Subgroups — C₂×#28408905 #44402145

Order 29056.

1232 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₂₂₇⋊C₄	C₂	1	2	yes	maximal, complemented
2	C₂²×C₂²⋊Dic₂₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₂²⋊Dic₂₂₇	C₂	1	1	yes	maximal, characteristic
4	#28408905	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₂₂₇⋊#95	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×C₂₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₂₂₇	C₂²	1	1	yes	characteristic
8	C₂³×Dic₂₂₇	C₂²	1	1	yes	characteristic
9	C₂³×Dic₂₂₇	C₂²	1	2	yes	—
10	C₂×Dic₂₂₇⋊C₄	—	2	8	no	—
11	C₂×Dic₂₂₇⋊C₄	C₂²	1	8	yes	complemented
12	C₂×C₂²⋊Dic₂₂₇	—	2	4	no	—
13	C₂×C₂²⋊Dic₂₂₇	C₄	1	8	yes	complemented
14	C₂×C₂²⋊Dic₂₂₇	C₂²	1	4	yes	—
15	C₂×C₂²⋊Dic₂₂₇	C₂²	1	4	yes	—
16	C₂₂₇⋊#95	C₂²	1	8	yes	complemented
17	C₂⁵×C₂₂₇	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₂₂₇	—	2	4	no	—
19	C₂³×C₄×C₂₂₇	—	2	8	no	—
20	C₂³×C₄×C₂₂₇	C₂²	1	4	yes	—
21	C₂³×C₄×C₂₂₇	C₂²	1	2	yes	—
22	C₂²×Dic₂₂₇	—	2	4	no	—
23	C₂²×Dic₂₂₇	—	2	2	no	—
24	C₂²×Dic₂₂₇	—	2	2	no	—
25	C₂²×Dic₂₂₇	—	2	4	no	—
26	C₂²×Dic₂₂₇	—	2	4	no	—
27	C₂²×Dic₂₂₇	—	2	4	no	—
28	C₂²×Dic₂₂₇	C₂×C₄	1	8	yes	complemented
29	C₂²×Dic₂₂₇	C₂³	1	2	yes	—
30	C₂²×Dic₂₂₇	C₂³	1	2	yes	—
31	C₂²×Dic₂₂₇	C₂³	1	4	yes	—
32	Dic₂₂₇⋊C₄	—	2	16	no	—
33	C₂²⋊Dic₂₂₇	—	2	8	no	—
34	C₂²⋊Dic₂₂₇	C₂×C₄	1	16	yes	complemented
35	C₂⁴×C₂₂₇	—	2	4	no	—
36	C₂⁴×C₂₂₇	—	2	4	no	—
37	C₂⁴×C₂₂₇	C₂×C₄	1	4	yes	—
38	C₂⁴×C₂₂₇	D₄	1	4	yes	—
39	C₂⁴×C₂₂₇	Q₈	1	4	yes	—
40	C₂⁴×C₂₂₇	C₂³	1	2	yes	—
41	C₂⁴×C₂₂₇	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₂₂₇	—	2	32	no	—
43	C₂²×C₄×C₂₂₇	—	2	8	no	—
44	C₂²×C₄×C₂₂₇	—	2	4	no	—
45	C₂²×C₄×C₂₂₇	—	2	4	no	—
46	C₂²×C₄×C₂₂₇	—	2	16	no	—
47	C₂²×C₄×C₂₂₇	D₄	1	8	yes	—
48	C₂²×C₄×C₂₂₇	C₂³	1	4	yes	—
49	C₂×Dic₂₂₇	—	2	16	no	—
50	C₂×Dic₂₂₇	—	2	4	no	—
51	C₂×Dic₂₂₇	—	2	4	no	—
52	C₂×Dic₂₂₇	—	2	2	no	—
53	C₂×Dic₂₂₇	—	2	2	no	—
54	C₂×Dic₂₂₇	—	2	4	no	—
55	C₂×Dic₂₂₇	—	2	4	no	—
56	C₂×Dic₂₂₇	—	4	4	no	—
57	C₂×Dic₂₂₇	—	4	4	no	—
58	C₂×Dic₂₂₇	C₂²×C₄	1	8	yes	complemented
59	C₂³×C₂₂₇	—	2	16	no	—
60	C₂³×C₂₂₇	—	2	8	no	—
61	C₂³×C₂₂₇	—	2	16	no	—
62	C₂³×C₂₂₇	—	2	8	no	—
63	C₂³×C₂₂₇	—	2	4	no	—
64	C₂³×C₂₂₇	—	2	4	no	—
65	C₂³×C₂₂₇	C₄⋊C₄	1	16	yes	complemented
66	C₂³×C₂₂₇	C₂²×C₄	1	4	yes	—
67	C₂³×C₂₂₇	C₂²×C₄	1	2	yes	—
68	C₂³×C₂₂₇	C₂×D₄	1	4	yes	—
69	C₂³×C₂₂₇	C₂×D₄	1	2	yes	—
70	C₂³×C₂₂₇	C₂×D₄	1	4	yes	—
71	C₂³×C₂₂₇	C₂×Q₈	1	4	yes	—
72	C₂³×C₂₂₇	C₂×Q₈	1	2	yes	—
73	C₂³×C₂₂₇	#41	1	4	yes	—
74	C₂³×C₂₂₇	C₂⁴	1	1	yes	characteristic
75	C₂×C₄×C₂₂₇	—	2	32	no	—
76	C₂×C₄×C₂₂₇	—	2	16	no	—
77	C₂×C₄×C₂₂₇	—	2	4	no	—
78	C₂×C₄×C₂₂₇	—	2	4	no	—
79	C₂×C₄×C₂₂₇	C₂×D₄	1	8	yes	—
80	C₂²×C₂₂₇	—	2	8	no	—
81	C₂²×C₂₂₇	—	2	4	no	—
82	C₂²×C₂₂₇	—	2	8	no	—
83	C₂²×C₂₂₇	—	2	4	no	—
84	C₂²×C₂₂₇	—	2	32	no	—
85	C₂²×C₂₂₇	C₂×C₄⋊C₄	1	4	yes	—
86	C₂²×C₂₂₇	C₂×C₄⋊C₄	1	8	yes	complemented
87	C₂²×C₂₂₇	C₄×D₄	1	8	yes	—
88	C₂²×C₂₂₇	#120	1	4	yes	—
89	C₂²×C₂₂₇	C₂²⋊Q₈	1	8	yes	—
90	C₂²×C₂₂₇	#123	1	4	yes	—
91	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
92	C₂²×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
93	C₂²×C₂₂₇	C₂²×D₄	1	2	yes	—
94	C₂²×C₂₂₇	C₂²×Q₈	1	1	yes	characteristic
95	C₂²×C₂₂₇	C₂×#41	1	2	yes	—
96	C₄×C₂₂₇	—	2	8	no	—
97	Dic₂₂₇	—	2	8	no	—
98	Dic₂₂₇	—	4	2	no	—
99	Dic₂₂₇	—	4	2	no	—
100	C₂×C₂₂₇	—	2	8	no	—
101	C₂×C₂₂₇	#385	1	8	yes	complemented
102	C₂×C₂₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₂₂₇	C₂×C₄×D₄	1	2	yes	—
104	C₂×C₂₂₇	C₂×#120	1	1	yes	characteristic
105	C₂×C₂₂₇	C₂×C₂²⋊Q₈	1	2	yes	—
106	C₂×C₂₂₇	C₂×#123	1	1	yes	characteristic
107	C₂₂₇	C₂×#385	1	1	yes	characteristic, complemented, Sylow
108	C₂×#385	—	227	1	no	maximal, Sylow
109	C₂⁴×C₄	D₂₂₇	1	1	yes	characteristic
110	C₂²×C₄⋊C₄	—	227	2	no	—
111	C₂²×C₂²⋊C₄	—	227	1	no	—
112	C₂²×C₂²⋊C₄	—	227	1	no	—
113	#385	—	227	8	no	—
114	C₂×#95	—	227	2	no	—
115	C₂⁵	C₂×D₂₂₇	1	1	yes	characteristic
116	C₂³×C₄	—	2	4	no	—
117	C₂³×C₄	—	2	8	no	—
118	C₂³×C₄	—	227	1	no	—
119	C₂³×C₄	—	227	1	no	—
120	C₂³×C₄	—	227	2	no	—
121	C₂³×C₄	C₂×D₂₂₇	1	4	yes	—
122	C₂³×C₄	C₂×D₂₂₇	1	2	yes	—
123	C₂×C₄⋊C₄	—	454	8	no	—
124	C₂×C₄⋊C₄	—	227	8	no	—
125	C₂×C₂²⋊C₄	—	454	4	no	—
126	C₂×C₂²⋊C₄	—	227	4	no	—
127	C₂×C₂²⋊C₄	—	227	8	no	—
128	C₂×C₂²⋊C₄	—	227	4	no	—
129	#95	—	227	8	no	—
130	C₂⁴	—	2	4	no	—
131	C₂⁴	—	2	4	no	—
132	C₂⁴	C₄×D₂₂₇	1	4	yes	—
133	C₂⁴	C₂²⋊D₂₂₇	1	4	yes	—
134	C₂⁴	Dic₄₅₄	1	4	yes	—
135	C₂⁴	C₂²×D₂₂₇	1	2	yes	—
136	C₂⁴	C₂²×D₂₂₇	1	1	yes	characteristic
137	C₂²×C₄	—	2	32	no	—
138	C₂²×C₄	—	2	8	no	—
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	—	2	16	no	—
142	C₂²×C₄	—	454	2	no	—
143	C₂²×C₄	—	454	2	no	—
144	C₂²×C₄	—	454	4	no	—
145	C₂²×C₄	—	454	4	no	—
146	C₂²×C₄	—	454	4	no	—
147	C₂²×C₄	—	454	4	no	—
148	C₂²×C₄	—	227	2	no	—
149	C₂²×C₄	—	227	2	no	—
150	C₂²×C₄	—	227	8	no	—
151	C₂²×C₄	—	227	4	no	—
152	C₂²×C₄	C₂²⋊D₂₂₇	1	8	yes	—
153	C₂²×C₄	C₂²×D₂₂₇	1	4	yes	—
154	C₄⋊C₄	—	454	16	no	—
155	C₂²⋊C₄	—	454	8	no	—
156	C₂²⋊C₄	—	227	16	no	—
157	C₂³	—	2	4	no	—
158	C₂³	—	2	4	no	—
159	C₂³	—	2	16	no	—
160	C₂³	—	2	8	no	—
161	C₂³	—	2	16	no	—
162	C₂³	—	2	8	no	—
163	C₂³	Dic₂₂₇⋊C₄	1	16	yes	complemented
164	C₂³	C₂×C₄×D₂₂₇	1	4	yes	—
165	C₂³	C₂×C₄×D₂₂₇	1	2	yes	—
166	C₂³	C₂×C₂²⋊D₂₂₇	1	4	yes	—
167	C₂³	C₂×C₂²⋊D₂₂₇	1	2	yes	—
168	C₂³	C₂×C₂²⋊D₂₂₇	1	4	yes	—
169	C₂³	C₂×Dic₄₅₄	1	4	yes	—
170	C₂³	C₂×Dic₄₅₄	1	2	yes	—
171	C₂³	#16813884	1	4	yes	—
172	C₂³	C₂³×D₂₂₇	1	1	yes	characteristic
173	C₂×C₄	—	2	32	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	16	no	—
177	C₂×C₄	—	908	4	no	—
178	C₂×C₄	—	908	4	no	—
179	C₂×C₄	—	454	2	no	—
180	C₂×C₄	—	454	2	no	—
181	C₂×C₄	—	454	4	no	—
182	C₂×C₄	—	454	4	no	—
183	C₂×C₄	—	454	4	no	—
184	C₂×C₄	—	454	4	no	—
185	C₂×C₄	—	454	16	no	—
186	C₂×C₄	—	227	8	no	—
187	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	8	yes	—
188	C₂²	—	2	32	no	—
189	C₂²	—	2	8	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	8	no	—
192	C₂²	—	2	4	no	—
193	C₂²	C₂×Dic₂₂₇⋊C₄	1	4	yes	—
194	C₂²	C₂×Dic₂₂₇⋊C₄	1	8	yes	complemented
195	C₂²	C₄×C₂²⋊D₂₂₇	1	8	yes	—
196	C₂²	#20300257	1	4	yes	—
197	C₂²	#20300267	1	8	yes	—
198	C₂²	#20300271	1	4	yes	—
199	C₂²	C₂²×C₄×D₂₂₇	1	1	yes	characteristic
200	C₂²	C₂²×C₂²⋊D₂₂₇	1	1	yes	characteristic
201	C₂²	C₂²×C₂²⋊D₂₂₇	1	2	yes	—
202	C₂²	C₂²×Dic₄₅₄	1	1	yes	characteristic
203	C₂²	C₂×#16813884	1	2	yes	—
204	C₄	—	2	8	no	—
205	C₄	—	908	2	no	—
206	C₄	—	908	2	no	—
207	C₄	—	454	8	no	—
208	C₂	—	2	8	no	—
209	C₂	#28408905	1	8	yes	direct product factor
210	C₂	C₂²×Dic₂₂₇⋊C₄	1	1	yes	characteristic
211	C₂	C₂×C₄×C₂²⋊D₂₂₇	1	2	yes	—
212	C₂	C₂×#20300257	1	1	yes	characteristic
213	C₂	C₂×#20300267	1	2	yes	—
214	C₂	C₂×#20300271	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₂₂₇
commutator	C₂²×C₂₂₇	C₂³×C₄
Frattini	C₂³	C₂³×D₂₂₇
Fitting	C₂⁴×C₄×C₂₂₇	C₂
socle	C₂⁴×C₂₂₇	C₂³
prime radical	C₂⁵×C₂₂₇	C₂²
cosocle	C₂³×C₂₂₇	C₂⁴
hypercenter	C₂⁴×C₄	D₂₂₇
hypocenter	C₂₂₇	C₂×#385
LCS	G ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₂₂₇⋊C₄	C₂	semidirect
4. #28408905	C₂	direct (N × H)
5. C₂×C₂₂₇⋊#95	C₂	semidirect
11. C₂×Dic₂₂₇⋊C₄	C₂²	semidirect
13. C₂×C₂²⋊Dic₂₂₇	C₄	semidirect
16. C₂₂₇⋊#95	C₂²	semidirect
28. C₂²×Dic₂₂₇	C₂×C₄	semidirect
34. C₂²⋊Dic₂₂₇	C₂×C₄	semidirect
58. C₂×Dic₂₂₇	C₂²×C₄	semidirect
65. C₂³×C₂₂₇	C₄⋊C₄	semidirect
86. C₂²×C₂₂₇	C₂×C₄⋊C₄	semidirect
101. C₂×C₂₂₇	#385	semidirect
107. C₂₂₇	C₂×#385	semidirect
163. C₂³	Dic₂₂₇⋊C₄	semidirect
194. C₂²	C₂×Dic₂₂₇⋊C₄	semidirect
209. C₂	#28408905	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×Dic₂₂₇⋊C₄	2
C₂²×C₂²⋊Dic₂₂₇	2
#28408905	2
C₂×C₂₂₇⋊#95	2
C₂⁴×C₄×C₂₂₇	2
C₂×#385	227

Subgroups — C2×#28408905 #44402145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#28408905 #44402145