Subgroups — C₂×#28409103 #44408413

Order 29056.

564 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#16813867	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×C₂²⋊D₂₂₇	C₂	1	1	yes	maximal, characteristic
3	#28409103	C₂	1	8	yes	maximal, direct product factor
4	C₂×Dic₂₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊(C₂₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×(C₂×D₂₂₇)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#20300198	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₈×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₂²⋊D₂₂₇	C₄	1	1	yes	characteristic
10	C₂²×C₄×D₂₂₇	C₂²	1	1	yes	characteristic
11	C₂×#16813867	—	2	4	no	—
12	C₂×#16813867	C₂²	1	4	yes	complemented
13	C₂²×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₄×C₂²⋊D₂₂₇	C₄	1	4	yes	—
16	C₄×C₂²⋊D₂₂₇	C₂²	1	4	yes	—
17	C₂×Dic₂₂₇⋊C₄	C₄	1	1	yes	characteristic
18	C₂×C₂²⋊Dic₂₂₇	C₄	1	1	yes	characteristic
19	C₂×(C₂×D₂₂₇)⋊C₄	C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
21	Dic₂₂₇⋊C₈	C₂²	1	4	yes	complemented
22	C₂²⋊(C₂₂₇⋊C₈)	C₂²	1	4	yes	complemented
23	(C₂×D₂₂₇)⋊C₈	C₂²	1	4	yes	complemented
24	#20300198	C₂²	1	4	yes	complemented
25	C₂³×C₄×C₂₂₇	C₂²	1	1	yes	characteristic
26	C₂²×C₈×C₂₂₇	—	2	4	no	—
27	C₂²×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
28	C₂²×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
29	C₂²×C₈×C₂₂₇	C₂²	1	4	yes	complemented
30	C₂³×D₂₂₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₂²⋊D₂₂₇	—	2	4	no	—
32	C₂×C₂²⋊D₂₂₇	C₂×C₄	1	4	yes	—
33	C₂²×Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₂₂₇	—	2	2	no	—
37	C₂×C₄×D₂₂₇	—	2	2	no	—
38	C₂×C₄×D₂₂₇	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₂₂₇	C₂³	1	2	yes	—
40	#16813867	—	2	8	no	—
41	C₂×C₂₂₇⋊C₈	—	2	2	no	—
42	C₂×C₂₂₇⋊C₈	—	2	2	no	—
43	C₂×C₂₂₇⋊C₈	C₂³	1	2	yes	—
44	C₂×C₂₂₇⋊C₈	C₂³	1	2	yes	—
45	Dic₂₂₇⋊C₄	C₂×C₄	1	4	yes	—
46	C₂²⋊Dic₂₂₇	C₂×C₄	1	4	yes	—
47	(C₂×D₂₂₇)⋊C₄	C₂×C₄	1	4	yes	—
48	C₄×Dic₂₂₇	C₂×C₄	1	2	yes	—
49	C₄×Dic₂₂₇	C₂³	1	2	yes	—
50	C₂⁴×C₂₂₇	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₂₇	—	2	2	no	—
52	C₂²×C₄×C₂₂₇	—	2	2	no	—
53	C₂²×C₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₂₇	C₂×C₄	1	2	yes	—
55	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
56	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₂₂₇	—	2	2	no	—
58	C₂×C₈×C₂₂₇	—	2	8	no	—
59	C₂×C₈×C₂₂₇	D₄	1	4	yes	complemented
60	C₂×C₈×C₂₂₇	C₂³	1	2	yes	—
61	C₂×C₈×C₂₂₇	C₂³	1	2	yes	—
62	C₂²×D₂₂₇	—	2	2	no	—
63	C₂²×D₂₂₇	—	2	2	no	—
64	C₂²×D₂₂₇	—	2	2	no	—
65	C₂²×D₂₂₇	C₂²×C₄	1	2	yes	—
66	C₂²⋊D₂₂₇	—	2	8	no	—
67	C₂×Dic₂₂₇	—	2	2	no	—
68	C₂×Dic₂₂₇	—	2	2	no	—
69	C₂×Dic₂₂₇	—	2	2	no	—
70	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
71	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
72	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
73	C₄×D₂₂₇	—	2	4	no	—
74	C₄×D₂₂₇	—	2	2	no	—
75	C₄×D₂₂₇	—	2	2	no	—
76	C₂₂₇⋊C₈	—	2	2	no	—
77	C₂₂₇⋊C₈	—	2	2	no	—
78	C₂³×C₂₂₇	—	2	2	no	—
79	C₂³×C₂₂₇	—	2	2	no	—
80	C₂³×C₂₂₇	M₁₆	1	4	yes	complemented
81	C₂³×C₂₂₇	C₂²×C₄	1	2	yes	—
82	C₂³×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₂₂₇	—	2	2	no	—
84	C₂×C₄×C₂₂₇	—	2	4	no	—
85	C₂×C₄×C₂₂₇	—	2	2	no	—
86	C₂×C₄×C₂₂₇	—	2	2	no	—
87	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₂₂₇	#41	1	2	yes	—
91	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
92	C₈×C₂₂₇	—	2	2	no	—
93	C₈×C₂₂₇	C₂×D₄	1	4	yes	complemented
94	C₂²×C₂₂₇	—	2	2	no	—
95	C₂²×C₂₂₇	—	2	2	no	—
96	C₂²×C₂₂₇	—	2	8	no	—
97	C₂²×C₂₂₇	C₄×D₄	1	2	yes	—
98	C₂²×C₂₂₇	C₂×M₁₆	1	2	yes	—
99	C₂²×C₂₂₇	C₂×M₁₆	1	4	yes	complemented
100	C₂²×C₂₂₇	#131	1	2	yes	—
101	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
102	C₂×D₂₂₇	—	2	2	no	—
103	C₂×D₂₂₇	—	2	2	no	—
104	C₂×D₂₂₇	—	2	2	no	—
105	C₂×D₂₂₇	—	4	4	no	—
106	C₄×C₂₂₇	—	2	2	no	—
107	C₄×C₂₂₇	C₄×D₄	1	2	yes	—
108	C₄×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
109	C₄×C₂₂₇	C₂×#41	1	1	yes	characteristic
110	Dic₂₂₇	—	2	2	no	—
111	Dic₂₂₇	—	2	2	no	—
112	Dic₂₂₇	—	2	2	no	—
113	C₂×C₂₂₇	—	2	4	no	—
114	C₂×C₂₂₇	#435	1	4	yes	complemented
115	C₂×C₂₂₇	C₂×C₄×D₄	1	1	yes	characteristic
116	C₂×C₂₂₇	C₂²×M₁₆	1	1	yes	characteristic
117	C₂×C₂₂₇	C₂×#131	1	1	yes	characteristic
118	D₂₂₇	—	4	2	no	—
119	C₂₂₇	C₂×#435	1	1	yes	characteristic, complemented, Sylow
120	C₂×#435	—	227	1	no	maximal, Sylow
121	C₂²×M₁₆	—	227	1	no	—
122	C₂³×C₈	D₂₂₇	1	1	yes	characteristic, complemented
123	C₂×C₄×D₄	—	227	1	no	—
124	#435	—	227	8	no	—
125	C₂×C₄⋊C₈	—	227	1	no	—
126	C₂×C₂²⋊C₈	—	227	1	no	—
127	C₂×C₂²⋊C₈	—	227	1	no	—
128	C₂×#97	—	227	1	no	—
129	C₂²×D₄	—	227	1	no	—
130	C₂³×C₄	—	227	1	no	—
131	C₂³×C₄	C₂×D₂₂₇	1	1	yes	characteristic
132	C₂×M₁₆	—	454	4	no	—
133	C₂×M₁₆	—	227	4	no	—
134	C₂²×C₈	—	2	4	no	—
135	C₂²×C₈	—	227	1	no	—
136	C₂²×C₈	—	227	1	no	—
137	C₂²×C₈	C₂×D₂₂₇	1	1	yes	characteristic
138	C₂²×C₈	C₂×D₂₂₇	1	1	yes	characteristic
139	C₂²×C₈	C₂×D₂₂₇	1	4	yes	complemented
140	C₄×D₄	—	227	4	no	—
141	C₄×D₄	—	227	4	no	—
142	C₂×C₄⋊C₄	—	227	1	no	—
143	C₂×C₂²⋊C₄	—	227	1	no	—
144	C₂×C₂²⋊C₄	—	227	1	no	—
145	C₂×C₄²	—	227	1	no	—
146	C₄⋊C₈	—	227	4	no	—
147	C₂²⋊C₈	—	227	4	no	—
148	C₂²⋊C₈	—	227	4	no	—
149	#97	—	227	4	no	—
150	C₂⁴	—	227	1	no	—
151	C₂⁴	C₄×D₂₂₇	1	1	yes	characteristic
152	C₂×D₄	—	454	4	no	—
153	C₂×D₄	—	227	4	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	454	2	no	—
157	C₂²×C₄	—	454	2	no	—
158	C₂²×C₄	—	227	1	no	—
159	C₂²×C₄	—	227	1	no	—
160	C₂²×C₄	—	227	1	no	—
161	C₂²×C₄	—	227	2	no	—
162	C₂²×C₄	—	227	2	no	—
163	C₂²×C₄	C₄×D₂₂₇	1	1	yes	characteristic
164	C₂²×C₄	C₄×D₂₂₇	1	2	yes	—
165	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
166	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
167	M₁₆	—	454	8	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	8	no	—
170	C₂×C₈	—	454	2	no	—
171	C₂×C₈	—	454	2	no	—
172	C₂×C₈	—	227	2	no	—
173	C₂×C₈	—	227	2	no	—
174	C₂×C₈	C₂²⋊D₂₂₇	1	4	yes	complemented
175	C₂×C₈	C₂²×D₂₂₇	1	2	yes	—
176	C₂×C₈	C₂²×D₂₂₇	1	2	yes	—
177	C₄⋊C₄	—	227	4	no	—
178	C₂²⋊C₄	—	227	4	no	—
179	C₂²⋊C₄	—	227	4	no	—
180	C₄²	—	227	2	no	—
181	C₄²	—	227	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	454	2	no	—
185	C₂³	—	454	2	no	—
186	C₂³	—	454	2	no	—
187	C₂³	—	227	2	no	—
188	C₂³	#16813867	1	4	yes	complemented
189	C₂³	C₂×C₄×D₂₂₇	1	2	yes	—
190	C₂³	C₂×C₄×D₂₂₇	1	1	yes	characteristic
191	D₄	—	454	8	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	454	2	no	—
197	C₂×C₄	—	454	2	no	—
198	C₂×C₄	—	454	2	no	—
199	C₂×C₄	—	454	2	no	—
200	C₂×C₄	—	454	2	no	—
201	C₂×C₄	—	454	4	no	—
202	C₂×C₄	—	227	2	no	—
203	C₂×C₄	—	227	2	no	—
204	C₂×C₄	—	227	2	no	—
205	C₂×C₄	C₂×C₄×D₂₂₇	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
207	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	2	yes	—
208	C₂×C₄	#16813884	1	2	yes	—
209	C₂×C₄	C₂³×D₂₂₇	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	454	2	no	—
212	C₈	—	454	2	no	—
213	C₈	C₂×C₂²⋊D₂₂₇	1	4	yes	complemented
214	C₂²	—	2	8	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	908	4	no	—
218	C₂²	—	454	2	no	—
219	C₂²	—	454	2	no	—
220	C₂²	—	454	2	no	—
221	C₂²	C₄×C₂²⋊D₂₂₇	1	2	yes	—
222	C₂²	C₂×#16813867	1	2	yes	—
223	C₂²	C₂×#16813867	1	4	yes	complemented
224	C₂²	#20300294	1	2	yes	—
225	C₂²	C₂²×C₄×D₂₂₇	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	454	2	no	—
228	C₄	—	454	2	no	—
229	C₄	—	454	2	no	—
230	C₄	C₄×C₂²⋊D₂₂₇	1	2	yes	—
231	C₄	C₂²×C₂²⋊D₂₂₇	1	1	yes	characteristic
232	C₄	C₂×#16813884	1	1	yes	characteristic
233	C₂	—	2	4	no	—
234	C₂	—	908	2	no	—
235	C₂	#28409103	1	4	yes	direct product factor
236	C₂	C₂×C₄×C₂²⋊D₂₂₇	1	1	yes	characteristic
237	C₂	C₂²×#16813867	1	1	yes	characteristic
238	C₂	C₂×#20300294	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₂₇
commutator	C₂²×C₂₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₂₇
Fitting	C₂³×C₈×C₂₂₇	C₂
socle	C₂³×C₂₂₇	C₂²×C₄
prime radical	C₂²×C₂²⋊D₂₂₇	C₄
cosocle	C₂×C₄×C₂₂₇	C₂⁴
hypercenter	C₂³×C₈	D₂₂₇
hypocenter	C₂₂₇	C₂×#435
LCS	G ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#16813867	C₂	semidirect
3. #28409103	C₂	direct (N × H)
4. C₂×Dic₂₂₇⋊C₈	C₂	semidirect
5. C₂×C₂²⋊(C₂₂₇⋊C₈)	C₂	semidirect
6. C₂×(C₂×D₂₂₇)⋊C₈	C₂	semidirect
7. C₂×#20300198	C₂	semidirect
8. C₂³×C₈×C₂₂₇	C₂	semidirect
12. C₂×#16813867	C₂²	semidirect
21. Dic₂₂₇⋊C₈	C₂²	semidirect
22. C₂²⋊(C₂₂₇⋊C₈)	C₂²	semidirect
23. (C₂×D₂₂₇)⋊C₈	C₂²	semidirect
24. #20300198	C₂²	semidirect
29. C₂²×C₈×C₂₂₇	C₂²	semidirect
59. C₂×C₈×C₂₂₇	D₄	semidirect
80. C₂³×C₂₂₇	M₁₆	semidirect
93. C₈×C₂₂₇	C₂×D₄	semidirect
99. C₂²×C₂₂₇	C₂×M₁₆	semidirect
114. C₂×C₂₂₇	#435	semidirect
119. C₂₂₇	C₂×#435	semidirect
122. C₂³×C₈	D₂₂₇	semidirect
139. C₂²×C₈	C₂×D₂₂₇	semidirect
174. C₂×C₈	C₂²⋊D₂₂₇	semidirect
188. C₂³	#16813867	semidirect
213. C₈	C₂×C₂²⋊D₂₂₇	semidirect
223. C₂²	C₂×#16813867	semidirect
235. C₂	#28409103	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#16813867	2
C₂×C₄×C₂²⋊D₂₂₇	2
#28409103	2
C₂×Dic₂₂₇⋊C₈	2
C₂×C₂²⋊(C₂₂₇⋊C₈)	2
C₂×(C₂×D₂₂₇)⋊C₈	2
C₂×#20300198	2
C₂³×C₈×C₂₂₇	2
C₂×#435	227

Subgroups — C2×#28409103 #44408413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#28409103 #44408413