Subgroups — #3054×C₂₁₁ #41010674

Order 27008.

500 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#135×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×M₁₆×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#525×C₂₁₁	C₂	1	2	yes	maximal, complemented
4	#470×C₂₁₁	C₂	1	2	yes	maximal, complemented
5	#469×C₂₁₁	C₂	1	2	yes	maximal, complemented
6	#468×C₂₁₁	C₂	1	4	yes	maximal, complemented
7	#427×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#416×C₂₁₁	C₂	1	2	yes	maximal, complemented
9	C₂×#41×C₂₁₁	C₂²	1	2	yes	—
10	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
11	#135×C₂₁₁	—	2	4	no	—
12	C₂×Q₁₆×C₂₁₁	C₂²	1	1	yes	characteristic, complemented
13	C₂×SD₁₆×C₂₁₁	C₂²	1	2	yes	complemented
14	C₂×D₈×C₂₁₁	C₂²	1	1	yes	characteristic, complemented
15	C₂×M₁₆×C₂₁₁	—	2	4	no	—
16	C₂×M₁₆×C₂₁₁	C₂²	1	2	yes	—
17	C₂×M₁₆×C₂₁₁	C₂²	1	2	yes	—
18	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
20	#123×C₂₁₁	—	2	2	no	—
21	C₂²⋊Q₈×C₂₁₁	C₂²	1	4	yes	—
22	#121×C₂₁₁	C₂²	1	4	yes	—
23	#120×C₂₁₁	—	2	2	no	—
24	C₄×D₄×C₂₁₁	—	2	4	no	—
25	#117×C₂₁₁	C₂²	1	2	yes	—
26	#107×C₂₁₁	C₂²	1	2	yes	complemented
27	#106×C₂₁₁	C₂²	1	2	yes	complemented
28	#103×C₂₁₁	C₂²	1	4	yes	complemented
29	#102×C₂₁₁	C₂²	1	4	yes	complemented
30	C₂⁴×C₂₁₁	D₄	1	1	yes	characteristic
31	#41×C₂₁₁	—	2	4	no	—
32	#41×C₂₁₁	—	2	4	no	—
33	C₂×Q₈×C₂₁₁	C₂³	1	2	yes	—
34	C₂×D₄×C₂₁₁	—	2	4	no	—
35	C₂×D₄×C₂₁₁	—	2	2	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	2	no	—
40	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
42	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
44	Q₁₆×C₂₁₁	—	2	2	no	—
45	SD₁₆×C₂₁₁	—	2	4	no	—
46	D₈×C₂₁₁	—	2	2	no	—
47	M₁₆×C₂₁₁	—	2	4	no	—
48	M₁₆×C₂₁₁	—	2	4	no	—
49	C₂×C₈×C₂₁₁	—	2	2	no	—
50	C₂×C₈×C₂₁₁	D₄	1	4	yes	complemented
51	C₂×C₈×C₂₁₁	C₂³	1	2	yes	—
52	C₂×C₈×C₂₁₁	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₂₁₁	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₂₁₁	—	2	4	no	—
55	C₄⋊C₄×C₂₁₁	C₂³	1	4	yes	—
56	C₂²⋊C₄×C₂₁₁	—	2	2	no	—
57	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
58	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
59	C₄²×C₂₁₁	—	2	2	no	—
60	C₂³×C₂₁₁	—	2	2	no	—
61	C₂³×C₂₁₁	—	2	2	no	—
62	C₂³×C₂₁₁	—	4	2	no	—
63	C₂³×C₂₁₁	C₂×D₄	1	2	yes	—
64	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
65	Q₈×C₂₁₁	—	2	2	no	—
66	Q₈×C₂₁₁	—	2	2	no	—
67	D₄×C₂₁₁	—	2	2	no	—
68	D₄×C₂₁₁	—	2	2	no	—
69	D₄×C₂₁₁	—	4	8	no	—
70	D₄×C₂₁₁	—	4	4	no	—
71	C₂×C₄×C₂₁₁	—	2	2	no	—
72	C₂×C₄×C₂₁₁	—	2	4	no	—
73	C₂×C₄×C₂₁₁	—	2	2	no	—
74	C₂×C₄×C₂₁₁	—	2	2	no	—
75	C₂×C₄×C₂₁₁	—	2	4	no	—
76	C₂×C₄×C₂₁₁	—	2	2	no	—
77	C₂×C₄×C₂₁₁	—	4	4	no	—
78	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₂₁₁	#41	1	2	yes	—
82	C₂×C₄×C₂₁₁	C₂⁴	1	1	yes	characteristic
83	C₈×C₂₁₁	—	2	2	no	—
84	C₈×C₂₁₁	C₂×D₄	1	2	yes	—
85	C₈×C₂₁₁	C₂×D₄	1	2	yes	—
86	C₂²×C₂₁₁	—	2	2	no	—
87	C₂²×C₂₁₁	—	2	2	no	—
88	C₂²×C₂₁₁	—	2	4	no	—
89	C₂²×C₂₁₁	—	4	4	no	—
90	C₂²×C₂₁₁	—	4	2	no	—
91	C₂²×C₂₁₁	—	4	4	no	—
92	C₂²×C₂₁₁	#121	1	2	yes	—
93	C₂²×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
94	C₄×C₂₁₁	—	2	2	no	—
95	C₄×C₂₁₁	—	4	2	no	—
96	C₄×C₂₁₁	—	4	4	no	—
97	C₄×C₂₁₁	#121	1	2	yes	—
98	C₄×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
99	C₄×C₂₁₁	C₂×#41	1	1	yes	characteristic
100	C₂×C₂₁₁	—	2	2	no	—
101	C₂×C₂₁₁	—	4	2	no	—
102	C₂×C₂₁₁	—	8	2	no	—
103	C₂×C₂₁₁	C₂×#121	1	1	yes	characteristic
104	C₂×C₂₁₁	#575	1	2	yes	—
105	C₂₁₁	#3054	1	1	yes	characteristic, direct product factor, Sylow
106	#3054	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂×#135	C₂×C₂₁₁	1	1	yes	characteristic, complemented
108	C₂²×M₁₆	C₂×C₂₁₁	1	1	yes	characteristic, complemented
109	#525	C₂×C₂₁₁	1	2	yes	complemented
110	#470	C₂×C₂₁₁	1	2	yes	complemented
111	#469	C₂×C₂₁₁	1	2	yes	complemented
112	#468	C₂×C₂₁₁	1	4	yes	complemented
113	#427	C₂×C₂₁₁	1	1	yes	characteristic, complemented
114	#416	C₂×C₂₁₁	1	2	yes	complemented
115	C₂×#41	C₂²×C₂₁₁	1	2	yes	—
116	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
117	#135	—	2	4	no	—
118	C₂×Q₁₆	C₂²×C₂₁₁	1	1	yes	characteristic, complemented
119	C₂×SD₁₆	C₂²×C₂₁₁	1	2	yes	complemented
120	C₂×D₈	C₂²×C₂₁₁	1	1	yes	characteristic, complemented
121	C₂×M₁₆	—	2	4	no	—
122	C₂×M₁₆	C₂²×C₂₁₁	1	2	yes	—
123	C₂×M₁₆	C₂²×C₂₁₁	1	2	yes	—
124	C₂²×C₈	C₂²×C₂₁₁	1	1	yes	characteristic
125	C₂²×C₈	C₂²×C₂₁₁	1	1	yes	characteristic
126	#123	—	2	2	no	—
127	C₂²⋊Q₈	C₂²×C₂₁₁	1	4	yes	—
128	#121	C₂²×C₂₁₁	1	4	yes	—
129	#120	—	2	2	no	—
130	C₄×D₄	—	2	4	no	—
131	#117	C₂²×C₂₁₁	1	2	yes	—
132	#107	C₂²×C₂₁₁	1	2	yes	complemented
133	#106	C₂²×C₂₁₁	1	2	yes	complemented
134	#103	C₂²×C₂₁₁	1	4	yes	complemented
135	#102	C₂²×C₂₁₁	1	4	yes	complemented
136	C₂⁴	D₄×C₂₁₁	1	1	yes	characteristic
137	#41	—	2	4	no	—
138	#41	—	2	4	no	—
139	C₂×Q₈	C₂³×C₂₁₁	1	2	yes	—
140	C₂×D₄	—	2	4	no	—
141	C₂×D₄	—	2	2	no	—
142	C₂×D₄	C₂³×C₂₁₁	1	2	yes	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	D₄×C₂₁₁	1	2	yes	—
147	C₂²×C₄	D₄×C₂₁₁	1	1	yes	characteristic
148	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
149	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
150	Q₁₆	—	2	2	no	—
151	SD₁₆	—	2	4	no	—
152	D₈	—	2	2	no	—
153	M₁₆	—	2	4	no	—
154	M₁₆	—	2	4	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	D₄×C₂₁₁	1	4	yes	complemented
157	C₂×C₈	C₂³×C₂₁₁	1	2	yes	—
158	C₂×C₈	C₂³×C₂₁₁	1	1	yes	characteristic
159	C₂×C₈	C₂³×C₂₁₁	1	1	yes	characteristic
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	C₂³×C₂₁₁	1	4	yes	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	2	4	no	—
164	C₂²⋊C₄	—	2	4	no	—
165	C₄²	—	2	2	no	—
166	C₂³	—	2	2	no	—
167	C₂³	—	2	2	no	—
168	C₂³	—	4	2	no	—
169	C₂³	C₂×D₄×C₂₁₁	1	2	yes	—
170	C₂³	C₂×D₄×C₂₁₁	1	1	yes	characteristic
171	Q₈	—	2	2	no	—
172	Q₈	—	2	2	no	—
173	D₄	—	2	2	no	—
174	D₄	—	2	2	no	—
175	D₄	—	4	8	no	—
176	D₄	—	4	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
185	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
186	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
187	C₂×C₄	#41×C₂₁₁	1	2	yes	—
188	C₂×C₄	C₂⁴×C₂₁₁	1	1	yes	characteristic
189	C₈	—	2	2	no	—
190	C₈	C₂×D₄×C₂₁₁	1	2	yes	—
191	C₈	C₂×D₄×C₂₁₁	1	2	yes	—
192	C₂²	—	2	2	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	4	no	—
195	C₂²	—	4	4	no	—
196	C₂²	—	4	2	no	—
197	C₂²	—	4	4	no	—
198	C₂²	#121×C₂₁₁	1	2	yes	—
199	C₂²	C₂²×D₄×C₂₁₁	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	4	2	no	—
202	C₄	—	4	4	no	—
203	C₄	#121×C₂₁₁	1	2	yes	—
204	C₄	C₂²×D₄×C₂₁₁	1	1	yes	characteristic
205	C₄	C₂×#41×C₂₁₁	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	—	4	2	no	—
208	C₂	—	8	2	no	—
209	C₂	C₂×#121×C₂₁₁	1	1	yes	characteristic
210	C₂	#575×C₂₁₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₁₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₁₁
socle	C₂²×C₂₁₁	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₁₁ ◁ C₂³×C₄×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#135×C₂₁₁	C₂	semidirect
2. C₂²×M₁₆×C₂₁₁	C₂	semidirect
3. #525×C₂₁₁	C₂	semidirect
4. #470×C₂₁₁	C₂	semidirect
5. #469×C₂₁₁	C₂	semidirect
6. #468×C₂₁₁	C₂	semidirect
7. #427×C₂₁₁	C₂	semidirect
8. #416×C₂₁₁	C₂	semidirect
12. C₂×Q₁₆×C₂₁₁	C₂²	semidirect
13. C₂×SD₁₆×C₂₁₁	C₂²	semidirect
14. C₂×D₈×C₂₁₁	C₂²	semidirect
26. #107×C₂₁₁	C₂²	semidirect
27. #106×C₂₁₁	C₂²	semidirect
28. #103×C₂₁₁	C₂²	semidirect
29. #102×C₂₁₁	C₂²	semidirect
50. C₂×C₈×C₂₁₁	D₄	semidirect
105. C₂₁₁	#3054	direct (N × H)
106. #3054	C₂₁₁	direct (N × H)
107. C₂×#135	C₂×C₂₁₁	semidirect
108. C₂²×M₁₆	C₂×C₂₁₁	semidirect
109. #525	C₂×C₂₁₁	semidirect
110. #470	C₂×C₂₁₁	semidirect
111. #469	C₂×C₂₁₁	semidirect
112. #468	C₂×C₂₁₁	semidirect
113. #427	C₂×C₂₁₁	semidirect
114. #416	C₂×C₂₁₁	semidirect
118. C₂×Q₁₆	C₂²×C₂₁₁	semidirect
119. C₂×SD₁₆	C₂²×C₂₁₁	semidirect
120. C₂×D₈	C₂²×C₂₁₁	semidirect
132. #107	C₂²×C₂₁₁	semidirect
133. #106	C₂²×C₂₁₁	semidirect
134. #103	C₂²×C₂₁₁	semidirect
135. #102	C₂²×C₂₁₁	semidirect
156. C₂×C₈	D₄×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#135×C₂₁₁	2
C₂²×M₁₆×C₂₁₁	2
#525×C₂₁₁	2
#470×C₂₁₁	2
#469×C₂₁₁	2
#468×C₂₁₁	2
#427×C₂₁₁	2
#416×C₂₁₁	2
#3054	211

Subgroups — #3054×C211 #41010674

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3054×C₂₁₁ #41010674