Subgroups — #1866×C₂₁₁ #41009486

Order 27008.

270 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#131×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#424×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#104×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	#416×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	#339×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#336×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41×C₂₁₁	C₂²	1	1	yes	characteristic
9	#131×C₂₁₁	—	2	2	no	—
10	#131×C₂₁₁	—	2	2	no	—
11	C₂×M₁₆×C₂₁₁	—	2	1	no	—
12	C₂×M₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₂₁₁	—	2	1	no	—
15	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
17	#117×C₂₁₁	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
19	#108×C₂₁₁	—	2	2	no	—
20	C₄⋊C₈×C₂₁₁	—	2	1	no	—
21	C₄⋊C₈×C₂₁₁	—	2	1	no	—
22	#104×C₂₁₁	—	2	2	no	—
23	#103×C₂₁₁	C₄	1	1	yes	characteristic, complemented
24	#103×C₂₁₁	C₄	1	1	yes	characteristic, complemented
25	#102×C₂₁₁	C₄	1	1	yes	characteristic, complemented
26	#102×C₂₁₁	C₄	1	1	yes	characteristic, complemented
27	#41×C₂₁₁	—	2	1	no	—
28	#41×C₂₁₁	—	2	1	no	—
29	#41×C₂₁₁	D₄	1	2	yes	—
30	#41×C₂₁₁	Q₈	1	2	yes	—
31	C₂×Q₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
32	C₂×D₄×C₂₁₁	—	2	1	no	—
33	C₂×D₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₁₁	—	2	1	no	—
35	C₂²×C₄×C₂₁₁	—	2	1	no	—
36	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
37	M₁₆×C₂₁₁	—	2	4	no	—
38	M₁₆×C₂₁₁	—	2	1	no	—
39	M₁₆×C₂₁₁	—	2	1	no	—
40	M₁₆×C₂₁₁	—	2	1	no	—
41	M₁₆×C₂₁₁	D₄	1	2	yes	—
42	C₂×C₈×C₂₁₁	—	2	4	no	—
43	C₂×C₈×C₂₁₁	—	2	1	no	—
44	C₂×C₈×C₂₁₁	—	2	1	no	—
45	C₂×C₈×C₂₁₁	—	2	1	no	—
46	C₂×C₈×C₂₁₁	—	2	1	no	—
47	C₂×C₈×C₂₁₁	—	2	1	no	—
48	C₂×C₈×C₂₁₁	—	2	1	no	—
49	C₂×C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
52	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
53	C₄⋊C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
54	C₄⋊C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
55	C₂²⋊C₄×C₂₁₁	—	2	1	no	—
56	C₄²×C₂₁₁	—	2	1	no	—
57	C₄²×C₂₁₁	—	2	1	no	—
58	C₄²×C₂₁₁	—	2	1	no	—
59	C₂³×C₂₁₁	—	2	1	no	—
60	C₂³×C₂₁₁	#41	1	1	yes	characteristic
61	Q₈×C₂₁₁	—	2	1	no	—
62	Q₈×C₂₁₁	C₄⋊C₄	1	2	yes	—
63	D₄×C₂₁₁	—	2	2	no	—
64	D₄×C₂₁₁	—	2	2	no	—
65	D₄×C₂₁₁	—	2	1	no	—
66	D₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×C₂₁₁	—	2	1	no	—
68	C₂×C₄×C₂₁₁	—	2	2	no	—
69	C₂×C₄×C₂₁₁	—	2	2	no	—
70	C₂×C₄×C₂₁₁	—	2	2	no	—
71	C₂×C₄×C₂₁₁	—	2	1	no	—
72	C₂×C₄×C₂₁₁	—	2	1	no	—
73	C₂×C₄×C₂₁₁	—	4	1	no	—
74	C₂×C₄×C₂₁₁	—	4	1	no	—
75	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
81	C₈×C₂₁₁	—	2	1	no	—
82	C₈×C₂₁₁	—	2	1	no	—
83	C₈×C₂₁₁	—	2	1	no	—
84	C₈×C₂₁₁	—	2	2	no	—
85	C₈×C₂₁₁	—	2	1	no	—
86	C₈×C₂₁₁	—	4	1	no	—
87	C₂²×C₂₁₁	—	2	1	no	—
88	C₂²×C₂₁₁	—	2	1	no	—
89	C₂²×C₂₁₁	—	2	2	no	—
90	C₂²×C₂₁₁	—	4	1	no	—
91	C₂²×C₂₁₁	—	4	1	no	—
92	C₂²×C₂₁₁	C₄×D₄	1	1	yes	characteristic
93	C₂²×C₂₁₁	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂²×C₂₁₁	#123	1	1	yes	characteristic
95	C₄×C₂₁₁	—	2	2	no	—
96	C₄×C₂₁₁	—	4	2	no	—
97	C₄×C₂₁₁	—	4	1	no	—
98	C₄×C₂₁₁	—	4	1	no	—
99	C₄×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₄×C₂₁₁	C₄×D₄	1	1	yes	characteristic
101	C₄×C₂₁₁	#120	1	1	yes	characteristic
102	C₄×C₂₁₁	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₂₁₁	—	2	1	no	—
104	C₂×C₂₁₁	—	2	1	no	—
105	C₂×C₂₁₁	—	4	2	no	—
106	C₂×C₂₁₁	#385	1	1	yes	characteristic
107	C₂×C₂₁₁	#443	1	1	yes	characteristic
108	C₂×C₂₁₁	#444	1	1	yes	characteristic
109	C₂₁₁	#1866	1	1	yes	characteristic, direct product factor, Sylow
110	#1866	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂×#131	C₂×C₂₁₁	1	1	yes	characteristic
112	C₂×#108	C₂×C₂₁₁	1	1	yes	characteristic, complemented
113	#424	C₂×C₂₁₁	1	1	yes	characteristic, complemented
114	C₂×#104	C₂×C₂₁₁	1	1	yes	characteristic
115	#416	C₂×C₂₁₁	1	1	yes	characteristic
116	#339	C₂×C₂₁₁	1	1	yes	characteristic, complemented
117	#336	C₂×C₂₁₁	1	1	yes	characteristic, complemented
118	C₂×#41	C₂²×C₂₁₁	1	1	yes	characteristic
119	#131	—	2	2	no	—
120	#131	—	2	2	no	—
121	C₂×M₁₆	—	2	1	no	—
122	C₂×M₁₆	C₂²×C₂₁₁	1	1	yes	characteristic
123	C₂×M₁₆	C₂²×C₂₁₁	1	1	yes	characteristic
124	C₂²×C₈	—	2	1	no	—
125	C₂²×C₈	C₂²×C₂₁₁	1	1	yes	characteristic
126	C₂²×C₈	C₂²×C₂₁₁	1	1	yes	characteristic
127	#117	C₂²×C₂₁₁	1	1	yes	characteristic
128	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
129	#108	—	2	2	no	—
130	C₄⋊C₈	—	2	1	no	—
131	C₄⋊C₈	—	2	1	no	—
132	#104	—	2	2	no	—
133	#103	C₄×C₂₁₁	1	1	yes	characteristic, complemented
134	#103	C₄×C₂₁₁	1	1	yes	characteristic, complemented
135	#102	C₄×C₂₁₁	1	1	yes	characteristic, complemented
136	#102	C₄×C₂₁₁	1	1	yes	characteristic, complemented
137	#41	—	2	1	no	—
138	#41	—	2	1	no	—
139	#41	D₄×C₂₁₁	1	2	yes	—
140	#41	Q₈×C₂₁₁	1	2	yes	—
141	C₂×Q₈	C₂×C₄×C₂₁₁	1	1	yes	characteristic
142	C₂×D₄	—	2	1	no	—
143	C₂×D₄	C₂×C₄×C₂₁₁	1	1	yes	characteristic
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	2	1	no	—
146	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
147	M₁₆	—	2	4	no	—
148	M₁₆	—	2	1	no	—
149	M₁₆	—	2	1	no	—
150	M₁₆	—	2	1	no	—
151	M₁₆	D₄×C₂₁₁	1	2	yes	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	C₂×C₄×C₂₁₁	1	1	yes	characteristic
160	C₂×C₈	C₂×C₄×C₂₁₁	1	1	yes	characteristic
161	C₂×C₈	D₄×C₂₁₁	1	1	yes	characteristic
162	C₂×C₈	D₄×C₂₁₁	1	1	yes	characteristic
163	C₄⋊C₄	C₂×C₄×C₂₁₁	1	1	yes	characteristic
164	C₄⋊C₄	C₂×C₄×C₂₁₁	1	1	yes	characteristic
165	C₂²⋊C₄	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	2	1	no	—
169	C₂³	—	2	1	no	—
170	C₂³	#41×C₂₁₁	1	1	yes	characteristic
171	Q₈	—	2	1	no	—
172	Q₈	C₄⋊C₄×C₂₁₁	1	2	yes	—
173	D₄	—	2	2	no	—
174	D₄	—	2	2	no	—
175	D₄	—	2	1	no	—
176	D₄	C₄⋊C₄×C₂₁₁	1	2	yes	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	—	4	1	no	—
185	C₂×C₄	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
186	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
187	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
188	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
189	C₂×C₄	C₂×Q₈×C₂₁₁	1	1	yes	characteristic
190	C₂×C₄	#41×C₂₁₁	1	1	yes	characteristic
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	2	1	no	—
194	C₈	—	2	2	no	—
195	C₈	—	2	1	no	—
196	C₈	—	4	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	1	no	—
201	C₂²	—	4	1	no	—
202	C₂²	C₄×D₄×C₂₁₁	1	1	yes	characteristic
203	C₂²	C₂²⋊Q₈×C₂₁₁	1	1	yes	characteristic
204	C₂²	#123×C₂₁₁	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	4	2	no	—
207	C₄	—	4	1	no	—
208	C₄	—	4	1	no	—
209	C₄	C₂×C₄⋊C₄×C₂₁₁	1	1	yes	characteristic
210	C₄	C₄×D₄×C₂₁₁	1	1	yes	characteristic
211	C₄	#120×C₂₁₁	1	1	yes	characteristic
212	C₄	C₂²⋊Q₈×C₂₁₁	1	1	yes	characteristic
213	C₂	—	2	1	no	—
214	C₂	—	2	1	no	—
215	C₂	—	4	2	no	—
216	C₂	#385×C₂₁₁	1	1	yes	characteristic
217	C₂	#443×C₂₁₁	1	1	yes	characteristic
218	C₂	#444×C₂₁₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₁₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₁₁
Frattini, cosocle	C₂²×C₄	C₂³×C₂₁₁
socle	C₂²×C₂₁₁	C₄×D₄
prime radical	C₂×#41×C₂₁₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₁₁ ◁ C₂²×C₈×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#108×C₂₁₁	C₂	semidirect
3. #424×C₂₁₁	C₂	semidirect
6. #339×C₂₁₁	C₂	semidirect
7. #336×C₂₁₁	C₂	semidirect
23. #103×C₂₁₁	C₄	semidirect
24. #103×C₂₁₁	C₄	semidirect
25. #102×C₂₁₁	C₄	semidirect
26. #102×C₂₁₁	C₄	semidirect
109. C₂₁₁	#1866	direct (N × H)
110. #1866	C₂₁₁	direct (N × H)
112. C₂×#108	C₂×C₂₁₁	semidirect
113. #424	C₂×C₂₁₁	semidirect
116. #339	C₂×C₂₁₁	semidirect
117. #336	C₂×C₂₁₁	semidirect
133. #103	C₄×C₂₁₁	semidirect
134. #103	C₄×C₂₁₁	semidirect
135. #102	C₄×C₂₁₁	semidirect
136. #102	C₄×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#131×C₂₁₁	2
C₂×#108×C₂₁₁	2
#424×C₂₁₁	2
C₂×#104×C₂₁₁	2
#416×C₂₁₁	2
#339×C₂₁₁	2
#336×C₂₁₁	2
#1866	211

Subgroups — #1866×C211 #41009486

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1866×C₂₁₁ #41009486