Subgroups — #2297×C₂₁₁ #41009917

Order 27008.

936 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	C₂²×#41×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#117×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄²×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#391×C₂₁₁	C₂	1	2	yes	maximal, complemented
5	#390×C₂₁₁	C₂	1	2	yes	maximal, complemented
6	#386×C₂₁₁	C₂	1	4	yes	maximal, complemented
7	C₄×C₂²⋊C₄×C₂₁₁	C₂	1	4	yes	maximal, complemented
8	C₂×#41×C₂₁₁	—	2	4	no	—
9	C₂×#41×C₂₁₁	—	2	4	no	—
10	C₂×#41×C₂₁₁	C₄	1	8	yes	complemented
11	C₂²×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₂₁₁	C₂²	1	2	yes	—
13	C₂²×D₄×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
15	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
16	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
17	#117×C₂₁₁	—	2	4	no	—
18	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	8	yes	—
20	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₁₁	—	2	4	no	—
22	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
23	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
24	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
25	#95×C₂₁₁	C₂²	1	8	yes	—
26	C₂⁴×C₂₁₁	C₂³	1	2	yes	—
27	C₂⁴×C₂₁₁	C₂³	1	1	yes	characteristic
28	#41×C₂₁₁	—	2	32	no	—
29	C₂×Q₈×C₂₁₁	—	2	2	no	—
30	C₂×Q₈×C₂₁₁	—	2	2	no	—
31	C₂×Q₈×C₂₁₁	C₂×C₄	1	4	yes	complemented
32	C₂×D₄×C₂₁₁	—	2	4	no	—
33	C₂×D₄×C₂₁₁	—	2	4	no	—
34	C₂×D₄×C₂₁₁	—	2	2	no	—
35	C₂×D₄×C₂₁₁	—	2	2	no	—
36	C₂×D₄×C₂₁₁	C₂×C₄	1	8	yes	—
37	C₂×D₄×C₂₁₁	C₂×C₄	1	4	yes	complemented
38	C₂²×C₄×C₂₁₁	—	2	4	no	—
39	C₂²×C₄×C₂₁₁	—	2	4	no	—
40	C₂²×C₄×C₂₁₁	—	2	4	no	—
41	C₂²×C₄×C₂₁₁	—	2	8	no	—
42	C₂²×C₄×C₂₁₁	C₂×C₄	1	4	yes	—
43	C₂²×C₄×C₂₁₁	C₂×C₄	1	8	yes	complemented
44	C₂²×C₄×C₂₁₁	D₄	1	4	yes	—
45	C₂²×C₄×C₂₁₁	D₄	1	4	yes	—
46	C₂²×C₄×C₂₁₁	C₂³	1	4	yes	—
47	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
48	C₂²×C₄×C₂₁₁	C₂³	1	4	yes	—
49	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₂₁₁	—	2	4	no	—
52	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
53	C₂²⋊C₄×C₂₁₁	—	2	16	no	—
54	C₄²×C₂₁₁	—	2	4	no	—
55	C₄²×C₂₁₁	—	2	4	no	—
56	C₄²×C₂₁₁	—	2	4	no	—
57	C₂³×C₂₁₁	—	2	4	no	—
58	C₂³×C₂₁₁	—	2	2	no	—
59	C₂³×C₂₁₁	—	2	2	no	—
60	C₂³×C₂₁₁	—	2	8	no	—
61	C₂³×C₂₁₁	C₂²×C₄	1	4	yes	—
62	C₂³×C₂₁₁	C₂²×C₄	1	2	yes	—
63	C₂³×C₂₁₁	C₂×D₄	1	2	yes	—
64	C₂³×C₂₁₁	C₂×D₄	1	2	yes	—
65	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
66	Q₈×C₂₁₁	—	2	8	no	—
67	D₄×C₂₁₁	—	2	16	no	—
68	D₄×C₂₁₁	—	2	8	no	—
69	C₂×C₄×C₂₁₁	—	2	16	no	—
70	C₂×C₄×C₂₁₁	—	2	4	no	—
71	C₂×C₄×C₂₁₁	—	2	2	no	—
72	C₂×C₄×C₂₁₁	—	2	2	no	—
73	C₂×C₄×C₂₁₁	—	2	4	no	—
74	C₂×C₄×C₂₁₁	—	2	16	no	—
75	C₂×C₄×C₂₁₁	—	2	4	no	—
76	C₂×C₄×C₂₁₁	—	2	8	no	—
77	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	16	yes	complemented
78	C₂×C₄×C₂₁₁	C₂²×C₄	1	4	yes	—
79	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
82	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
83	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₂₁₁	#41	1	8	yes	complemented
86	C₂²×C₂₁₁	—	2	4	no	—
87	C₂²×C₂₁₁	—	2	4	no	—
88	C₂²×C₂₁₁	—	2	4	no	—
89	C₂²×C₂₁₁	—	2	4	no	—
90	C₂²×C₂₁₁	—	2	8	no	—
91	C₂²×C₂₁₁	—	4	4	no	—
92	C₂²×C₂₁₁	—	4	4	no	—
93	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	4	yes	—
94	C₂²×C₂₁₁	C₂³×C₄	1	1	yes	characteristic
95	C₂²×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
96	C₂²×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
97	C₂²×C₂₁₁	C₂×#41	1	4	yes	—
98	C₄×C₂₁₁	—	2	4	no	—
99	C₄×C₂₁₁	—	2	8	no	—
100	C₄×C₂₁₁	—	4	4	no	—
101	C₄×C₂₁₁	C₂×C₂²⋊C₄	1	4	yes	—
102	C₄×C₂₁₁	C₂×C₂²⋊C₄	1	4	yes	—
103	C₂×C₂₁₁	—	2	4	no	—
104	C₂×C₂₁₁	—	4	4	no	—
105	C₂×C₂₁₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₂₁₁	C₄×#41	1	2	yes	—
107	C₂×C₂₁₁	#525	1	2	yes	—
108	C₂×C₂₁₁	#532	1	2	yes	—
109	C₂₁₁	#2297	1	1	yes	characteristic, direct product factor, Sylow
110	#2297	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂²×#41	C₂×C₂₁₁	1	1	yes	characteristic
112	C₂×#117	C₂×C₂₁₁	1	1	yes	characteristic, complemented
113	C₂²×C₄²	C₂×C₂₁₁	1	1	yes	characteristic, complemented
114	#391	C₂×C₂₁₁	1	2	yes	complemented
115	#390	C₂×C₂₁₁	1	2	yes	complemented
116	#386	C₂×C₂₁₁	1	4	yes	complemented
117	C₄×C₂²⋊C₄	C₂×C₂₁₁	1	4	yes	complemented
118	C₂×#41	—	2	4	no	—
119	C₂×#41	—	2	4	no	—
120	C₂×#41	C₄×C₂₁₁	1	8	yes	complemented
121	C₂²×Q₈	C₂²×C₂₁₁	1	1	yes	characteristic
122	C₂²×D₄	C₂²×C₂₁₁	1	2	yes	—
123	C₂²×D₄	C₂²×C₂₁₁	1	1	yes	characteristic
124	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
125	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
126	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
127	#117	—	2	4	no	—
128	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
129	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
130	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	8	yes	—
131	C₂×C₄²	—	2	4	no	—
132	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
133	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
134	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
135	#95	C₂²×C₂₁₁	1	8	yes	—
136	C₂⁴	C₂³×C₂₁₁	1	2	yes	—
137	C₂⁴	C₂³×C₂₁₁	1	1	yes	characteristic
138	#41	—	2	32	no	—
139	C₂×Q₈	—	2	2	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	C₂×C₄×C₂₁₁	1	4	yes	complemented
142	C₂×D₄	—	2	4	no	—
143	C₂×D₄	—	2	4	no	—
144	C₂×D₄	—	2	2	no	—
145	C₂×D₄	—	2	2	no	—
146	C₂×D₄	C₂×C₄×C₂₁₁	1	8	yes	—
147	C₂×D₄	C₂×C₄×C₂₁₁	1	4	yes	complemented
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	2	8	no	—
152	C₂²×C₄	C₂×C₄×C₂₁₁	1	4	yes	—
153	C₂²×C₄	C₂×C₄×C₂₁₁	1	8	yes	complemented
154	C₂²×C₄	D₄×C₂₁₁	1	4	yes	—
155	C₂²×C₄	D₄×C₂₁₁	1	4	yes	—
156	C₂²×C₄	C₂³×C₂₁₁	1	4	yes	—
157	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
158	C₂²×C₄	C₂³×C₂₁₁	1	4	yes	—
159	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
160	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
161	C₄⋊C₄	—	2	4	no	—
162	C₂²⋊C₄	—	2	4	no	—
163	C₂²⋊C₄	—	2	16	no	—
164	C₄²	—	2	4	no	—
165	C₄²	—	2	4	no	—
166	C₄²	—	2	4	no	—
167	C₂³	—	2	4	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	2	2	no	—
170	C₂³	—	2	8	no	—
171	C₂³	C₂²×C₄×C₂₁₁	1	4	yes	—
172	C₂³	C₂²×C₄×C₂₁₁	1	2	yes	—
173	C₂³	C₂×D₄×C₂₁₁	1	2	yes	—
174	C₂³	C₂×D₄×C₂₁₁	1	2	yes	—
175	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
176	Q₈	—	2	8	no	—
177	D₄	—	2	16	no	—
178	D₄	—	2	8	no	—
179	C₂×C₄	—	2	16	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	16	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	8	no	—
187	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	16	yes	complemented
188	C₂×C₄	C₂²×C₄×C₂₁₁	1	4	yes	—
189	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
190	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
191	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
192	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
193	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
194	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
195	C₂×C₄	#41×C₂₁₁	1	8	yes	complemented
196	C₂²	—	2	4	no	—
197	C₂²	—	2	4	no	—
198	C₂²	—	2	4	no	—
199	C₂²	—	2	4	no	—
200	C₂²	—	2	8	no	—
201	C₂²	—	4	4	no	—
202	C₂²	—	4	4	no	—
203	C₂²	C₂×C₂²⋊C₄×C₂₁₁	1	4	yes	—
204	C₂²	C₂³×C₄×C₂₁₁	1	1	yes	characteristic
205	C₂²	C₂²×D₄×C₂₁₁	1	1	yes	characteristic
206	C₂²	C₂²×D₄×C₂₁₁	1	1	yes	characteristic
207	C₂²	C₂×#41×C₂₁₁	1	4	yes	—
208	C₄	—	2	8	no	—
209	C₄	—	2	4	no	—
210	C₄	—	4	4	no	—
211	C₄	C₂×C₂²⋊C₄×C₂₁₁	1	4	yes	—
212	C₄	C₂×C₂²⋊C₄×C₂₁₁	1	4	yes	—
213	C₂	—	2	4	no	—
214	C₂	—	4	4	no	—
215	C₂	C₂²×C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
216	C₂	C₄×#41×C₂₁₁	1	2	yes	—
217	C₂	#525×C₂₁₁	1	2	yes	—
218	C₂	#532×C₂₁₁	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₁₁	C₂³
commutator	C₂²	C₂³×C₄×C₂₁₁
Frattini, cosocle	C₂³	C₂⁴×C₂₁₁
socle	C₂³×C₂₁₁	C₂⁴
prime radical	C₂²×#41×C₂₁₁	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×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. C₂²×C₄²×C₂₁₁	C₂	semidirect
4. #391×C₂₁₁	C₂	semidirect
5. #390×C₂₁₁	C₂	semidirect
6. #386×C₂₁₁	C₂	semidirect
7. C₄×C₂²⋊C₄×C₂₁₁	C₂	semidirect
10. C₂×#41×C₂₁₁	C₄	semidirect
31. C₂×Q₈×C₂₁₁	C₂×C₄	semidirect
37. C₂×D₄×C₂₁₁	C₂×C₄	semidirect
43. C₂²×C₄×C₂₁₁	C₂×C₄	semidirect
77. C₂×C₄×C₂₁₁	C₂²⋊C₄	semidirect
85. C₂×C₄×C₂₁₁	#41	semidirect
109. C₂₁₁	#2297	direct (N × H)
110. #2297	C₂₁₁	direct (N × H)
112. C₂×#117	C₂×C₂₁₁	semidirect
113. C₂²×C₄²	C₂×C₂₁₁	semidirect
114. #391	C₂×C₂₁₁	semidirect
115. #390	C₂×C₂₁₁	semidirect
116. #386	C₂×C₂₁₁	semidirect
117. C₄×C₂²⋊C₄	C₂×C₂₁₁	semidirect
120. C₂×#41	C₄×C₂₁₁	semidirect
141. C₂×Q₈	C₂×C₄×C₂₁₁	semidirect
147. C₂×D₄	C₂×C₄×C₂₁₁	semidirect
153. C₂²×C₄	C₂×C₄×C₂₁₁	semidirect
187. C₂×C₄	C₂²⋊C₄×C₂₁₁	semidirect
195. C₂×C₄	#41×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#41×C₂₁₁	2
C₂×#117×C₂₁₁	2
C₂²×C₄²×C₂₁₁	2
#391×C₂₁₁	2
#390×C₂₁₁	2
#386×C₂₁₁	2
C₄×C₂²⋊C₄×C₂₁₁	2
#2297	211

Subgroups — #2297×C211 #41009917

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2297×C₂₁₁ #41009917