Subgroups — C₂×#394×C₂₂₇ #44396342

Order 29056.

1252 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×C₂₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₄×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₂²⋊C₄×C₂₂₇	C₂	1	2	yes	maximal, complemented
4	C₂²×C₂²⋊C₄×C₂₂₇	C₂	1	2	yes	maximal, complemented
5	#394×C₂₂₇	C₂	1	8	yes	maximal, direct product factor
6	C₂×#95×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×C₂₂₇	C₂²	1	2	yes	—
8	C₂²×D₄×C₂₂₇	—	2	8	no	—
9	C₂²×D₄×C₂₂₇	—	2	4	no	—
10	C₂²×D₄×C₂₂₇	C₂²	1	4	yes	—
11	C₂³×C₄×C₂₂₇	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₂₂₇	C₂²	1	2	yes	—
13	C₂³×C₄×C₂₂₇	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₂₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₂₂₇	—	2	4	no	—
16	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	4	yes	complemented
17	C₂×C₂²⋊C₄×C₂₂₇	—	2	8	no	—
18	C₂×C₂²⋊C₄×C₂₂₇	—	2	8	no	—
19	C₂×C₂²⋊C₄×C₂₂₇	C₂²	1	8	yes	complemented
20	C₂×C₂²⋊C₄×C₂₂₇	C₂²	1	8	yes	complemented
21	#95×C₂₂₇	C₂²	1	4	yes	complemented
22	C₂⁴×C₂₂₇	—	2	4	no	—
23	C₂⁴×C₂₂₇	—	2	4	no	—
24	C₂⁴×C₂₂₇	—	2	4	no	—
25	C₂⁴×C₂₂₇	—	2	4	no	—
26	C₂⁴×C₂₂₇	—	2	4	no	—
27	C₂⁴×C₂₂₇	—	2	4	no	—
28	C₂⁴×C₂₂₇	D₄	1	8	yes	—
29	C₂⁴×C₂₂₇	C₂³	1	4	yes	—
30	C₂⁴×C₂₂₇	C₂³	1	1	yes	characteristic
31	C₂×D₄×C₂₂₇	—	2	32	no	—
32	C₂×D₄×C₂₂₇	—	2	16	no	—
33	C₂×D₄×C₂₂₇	—	2	8	no	—
34	C₂²×C₄×C₂₂₇	—	2	2	no	—
35	C₂²×C₄×C₂₂₇	—	2	4	no	—
36	C₂²×C₄×C₂₂₇	—	2	2	no	—
37	C₂²×C₄×C₂₂₇	—	2	4	no	—
38	C₂²×C₄×C₂₂₇	—	2	4	no	—
39	C₂²×C₄×C₂₂₇	—	2	4	no	—
40	C₂²×C₄×C₂₂₇	—	2	2	no	—
41	C₂²×C₄×C₂₂₇	—	2	4	no	—
42	C₂²×C₄×C₂₂₇	D₄	1	4	yes	—
43	C₂²×C₄×C₂₂₇	D₄	1	4	yes	complemented
44	C₂²×C₄×C₂₂₇	C₂³	1	4	yes	—
45	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
47	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
48	C₄⋊C₄×C₂₂₇	—	2	8	no	—
49	C₂²⋊C₄×C₂₂₇	—	2	16	no	—
50	C₂²⋊C₄×C₂₂₇	—	2	16	no	—
51	C₂³×C₂₂₇	—	2	16	no	—
52	C₂³×C₂₂₇	—	2	16	no	—
53	C₂³×C₂₂₇	—	2	16	no	—
54	C₂³×C₂₂₇	—	2	4	no	—
55	C₂³×C₂₂₇	—	2	4	no	—
56	C₂³×C₂₂₇	—	2	4	no	—
57	C₂³×C₂₂₇	—	2	4	no	—
58	C₂³×C₂₂₇	—	2	4	no	—
59	C₂³×C₂₂₇	—	2	4	no	—
60	C₂³×C₂₂₇	—	4	8	no	—
61	C₂³×C₂₂₇	—	4	8	no	—
62	C₂³×C₂₂₇	—	4	8	no	—
63	C₂³×C₂₂₇	—	4	8	no	—
64	C₂³×C₂₂₇	C₂×D₄	1	8	yes	—
65	C₂³×C₂₂₇	C₂×D₄	1	4	yes	—
66	C₂³×C₂₂₇	C₂×D₄	1	2	yes	—
67	C₂³×C₂₂₇	C₂×D₄	1	2	yes	—
68	C₂³×C₂₂₇	#41	1	2	yes	—
69	C₂³×C₂₂₇	#41	1	4	yes	—
70	C₂³×C₂₂₇	C₂⁴	1	1	yes	characteristic
71	D₄×C₂₂₇	—	2	32	no	—
72	C₂×C₄×C₂₂₇	—	2	8	no	—
73	C₂×C₄×C₂₂₇	—	2	8	no	—
74	C₂×C₄×C₂₂₇	—	2	2	no	—
75	C₂×C₄×C₂₂₇	—	2	4	no	—
76	C₂×C₄×C₂₂₇	—	2	2	no	—
77	C₂×C₄×C₂₂₇	—	2	4	no	—
78	C₂×C₄×C₂₂₇	—	2	4	no	—
79	C₂×C₄×C₂₂₇	—	2	4	no	—
80	C₂×C₄×C₂₂₇	—	2	2	no	—
81	C₂×C₄×C₂₂₇	—	2	4	no	—
82	C₂×C₄×C₂₂₇	—	4	4	no	—
83	C₂×C₄×C₂₂₇	—	4	8	no	—
84	C₂×C₄×C₂₂₇	C₂×D₄	1	4	yes	—
85	C₂×C₄×C₂₂₇	C₂×D₄	1	4	yes	complemented
86	C₂²×C₂₂₇	—	2	8	no	—
87	C₂²×C₂₂₇	—	2	8	no	—
88	C₂²×C₂₂₇	—	2	8	no	—
89	C₂²×C₂₂₇	—	4	4	no	—
90	C₂²×C₂₂₇	—	4	4	no	—
91	C₂²×C₂₂₇	—	4	4	no	—
92	C₂²×C₂₂₇	—	4	4	no	—
93	C₂²×C₂₂₇	—	4	32	no	—
94	C₂²×C₂₂₇	#120	1	4	yes	—
95	C₂²×C₂₂₇	#121	1	4	yes	—
96	C₂²×C₂₂₇	#121	1	8	yes	—
97	C₂²×C₂₂₇	#123	1	8	yes	—
98	C₂²×C₂₂₇	#124	1	4	yes	—
99	C₂²×C₂₂₇	C₂²×D₄	1	2	yes	—
100	C₂²×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
101	C₂²×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
102	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
103	C₂²×C₂₂₇	C₂×#41	1	2	yes	—
104	C₄×C₂₂₇	—	2	4	no	—
105	C₄×C₂₂₇	—	2	4	no	—
106	C₄×C₂₂₇	—	4	2	no	—
107	C₄×C₂₂₇	—	4	4	no	—
108	C₂×C₂₂₇	—	4	8	no	—
109	C₂×C₂₂₇	#394	1	8	yes	direct product factor
110	C₂×C₂₂₇	C₂×#120	1	1	yes	characteristic
111	C₂×C₂₂₇	C₂×#121	1	1	yes	characteristic
112	C₂×C₂₂₇	C₂×#121	1	2	yes	—
113	C₂×C₂₂₇	C₂×#123	1	2	yes	—
114	C₂×C₂₂₇	C₂×#124	1	1	yes	characteristic
115	C₂₂₇	C₂×#394	1	1	yes	characteristic, direct product factor, Sylow
116	C₂×#394	C₂₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂³×D₄	C₂×C₂₂₇	1	1	yes	characteristic
118	C₂²×C₄⋊C₄	C₂×C₂₂₇	1	1	yes	characteristic, complemented
119	C₂²×C₂²⋊C₄	C₂×C₂₂₇	1	2	yes	complemented
120	C₂²×C₂²⋊C₄	C₂×C₂₂₇	1	2	yes	complemented
121	#394	C₂×C₂₂₇	1	8	yes	direct product factor
122	C₂×#95	C₂×C₂₂₇	1	1	yes	characteristic, complemented
123	C₂⁵	C₂²×C₂₂₇	1	2	yes	—
124	C₂²×D₄	—	2	8	no	—
125	C₂²×D₄	—	2	4	no	—
126	C₂²×D₄	C₂²×C₂₂₇	1	4	yes	—
127	C₂³×C₄	C₂²×C₂₂₇	1	1	yes	characteristic
128	C₂³×C₄	C₂²×C₂₂₇	1	2	yes	—
129	C₂³×C₄	C₂²×C₂₂₇	1	1	yes	characteristic
130	C₂³×C₄	C₂²×C₂₂₇	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	2	4	no	—
132	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	4	yes	complemented
133	C₂×C₂²⋊C₄	—	2	8	no	—
134	C₂×C₂²⋊C₄	—	2	8	no	—
135	C₂×C₂²⋊C₄	C₂²×C₂₂₇	1	8	yes	complemented
136	C₂×C₂²⋊C₄	C₂²×C₂₂₇	1	8	yes	complemented
137	#95	C₂²×C₂₂₇	1	4	yes	complemented
138	C₂⁴	—	2	4	no	—
139	C₂⁴	—	2	4	no	—
140	C₂⁴	—	2	4	no	—
141	C₂⁴	—	2	4	no	—
142	C₂⁴	—	2	4	no	—
143	C₂⁴	—	2	4	no	—
144	C₂⁴	D₄×C₂₂₇	1	8	yes	—
145	C₂⁴	C₂³×C₂₂₇	1	4	yes	—
146	C₂⁴	C₂³×C₂₂₇	1	1	yes	characteristic
147	C₂×D₄	—	2	32	no	—
148	C₂×D₄	—	2	16	no	—
149	C₂×D₄	—	2	8	no	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	—	2	4	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	D₄×C₂₂₇	1	4	yes	—
159	C₂²×C₄	D₄×C₂₂₇	1	4	yes	complemented
160	C₂²×C₄	C₂³×C₂₂₇	1	4	yes	—
161	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
162	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
163	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
164	C₄⋊C₄	—	2	8	no	—
165	C₂²⋊C₄	—	2	16	no	—
166	C₂²⋊C₄	—	2	16	no	—
167	C₂³	—	2	16	no	—
168	C₂³	—	2	16	no	—
169	C₂³	—	2	16	no	—
170	C₂³	—	2	4	no	—
171	C₂³	—	2	4	no	—
172	C₂³	—	2	4	no	—
173	C₂³	—	2	4	no	—
174	C₂³	—	2	4	no	—
175	C₂³	—	2	4	no	—
176	C₂³	—	4	8	no	—
177	C₂³	—	4	8	no	—
178	C₂³	—	4	8	no	—
179	C₂³	—	4	8	no	—
180	C₂³	C₂×D₄×C₂₂₇	1	8	yes	—
181	C₂³	C₂×D₄×C₂₂₇	1	4	yes	—
182	C₂³	C₂×D₄×C₂₂₇	1	2	yes	—
183	C₂³	C₂×D₄×C₂₂₇	1	2	yes	—
184	C₂³	#41×C₂₂₇	1	2	yes	—
185	C₂³	#41×C₂₂₇	1	4	yes	—
186	C₂³	C₂⁴×C₂₂₇	1	1	yes	characteristic
187	D₄	—	2	32	no	—
188	C₂×C₄	—	2	8	no	—
189	C₂×C₄	—	2	8	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	4	8	no	—
200	C₂×C₄	C₂×D₄×C₂₂₇	1	4	yes	—
201	C₂×C₄	C₂×D₄×C₂₂₇	1	4	yes	complemented
202	C₂²	—	2	8	no	—
203	C₂²	—	2	8	no	—
204	C₂²	—	2	8	no	—
205	C₂²	—	4	4	no	—
206	C₂²	—	4	4	no	—
207	C₂²	—	4	4	no	—
208	C₂²	—	4	4	no	—
209	C₂²	—	4	32	no	—
210	C₂²	#120×C₂₂₇	1	4	yes	—
211	C₂²	#121×C₂₂₇	1	4	yes	—
212	C₂²	#121×C₂₂₇	1	8	yes	—
213	C₂²	#123×C₂₂₇	1	8	yes	—
214	C₂²	#124×C₂₂₇	1	4	yes	—
215	C₂²	C₂²×D₄×C₂₂₇	1	2	yes	—
216	C₂²	C₂²×D₄×C₂₂₇	1	1	yes	characteristic
217	C₂²	C₂²×D₄×C₂₂₇	1	1	yes	characteristic
218	C₂²	C₂×#41×C₂₂₇	1	1	yes	characteristic
219	C₂²	C₂×#41×C₂₂₇	1	2	yes	—
220	C₄	—	2	4	no	—
221	C₄	—	2	4	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	4	no	—
224	C₂	—	4	8	no	—
225	C₂	#394×C₂₂₇	1	8	yes	direct product factor
226	C₂	C₂×#120×C₂₂₇	1	1	yes	characteristic
227	C₂	C₂×#121×C₂₂₇	1	1	yes	characteristic
228	C₂	C₂×#121×C₂₂₇	1	2	yes	—
229	C₂	C₂×#123×C₂₂₇	1	2	yes	—
230	C₂	C₂×#124×C₂₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₂₂₇	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₂₇
prime radical	C₂³×D₄×C₂₂₇	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄⋊C₄×C₂₂₇	C₂	semidirect
3. C₂²×C₂²⋊C₄×C₂₂₇	C₂	semidirect
4. C₂²×C₂²⋊C₄×C₂₂₇	C₂	semidirect
5. #394×C₂₂₇	C₂	direct (N × H)
6. C₂×#95×C₂₂₇	C₂	semidirect
16. C₂×C₄⋊C₄×C₂₂₇	C₂²	semidirect
19. C₂×C₂²⋊C₄×C₂₂₇	C₂²	semidirect
20. C₂×C₂²⋊C₄×C₂₂₇	C₂²	semidirect
21. #95×C₂₂₇	C₂²	semidirect
43. C₂²×C₄×C₂₂₇	D₄	semidirect
85. C₂×C₄×C₂₂₇	C₂×D₄	semidirect
109. C₂×C₂₂₇	#394	direct (N × H)
115. C₂₂₇	C₂×#394	direct (N × H)
116. C₂×#394	C₂₂₇	direct (N × H)
118. C₂²×C₄⋊C₄	C₂×C₂₂₇	semidirect
119. C₂²×C₂²⋊C₄	C₂×C₂₂₇	semidirect
120. C₂²×C₂²⋊C₄	C₂×C₂₂₇	semidirect
121. #394	C₂×C₂₂₇	direct (N × H)
122. C₂×#95	C₂×C₂₂₇	semidirect
132. C₂×C₄⋊C₄	C₂²×C₂₂₇	semidirect
135. C₂×C₂²⋊C₄	C₂²×C₂₂₇	semidirect
136. C₂×C₂²⋊C₄	C₂²×C₂₂₇	semidirect
137. #95	C₂²×C₂₂₇	semidirect
159. C₂²×C₄	D₄×C₂₂₇	semidirect
201. C₂×C₄	C₂×D₄×C₂₂₇	semidirect
225. C₂	#394×C₂₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₄×C₂₂₇	2
C₂²×C₄⋊C₄×C₂₂₇	2
C₂²×C₂²⋊C₄×C₂₂₇	2
#394×C₂₂₇	2
C₂×#95×C₂₂₇	2
C₂×#394	227

Subgroups — C2×#394×C227 #44396342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#394×C₂₂₇ #44396342