Subgroups — C₂×#482×C₂₁₁ #41010707

Order 27008.

468 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#121×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#117×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#482×C₂₁₁	C₂	1	8	yes	maximal, direct product factor
4	C₂×#107×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#106×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#102×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#102×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂²⋊C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₂₁₁	—	2	1	no	—
10	C₂²×D₄×C₂₁₁	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
14	#121×C₂₁₁	—	2	2	no	—
15	#121×C₂₁₁	C₂²	1	4	yes	—
16	#117×C₂₁₁	—	2	2	no	—
17	#117×C₂₁₁	C₂²	1	4	yes	complemented
18	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₂₁₁	—	2	1	no	—
22	C₂×C₂²⋊C₄×C₂₁₁	—	2	1	no	—
23	C₂×C₄²×C₂₁₁	—	2	1	no	—
24	#107×C₂₁₁	C₂²	1	4	yes	complemented
25	#106×C₂₁₁	C₂²	1	4	yes	complemented
26	#102×C₂₁₁	C₂²	1	4	yes	complemented
27	#102×C₂₁₁	C₂²	1	4	yes	complemented
28	C₂²⋊C₈×C₂₁₁	C₂²	1	4	yes	complemented
29	C₂⁴×C₂₁₁	—	2	1	no	—
30	C₂⁴×C₂₁₁	D₄	1	1	yes	characteristic
31	C₂×D₄×C₂₁₁	—	2	4	no	—
32	C₂×D₄×C₂₁₁	—	2	1	no	—
33	C₂×D₄×C₂₁₁	—	2	2	no	—
34	C₂×D₄×C₂₁₁	—	2	2	no	—
35	C₂×D₄×C₂₁₁	—	4	4	no	—
36	C₂×D₄×C₂₁₁	C₂³	1	2	yes	—
37	C₂²×C₄×C₂₁₁	—	2	2	no	—
38	C₂²×C₄×C₂₁₁	—	2	2	no	—
39	C₂²×C₄×C₂₁₁	—	2	1	no	—
40	C₂²×C₄×C₂₁₁	—	2	1	no	—
41	C₂²×C₄×C₂₁₁	—	2	1	no	—
42	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
43	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
46	C₂×C₈×C₂₁₁	—	2	2	no	—
47	C₂×C₈×C₂₁₁	—	2	2	no	—
48	C₂×C₈×C₂₁₁	C₂³	1	2	yes	—
49	C₂×C₈×C₂₁₁	C₂³	1	2	yes	—
50	C₄⋊C₄×C₂₁₁	—	2	1	no	—
51	C₄⋊C₄×C₂₁₁	—	2	1	no	—
52	C₄⋊C₄×C₂₁₁	—	2	1	no	—
53	C₄⋊C₄×C₂₁₁	C₂³	1	2	yes	—
54	C₄⋊C₄×C₂₁₁	C₂³	1	2	yes	—
55	C₄⋊C₄×C₂₁₁	C₂³	1	2	yes	—
56	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
57	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
58	C₄²×C₂₁₁	—	2	4	no	—
59	C₂³×C₂₁₁	—	2	2	no	—
60	C₂³×C₂₁₁	—	2	2	no	—
61	C₂³×C₂₁₁	—	2	2	no	—
62	C₂³×C₂₁₁	—	2	2	no	—
63	C₂³×C₂₁₁	—	4	2	no	—
64	C₂³×C₂₁₁	—	4	2	no	—
65	C₂³×C₂₁₁	—	4	2	no	—
66	C₂³×C₂₁₁	C₂×D₄	1	2	yes	—
67	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
68	D₄×C₂₁₁	—	2	2	no	—
69	D₄×C₂₁₁	—	2	2	no	—
70	D₄×C₂₁₁	—	4	8	no	—
71	D₄×C₂₁₁	—	4	2	no	—
72	C₂×C₄×C₂₁₁	—	2	2	no	—
73	C₂×C₄×C₂₁₁	—	2	4	no	—
74	C₂×C₄×C₂₁₁	—	2	2	no	—
75	C₂×C₄×C₂₁₁	—	2	2	no	—
76	C₂×C₄×C₂₁₁	—	2	2	no	—
77	C₂×C₄×C₂₁₁	—	2	2	no	—
78	C₂×C₄×C₂₁₁	—	2	2	no	—
79	C₂×C₄×C₂₁₁	—	4	2	no	—
80	C₂×C₄×C₂₁₁	—	4	2	no	—
81	C₂×C₄×C₂₁₁	—	4	2	no	—
82	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
83	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₂₁₁	#41	1	2	yes	—
85	C₂×C₄×C₂₁₁	#41	1	2	yes	—
86	C₂×C₄×C₂₁₁	C₂⁴	1	1	yes	characteristic
87	C₈×C₂₁₁	—	2	2	no	—
88	C₈×C₂₁₁	—	2	2	no	—
89	C₂²×C₂₁₁	—	2	2	no	—
90	C₂²×C₂₁₁	—	2	2	no	—
91	C₂²×C₂₁₁	—	2	2	no	—
92	C₂²×C₂₁₁	—	4	4	no	—
93	C₂²×C₂₁₁	—	4	2	no	—
94	C₂²×C₂₁₁	—	4	2	no	—
95	C₂²×C₂₁₁	—	4	2	no	—
96	C₂²×C₂₁₁	—	8	4	no	—
97	C₂²×C₂₁₁	#123	1	2	yes	—
98	C₂²×C₂₁₁	#135	1	2	yes	—
99	C₂²×C₂₁₁	AGL(1,ℤ/8)	1	2	yes	—
100	C₂²×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
101	C₄×C₂₁₁	—	2	2	no	—
102	C₄×C₂₁₁	—	4	2	no	—
103	C₄×C₂₁₁	—	4	2	no	—
104	C₄×C₂₁₁	—	4	2	no	—
105	C₄×C₂₁₁	#123	1	2	yes	—
106	C₄×C₂₁₁	C₂×#41	1	1	yes	characteristic
107	C₄×C₂₁₁	C₂×#41	1	1	yes	characteristic
108	C₂×C₂₁₁	—	4	2	no	—
109	C₂×C₂₁₁	—	8	2	no	—
110	C₂×C₂₁₁	#482	1	4	yes	direct product factor
111	C₂×C₂₁₁	C₂×#123	1	1	yes	characteristic
112	C₂×C₂₁₁	C₂×#135	1	1	yes	characteristic
113	C₂×C₂₁₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂₁₁	C₂×#482	1	1	yes	characteristic, direct product factor, Sylow
115	C₂×#482	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂×#121	C₂×C₂₁₁	1	1	yes	characteristic
117	C₂×#117	C₂×C₂₁₁	1	1	yes	characteristic, complemented
118	#482	C₂×C₂₁₁	1	8	yes	direct product factor
119	C₂×#107	C₂×C₂₁₁	1	1	yes	characteristic, complemented
120	C₂×#106	C₂×C₂₁₁	1	1	yes	characteristic, complemented
121	C₂×#102	C₂×C₂₁₁	1	1	yes	characteristic, complemented
122	C₂×#102	C₂×C₂₁₁	1	1	yes	characteristic, complemented
123	C₂×C₂²⋊C₈	C₂×C₂₁₁	1	1	yes	characteristic, complemented
124	C₂²×D₄	—	2	1	no	—
125	C₂²×D₄	C₂²×C₂₁₁	1	1	yes	characteristic
126	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
127	C₂²×C₈	C₂²×C₂₁₁	1	1	yes	characteristic
128	C₂²×C₈	C₂²×C₂₁₁	1	1	yes	characteristic
129	#121	—	2	2	no	—
130	#121	C₂²×C₂₁₁	1	4	yes	—
131	#117	—	2	2	no	—
132	#117	C₂²×C₂₁₁	1	4	yes	complemented
133	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
134	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
135	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
136	C₂×C₂²⋊C₄	—	2	1	no	—
137	C₂×C₂²⋊C₄	—	2	1	no	—
138	C₂×C₄²	—	2	1	no	—
139	#107	C₂²×C₂₁₁	1	4	yes	complemented
140	#106	C₂²×C₂₁₁	1	4	yes	complemented
141	#102	C₂²×C₂₁₁	1	4	yes	complemented
142	#102	C₂²×C₂₁₁	1	4	yes	complemented
143	C₂²⋊C₈	C₂²×C₂₁₁	1	4	yes	complemented
144	C₂⁴	—	2	1	no	—
145	C₂⁴	D₄×C₂₁₁	1	1	yes	characteristic
146	C₂×D₄	—	2	4	no	—
147	C₂×D₄	—	2	1	no	—
148	C₂×D₄	—	2	2	no	—
149	C₂×D₄	—	2	2	no	—
150	C₂×D₄	—	4	4	no	—
151	C₂×D₄	C₂³×C₂₁₁	1	2	yes	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	1	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	D₄×C₂₁₁	1	1	yes	characteristic
158	C₂²×C₄	D₄×C₂₁₁	1	2	yes	—
159	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
160	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	C₂³×C₂₁₁	1	2	yes	—
164	C₂×C₈	C₂³×C₂₁₁	1	2	yes	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	C₂³×C₂₁₁	1	2	yes	—
169	C₄⋊C₄	C₂³×C₂₁₁	1	2	yes	—
170	C₄⋊C₄	C₂³×C₂₁₁	1	2	yes	—
171	C₂²⋊C₄	—	2	4	no	—
172	C₂²⋊C₄	—	2	4	no	—
173	C₄²	—	2	4	no	—
174	C₂³	—	2	2	no	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	2	no	—
177	C₂³	—	2	2	no	—
178	C₂³	—	4	2	no	—
179	C₂³	—	4	2	no	—
180	C₂³	—	4	2	no	—
181	C₂³	C₂×D₄×C₂₁₁	1	2	yes	—
182	C₂³	C₂×D₄×C₂₁₁	1	1	yes	characteristic
183	D₄	—	2	2	no	—
184	D₄	—	2	2	no	—
185	D₄	—	4	8	no	—
186	D₄	—	4	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
198	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
199	C₂×C₄	#41×C₂₁₁	1	2	yes	—
200	C₂×C₄	#41×C₂₁₁	1	2	yes	—
201	C₂×C₄	C₂⁴×C₂₁₁	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	2	2	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	4	4	no	—
208	C₂²	—	4	2	no	—
209	C₂²	—	4	2	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	8	4	no	—
212	C₂²	#123×C₂₁₁	1	2	yes	—
213	C₂²	#135×C₂₁₁	1	2	yes	—
214	C₂²	AGL(1,ℤ/8)×C₂₁₁	1	2	yes	—
215	C₂²	C₂²×D₄×C₂₁₁	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	4	2	no	—
218	C₄	—	4	2	no	—
219	C₄	—	4	2	no	—
220	C₄	#123×C₂₁₁	1	2	yes	—
221	C₄	C₂×#41×C₂₁₁	1	1	yes	characteristic
222	C₄	C₂×#41×C₂₁₁	1	1	yes	characteristic
223	C₂	—	4	2	no	—
224	C₂	—	8	2	no	—
225	C₂	#482×C₂₁₁	1	4	yes	direct product factor
226	C₂	C₂×#123×C₂₁₁	1	1	yes	characteristic
227	C₂	C₂×#135×C₂₁₁	1	1	yes	characteristic
228	C₂	C₂×AGL(1,ℤ/8)×C₂₁₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₁₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₁₁
prime radical	C₂×#121×C₂₁₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₁₁ ◁ C₂³×C₄×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#117×C₂₁₁	C₂	semidirect
3. #482×C₂₁₁	C₂	direct (N × H)
4. C₂×#107×C₂₁₁	C₂	semidirect
5. C₂×#106×C₂₁₁	C₂	semidirect
6. C₂×#102×C₂₁₁	C₂	semidirect
7. C₂×#102×C₂₁₁	C₂	semidirect
8. C₂×C₂²⋊C₈×C₂₁₁	C₂	semidirect
17. #117×C₂₁₁	C₂²	semidirect
24. #107×C₂₁₁	C₂²	semidirect
25. #106×C₂₁₁	C₂²	semidirect
26. #102×C₂₁₁	C₂²	semidirect
27. #102×C₂₁₁	C₂²	semidirect
28. C₂²⋊C₈×C₂₁₁	C₂²	semidirect
110. C₂×C₂₁₁	#482	direct (N × H)
114. C₂₁₁	C₂×#482	direct (N × H)
115. C₂×#482	C₂₁₁	direct (N × H)
117. C₂×#117	C₂×C₂₁₁	semidirect
118. #482	C₂×C₂₁₁	direct (N × H)
119. C₂×#107	C₂×C₂₁₁	semidirect
120. C₂×#106	C₂×C₂₁₁	semidirect
121. C₂×#102	C₂×C₂₁₁	semidirect
122. C₂×#102	C₂×C₂₁₁	semidirect
123. C₂×C₂²⋊C₈	C₂×C₂₁₁	semidirect
132. #117	C₂²×C₂₁₁	semidirect
139. #107	C₂²×C₂₁₁	semidirect
140. #106	C₂²×C₂₁₁	semidirect
141. #102	C₂²×C₂₁₁	semidirect
142. #102	C₂²×C₂₁₁	semidirect
143. C₂²⋊C₈	C₂²×C₂₁₁	semidirect
225. C₂	#482×C₂₁₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#121×C₂₁₁	2
C₂×#117×C₂₁₁	2
#482×C₂₁₁	2
C₂×#107×C₂₁₁	2
C₂×#106×C₂₁₁	2
C₂×#102×C₂₁₁	2
C₂×C₂²⋊C₈×C₂₁₁	2
C₂×#482	211

Subgroups — C2×#482×C211 #41010707

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#482×C₂₁₁ #41010707