	—	4	2	no	—
206	D₄	—	8	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	C₂×D₄×C₂₂₇	1	2	yes	—
214	C₂×C₄	C₂×D₄×C₂₂₇	1	1	yes	characteristic
215	C₂×C₄	C₂×D₄×C₂₂₇	1	2	yes	—
216	C₂×C₄	#41×C₂₂₇	1	2	yes	—
217	C₂×C₄	C₂⁴×C₂₂₇	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	C₂×D₄×C₂₂₇	1	2	yes	—
220	C₈	C₂×D₄×C₂₂₇	1	2	yes	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	4	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	4	no	—
226	C₂²	—	8	4	no	—
227	C₂²	#121×C₂₂₇	1	2	yes	—
228	C₂²	C₂²×D₄×C₂₂₇	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	4	2	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	2	no	—
233	C₄	#121×C₂₂₇	1	2	yes	—
234	C₄	C₂²×D₄×C₂₂₇	1	1	yes	characteristic
235	C₄	C₂×#41×C₂₂₇	1	1	yes	characteristic
236	C₂	—	2	2	no	—
237	C₂	—	4	2	no	—
238	C₂	—	8	2	no	—
239	C₂	C₂×#121×C₂₂₇	1	1	yes	characteristic
240	C₂	#577×C₂₂₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₂₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₂₇
prime radical	#535×C₂₂₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₇ ◁ C₂×#41×C₂₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×SD₁₆×C₂₂₇	C₂	semidirect
2. C₂×#131×C₂₂₇	C₂	semidirect
3. #536×C₂₂₇	C₂	semidirect
5. #468×C₂₂₇	C₂	semidirect
6. #465×C₂₂₇	C₂	semidirect
7. #427×C₂₂₇	C₂	semidirect
8. #419×C₂₂₇	C₂	semidirect
9. #418×C₂₂₇	C₂	semidirect
18. C₂×M₁₆×C₂₂₇	C₂²	semidirect
20. C₂²×C₈×C₂₂₇	C₂²	semidirect
26. C₂²⋊Q₈×C₂₂₇	C₂²	semidirect
31. #107×C₂₂₇	C₂²	semidirect
32. #106×C₂₂₇	C₂²	semidirect
33. #103×C₂₂₇	C₂²	semidirect
58. C₂×C₈×C₂₂₇	D₄	semidirect
120. C₂₂₇	#3057	direct (N × H)
121. #3057	C₂₂₇	direct (N × H)
122. C₂²×SD₁₆	C₂×C₂₂₇	semidirect
123. C₂×#131	C₂×C₂₂₇	semidirect
124. #536	C₂×C₂₂₇	semidirect
126. #468	C₂×C₂₂₇	semidirect
127. #465	C₂×C₂₂₇	semidirect
128. #427	C₂×C₂₂₇	semidirect
129. #419	C₂×C₂₂₇	semidirect
130. #418	C₂×C₂₂₇	semidirect
139. C₂×M₁₆	C₂²×C₂₂₇	semidirect
141. C₂²×C₈	C₂²×C₂₂₇	semidirect
147. C₂²⋊Q₈	C₂²×C₂₂₇	semidirect
152. #107	C₂²×C₂₂₇	semidirect
153. #106	C₂²×C₂₂₇	semidirect
154. #103	C₂²×C₂₂₇	semidirect
179. C₂×C₈	D₄×C₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×SD₁₆×C₂₂₇	2
C₂×#131×C₂₂₇	2
#536×C₂₂₇	2
#535×C₂₂₇	2
#468×C₂₂₇	2
#465×C₂₂₇	2
#427×C₂₂₇	2
#419×C₂₂₇	2
#418×C₂₂₇	2
#3057	227

Subgroups — #3057×C227 #44397013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3057×C₂₂₇ #44397013