Subgroups — #1806×C₂₁₁ #41009426

Order 27008.

312 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	C₄×#41×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	#423×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#104×C₂₁₁	C₂	1	2	yes	maximal
4	C₄×C₄⋊C₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#339×C₂₁₁	C₂	1	2	yes	maximal, complemented
6	C₂×#41×C₂₁₁	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₂₁₁	C₂²	1	2	yes	—
8	C₄×Q₈×C₂₁₁	C₄	1	1	yes	characteristic, complemented
9	C₄×Q₈×C₂₁₁	C₄	1	1	yes	characteristic, complemented
10	C₄×D₄×C₂₁₁	—	2	1	no	—
11	C₄×D₄×C₂₁₁	—	2	1	no	—
12	C₄×D₄×C₂₁₁	C₄	1	1	yes	characteristic, complemented
13	C₄×D₄×C₂₁₁	C₄	1	1	yes	characteristic, complemented
14	#117×C₂₁₁	—	2	1	no	—
15	#117×C₂₁₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₁₁	—	2	1	no	—
17	C₂×C₄²×C₂₁₁	—	2	1	no	—
18	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
19	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₂₁₁	—	2	1	no	—
21	C₄⋊C₈×C₂₁₁	—	2	1	no	—
22	#104×C₂₁₁	—	2	4	no	—
23	#95×C₂₁₁	—	2	1	no	—
24	#41×C₂₁₁	—	2	1	no	—
25	#41×C₂₁₁	—	2	1	no	—
26	#41×C₂₁₁	D₄	1	2	yes	—
27	#41×C₂₁₁	Q₈	1	2	yes	—
28	C₂×Q₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
29	C₂×D₄×C₂₁₁	—	2	1	no	—
30	C₂×D₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₁₁	—	2	1	no	—
32	C₂²×C₄×C₂₁₁	—	2	2	no	—
33	C₂²×C₄×C₂₁₁	—	2	1	no	—
34	C₂²×C₄×C₂₁₁	—	2	1	no	—
35	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
38	M₁₆×C₂₁₁	—	2	2	no	—
39	M₁₆×C₂₁₁	—	2	2	no	—
40	C₂×C₈×C₂₁₁	—	2	2	no	—
41	C₄⋊C₄×C₂₁₁	—	2	4	no	—
42	C₄⋊C₄×C₂₁₁	—	2	1	no	—
43	C₄⋊C₄×C₂₁₁	—	2	1	no	—
44	C₄⋊C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
45	C₄⋊C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
46	C₂²⋊C₄×C₂₁₁	—	2	1	no	—
47	C₂²⋊C₄×C₂₁₁	—	2	1	no	—
48	C₂²⋊C₄×C₂₁₁	—	2	1	no	—
49	C₄²×C₂₁₁	—	2	1	no	—
50	C₄²×C₂₁₁	—	2	1	no	—
51	C₄²×C₂₁₁	—	2	1	no	—
52	C₄²×C₂₁₁	—	2	2	no	—
53	C₄²×C₂₁₁	—	2	2	no	—
54	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₂₁₁	D₄	1	1	yes	characteristic
57	C₄²×C₂₁₁	D₄	1	1	yes	characteristic
58	C₂³×C₂₁₁	—	2	1	no	—
59	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
60	Q₈×C₂₁₁	—	2	1	no	—
61	Q₈×C₂₁₁	C₄⋊C₄	1	2	yes	complemented
62	D₄×C₂₁₁	—	2	2	no	—
63	D₄×C₂₁₁	—	2	2	no	—
64	D₄×C₂₁₁	—	2	1	no	—
65	D₄×C₂₁₁	C₄⋊C₄	1	2	yes	complemented
66	C₂×C₄×C₂₁₁	—	2	1	no	—
67	C₂×C₄×C₂₁₁	—	2	2	no	—
68	C₂×C₄×C₂₁₁	—	2	2	no	—
69	C₂×C₄×C₂₁₁	—	2	2	no	—
70	C₂×C₄×C₂₁₁	—	2	4	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	2	no	—
75	C₂×C₄×C₂₁₁	—	4	2	no	—
76	C₂×C₄×C₂₁₁	—	4	2	no	—
77	C₂×C₄×C₂₁₁	—	4	2	no	—
78	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
87	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
88	C₈×C₂₁₁	—	4	2	no	—
89	C₂²×C₂₁₁	—	2	1	no	—
90	C₂²×C₂₁₁	—	2	1	no	—
91	C₂²×C₂₁₁	—	2	2	no	—
92	C₂²×C₂₁₁	—	4	1	no	—
93	C₂²×C₂₁₁	—	4	1	no	—
94	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂²×C₂₁₁	#121	1	1	yes	characteristic
96	C₂²×C₂₁₁	C₂²⋊Q₈	1	1	yes	characteristic
97	C₄×C₂₁₁	—	2	2	no	—
98	C₄×C₂₁₁	—	2	1	no	—
99	C₄×C₂₁₁	—	2	1	no	—
100	C₄×C₂₁₁	—	2	1	no	—
101	C₄×C₂₁₁	—	4	4	no	—
102	C₄×C₂₁₁	—	4	1	no	—
103	C₄×C₂₁₁	—	4	1	no	—
104	C₄×C₂₁₁	#104	1	2	yes	complemented
105	C₄×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₄×C₂₁₁	#117	1	1	yes	characteristic
107	C₄×C₂₁₁	#121	1	1	yes	characteristic
108	C₄×C₂₁₁	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₂₁₁	—	2	1	no	—
110	C₂×C₂₁₁	—	2	1	no	—
111	C₂×C₂₁₁	—	4	2	no	—
112	C₂×C₂₁₁	#380	1	1	yes	characteristic
113	C₂×C₂₁₁	C₂×#104	1	1	yes	characteristic
114	C₂×C₂₁₁	#421	1	1	yes	characteristic
115	C₂₁₁	#1806	1	1	yes	characteristic, direct product factor, Sylow
116	#1806	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₄×#41	C₂×C₂₁₁	1	1	yes	characteristic
118	#423	C₂×C₂₁₁	1	1	yes	characteristic, complemented
119	C₂×#104	C₂×C₂₁₁	1	2	yes	—
120	C₄×C₄⋊C₄	C₂×C₂₁₁	1	1	yes	characteristic, complemented
121	#339	C₂×C₂₁₁	1	2	yes	complemented
122	C₂×#41	C₂²×C₂₁₁	1	1	yes	characteristic
123	C₂×M₁₆	C₂²×C₂₁₁	1	2	yes	—
124	C₄×Q₈	C₄×C₂₁₁	1	1	yes	characteristic, complemented
125	C₄×Q₈	C₄×C₂₁₁	1	1	yes	characteristic, complemented
126	C₄×D₄	—	2	1	no	—
127	C₄×D₄	—	2	1	no	—
128	C₄×D₄	C₄×C₂₁₁	1	1	yes	characteristic, complemented
129	C₄×D₄	C₄×C₂₁₁	1	1	yes	characteristic, complemented
130	#117	—	2	1	no	—
131	#117	C₂²×C₂₁₁	1	1	yes	characteristic
132	C₂×C₄⋊C₄	—	2	1	no	—
133	C₂×C₄²	—	2	1	no	—
134	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
135	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
136	C₄⋊C₈	—	2	1	no	—
137	C₄⋊C₈	—	2	1	no	—
138	#104	—	2	4	no	—
139	#95	—	2	1	no	—
140	#41	—	2	1	no	—
141	#41	—	2	1	no	—
142	#41	D₄×C₂₁₁	1	2	yes	—
143	#41	Q₈×C₂₁₁	1	2	yes	—
144	C₂×Q₈	C₂×C₄×C₂₁₁	1	1	yes	characteristic
145	C₂×D₄	—	2	1	no	—
146	C₂×D₄	C₂×C₄×C₂₁₁	1	1	yes	characteristic
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	D₄×C₂₁₁	1	1	yes	characteristic
152	C₂²×C₄	D₄×C₂₁₁	1	1	yes	characteristic
153	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
154	M₁₆	—	2	2	no	—
155	M₁₆	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	2	1	no	—
159	C₄⋊C₄	—	2	1	no	—
160	C₄⋊C₄	C₂×C₄×C₂₁₁	1	1	yes	characteristic
161	C₄⋊C₄	C₂×C₄×C₂₁₁	1	1	yes	characteristic
162	C₂²⋊C₄	—	2	1	no	—
163	C₂²⋊C₄	—	2	1	no	—
164	C₂²⋊C₄	—	2	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	2	2	no	—
169	C₄²	—	2	2	no	—
170	C₄²	C₂×C₄×C₂₁₁	1	1	yes	characteristic
171	C₄²	C₂×C₄×C₂₁₁	1	1	yes	characteristic
172	C₄²	D₄×C₂₁₁	1	1	yes	characteristic
173	C₄²	D₄×C₂₁₁	1	1	yes	characteristic
174	C₂³	—	2	1	no	—
175	C₂³	C₂×D₄×C₂₁₁	1	1	yes	characteristic
176	Q₈	—	2	1	no	—
177	Q₈	C₄⋊C₄×C₂₁₁	1	2	yes	complemented
178	D₄	—	2	2	no	—
179	D₄	—	2	2	no	—
180	D₄	—	2	1	no	—
181	D₄	C₄⋊C₄×C₂₁₁	1	2	yes	complemented
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
195	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
196	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
197	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
198	C₂×C₄	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
199	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
200	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
201	C₂×C₄	C₂×Q₈×C₂₁₁	1	1	yes	characteristic
202	C₂×C₄	#41×C₂₁₁	1	1	yes	characteristic
203	C₂×C₄	#41×C₂₁₁	1	1	yes	characteristic
204	C₈	—	4	2	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	4	1	no	—
209	C₂²	—	4	1	no	—
210	C₂²	C₂×C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
211	C₂²	#121×C₂₁₁	1	1	yes	characteristic
212	C₂²	C₂²⋊Q₈×C₂₁₁	1	1	yes	characteristic
213	C₄	—	2	2	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	4	4	no	—
218	C₄	—	4	1	no	—
219	C₄	—	4	1	no	—
220	C₄	#104×C₂₁₁	1	2	yes	complemented
221	C₄	C₂×C₄⋊C₄×C₂₁₁	1	1	yes	characteristic
222	C₄	#117×C₂₁₁	1	1	yes	characteristic
223	C₄	#121×C₂₁₁	1	1	yes	characteristic
224	C₄	C₂²⋊Q₈×C₂₁₁	1	1	yes	characteristic
225	C₂	—	2	1	no	—
226	C₂	—	2	1	no	—
227	C₂	—	4	2	no	—
228	C₂	#380×C₂₁₁	1	1	yes	characteristic
229	C₂	C₂×#104×C₂₁₁	1	1	yes	characteristic
230	C₂	#421×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	C₂×C₂²⋊C₄
prime radical	C₂×#41×C₂₁₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₁₁ ◁ C₂×C₄²×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #423×C₂₁₁	C₂	semidirect
4. C₄×C₄⋊C₄×C₂₁₁	C₂	semidirect
5. #339×C₂₁₁	C₂	semidirect
8. C₄×Q₈×C₂₁₁	C₄	semidirect
9. C₄×Q₈×C₂₁₁	C₄	semidirect
12. C₄×D₄×C₂₁₁	C₄	semidirect
13. C₄×D₄×C₂₁₁	C₄	semidirect
61. Q₈×C₂₁₁	C₄⋊C₄	semidirect
65. D₄×C₂₁₁	C₄⋊C₄	semidirect
104. C₄×C₂₁₁	#104	semidirect
115. C₂₁₁	#1806	direct (N × H)
116. #1806	C₂₁₁	direct (N × H)
118. #423	C₂×C₂₁₁	semidirect
120. C₄×C₄⋊C₄	C₂×C₂₁₁	semidirect
121. #339	C₂×C₂₁₁	semidirect
124. C₄×Q₈	C₄×C₂₁₁	semidirect
125. C₄×Q₈	C₄×C₂₁₁	semidirect
128. C₄×D₄	C₄×C₂₁₁	semidirect
129. C₄×D₄	C₄×C₂₁₁	semidirect
177. Q₈	C₄⋊C₄×C₂₁₁	semidirect
181. D₄	C₄⋊C₄×C₂₁₁	semidirect
220. C₄	#104×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×#41×C₂₁₁	2
#423×C₂₁₁	2
C₂×#104×C₂₁₁	2
C₄×C₄⋊C₄×C₂₁₁	2
#339×C₂₁₁	2
#1806	211

Subgroups — #1806×C211 #41009426

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1806×C₂₁₁ #41009426