Subgroups — C₂×#468×C₂₁₁ #41010671

Order 27008.

544 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×SD₁₆×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×M₁₆×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊Q₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#121×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	#468×C₂₁₁	C₂	1	8	yes	maximal, direct product factor
6	C₂×#107×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#103×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#102×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₂₁₁	—	2	1	no	—
11	C₂²×D₄×C₂₁₁	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×C₂₁₁	—	2	4	no	—
14	C₂×SD₁₆×C₂₁₁	C₂²	1	4	yes	complemented
15	C₂×M₁₆×C₂₁₁	—	2	4	no	—
16	C₂×M₁₆×C₂₁₁	C₂²	1	4	yes	complemented
17	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
19	C₂²⋊Q₈×C₂₁₁	—	2	2	no	—
20	C₂²⋊Q₈×C₂₁₁	C₂²	1	4	yes	complemented
21	#121×C₂₁₁	—	2	2	no	—
22	#121×C₂₁₁	C₂²	1	4	yes	—
23	C₂×C₄⋊C₄×C₂₁₁	—	2	1	no	—
24	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₂₁₁	—	2	1	no	—
27	C₂×C₂²⋊C₄×C₂₁₁	—	2	1	no	—
28	#107×C₂₁₁	C₂²	1	4	yes	complemented
29	#103×C₂₁₁	C₂²	1	4	yes	complemented
30	#102×C₂₁₁	C₂²	1	4	yes	complemented
31	C₂⁴×C₂₁₁	—	2	1	no	—
32	C₂⁴×C₂₁₁	D₄	1	1	yes	characteristic
33	C₂×Q₈×C₂₁₁	—	2	1	no	—
34	C₂×Q₈×C₂₁₁	—	2	2	no	—
35	C₂×Q₈×C₂₁₁	—	2	2	no	—
36	C₂×Q₈×C₂₁₁	C₂³	1	2	yes	—
37	C₂×D₄×C₂₁₁	—	2	4	no	—
38	C₂×D₄×C₂₁₁	—	2	2	no	—
39	C₂×D₄×C₂₁₁	—	2	1	no	—
40	C₂×D₄×C₂₁₁	—	2	2	no	—
41	C₂×D₄×C₂₁₁	—	4	4	no	—
42	C₂×D₄×C₂₁₁	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₁₁	—	2	2	no	—
44	C₂²×C₄×C₂₁₁	—	2	2	no	—
45	C₂²×C₄×C₂₁₁	—	2	1	no	—
46	C₂²×C₄×C₂₁₁	—	2	1	no	—
47	C₂²×C₄×C₂₁₁	—	2	1	no	—
48	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
49	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
51	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
52	SD₁₆×C₂₁₁	—	2	8	no	—
53	M₁₆×C₂₁₁	—	2	8	no	—
54	C₂×C₈×C₂₁₁	—	2	2	no	—
55	C₂×C₈×C₂₁₁	D₄	1	4	yes	complemented
56	C₂×C₈×C₂₁₁	C₂³	1	2	yes	—
57	C₂×C₈×C₂₁₁	C₂³	1	2	yes	—
58	C₄⋊C₄×C₂₁₁	—	2	4	no	—
59	C₄⋊C₄×C₂₁₁	—	2	1	no	—
60	C₄⋊C₄×C₂₁₁	—	2	1	no	—
61	C₄⋊C₄×C₂₁₁	C₂³	1	2	yes	—
62	C₄⋊C₄×C₂₁₁	C₂³	1	2	yes	—
63	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
64	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
65	C₂³×C₂₁₁	—	2	2	no	—
66	C₂³×C₂₁₁	—	2	2	no	—
67	C₂³×C₂₁₁	—	2	2	no	—
68	C₂³×C₂₁₁	—	2	2	no	—
69	C₂³×C₂₁₁	—	4	2	no	—
70	C₂³×C₂₁₁	—	4	2	no	—
71	C₂³×C₂₁₁	—	4	2	no	—
72	C₂³×C₂₁₁	C₂×D₄	1	2	yes	—
73	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
74	Q₈×C₂₁₁	—	2	2	no	—
75	Q₈×C₂₁₁	—	2	2	no	—
76	Q₈×C₂₁₁	—	4	2	no	—
77	D₄×C₂₁₁	—	2	2	no	—
78	D₄×C₂₁₁	—	2	2	no	—
79	D₄×C₂₁₁	—	4	8	no	—
80	D₄×C₂₁₁	—	4	2	no	—
81	C₂×C₄×C₂₁₁	—	2	2	no	—
82	C₂×C₄×C₂₁₁	—	2	4	no	—
83	C₂×C₄×C₂₁₁	—	2	2	no	—
84	C₂×C₄×C₂₁₁	—	2	2	no	—
85	C₂×C₄×C₂₁₁	—	2	2	no	—
86	C₂×C₄×C₂₁₁	—	2	2	no	—
87	C₂×C₄×C₂₁₁	—	2	2	no	—
88	C₂×C₄×C₂₁₁	—	4	2	no	—
89	C₂×C₄×C₂₁₁	—	4	2	no	—
90	C₂×C₄×C₂₁₁	—	4	2	no	—
91	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₂₁₁	#41	1	2	yes	—
95	C₂×C₄×C₂₁₁	C₂⁴	1	1	yes	characteristic
96	C₈×C₂₁₁	—	2	2	no	—
97	C₈×C₂₁₁	C₂×D₄	1	4	yes	complemented
98	C₂²×C₂₁₁	—	2	2	no	—
99	C₂²×C₂₁₁	—	2	2	no	—
100	C₂²×C₂₁₁	—	2	2	no	—
101	C₂²×C₂₁₁	—	4	4	no	—
102	C₂²×C₂₁₁	—	4	2	no	—
103	C₂²×C₂₁₁	—	4	2	no	—
104	C₂²×C₂₁₁	—	4	2	no	—
105	C₂²×C₂₁₁	—	8	4	no	—
106	C₂²×C₂₁₁	#121	1	2	yes	—
107	C₂²×C₂₁₁	AGL(1,ℤ/8)	1	2	yes	—
108	C₂²×C₂₁₁	#137	1	2	yes	—
109	C₂²×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
110	C₄×C₂₁₁	—	2	2	no	—
111	C₄×C₂₁₁	—	4	2	no	—
112	C₄×C₂₁₁	—	4	2	no	—
113	C₄×C₂₁₁	—	4	2	no	—
114	C₄×C₂₁₁	#121	1	2	yes	—
115	C₄×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
116	C₄×C₂₁₁	C₂×#41	1	1	yes	characteristic
117	C₂×C₂₁₁	—	4	2	no	—
118	C₂×C₂₁₁	—	8	2	no	—
119	C₂×C₂₁₁	#468	1	4	yes	direct product factor
120	C₂×C₂₁₁	C₂×#121	1	1	yes	characteristic
121	C₂×C₂₁₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₂₁₁	C₂×#137	1	1	yes	characteristic
123	C₂₁₁	C₂×#468	1	1	yes	characteristic, direct product factor, Sylow
124	C₂×#468	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂²×SD₁₆	C₂×C₂₁₁	1	1	yes	characteristic, complemented
126	C₂²×M₁₆	C₂×C₂₁₁	1	1	yes	characteristic, complemented
127	C₂×C₂²⋊Q₈	C₂×C₂₁₁	1	1	yes	characteristic, complemented
128	C₂×#121	C₂×C₂₁₁	1	1	yes	characteristic
129	#468	C₂×C₂₁₁	1	8	yes	direct product factor
130	C₂×#107	C₂×C₂₁₁	1	1	yes	characteristic, complemented
131	C₂×#103	C₂×C₂₁₁	1	1	yes	characteristic, complemented
132	C₂×#102	C₂×C₂₁₁	1	1	yes	characteristic, complemented
133	C₂²×Q₈	C₂²×C₂₁₁	1	1	yes	characteristic
134	C₂²×D₄	—	2	1	no	—
135	C₂²×D₄	C₂²×C₂₁₁	1	1	yes	characteristic
136	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
137	C₂×SD₁₆	—	2	4	no	—
138	C₂×SD₁₆	C₂²×C₂₁₁	1	4	yes	complemented
139	C₂×M₁₆	—	2	4	no	—
140	C₂×M₁₆	C₂²×C₂₁₁	1	4	yes	complemented
141	C₂²×C₈	C₂²×C₂₁₁	1	1	yes	characteristic
142	C₂²×C₈	C₂²×C₂₁₁	1	1	yes	characteristic
143	C₂²⋊Q₈	—	2	2	no	—
144	C₂²⋊Q₈	C₂²×C₂₁₁	1	4	yes	complemented
145	#121	—	2	2	no	—
146	#121	C₂²×C₂₁₁	1	4	yes	—
147	C₂×C₄⋊C₄	—	2	1	no	—
148	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
149	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
150	C₂×C₂²⋊C₄	—	2	1	no	—
151	C₂×C₂²⋊C₄	—	2	1	no	—
152	#107	C₂²×C₂₁₁	1	4	yes	complemented
153	#103	C₂²×C₂₁₁	1	4	yes	complemented
154	#102	C₂²×C₂₁₁	1	4	yes	complemented
155	C₂⁴	—	2	1	no	—
156	C₂⁴	D₄×C₂₁₁	1	1	yes	characteristic
157	C₂×Q₈	—	2	1	no	—
158	C₂×Q₈	—	2	2	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	C₂³×C₂₁₁	1	2	yes	—
161	C₂×D₄	—	2	4	no	—
162	C₂×D₄	—	2	2	no	—
163	C₂×D₄	—	2	1	no	—
164	C₂×D₄	—	2	2	no	—
165	C₂×D₄	—	4	4	no	—
166	C₂×D₄	C₂³×C₂₁₁	1	2	yes	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	D₄×C₂₁₁	1	1	yes	characteristic
173	C₂²×C₄	D₄×C₂₁₁	1	2	yes	—
174	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
175	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
176	SD₁₆	—	2	8	no	—
177	M₁₆	—	2	8	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	D₄×C₂₁₁	1	4	yes	complemented
180	C₂×C₈	C₂³×C₂₁₁	1	2	yes	—
181	C₂×C₈	C₂³×C₂₁₁	1	2	yes	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	C₂³×C₂₁₁	1	2	yes	—
186	C₄⋊C₄	C₂³×C₂₁₁	1	2	yes	—
187	C₂²⋊C₄	—	2	4	no	—
188	C₂²⋊C₄	—	2	4	no	—
189	C₂³	—	2	2	no	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	4	2	no	—
194	C₂³	—	4	2	no	—
195	C₂³	—	4	2	no	—
196	C₂³	C₂×D₄×C₂₁₁	1	2	yes	—
197	C₂³	C₂×D₄×C₂₁₁	1	1	yes	characteristic
198	Q₈	—	2	2	no	—
199	Q₈	—	2	2	no	—
200	Q₈	—	4	2	no	—
201	D₄	—	2	2	no	—
202	D₄	—	2	2	no	—
203	D₄	—	4	8	no	—
204	D₄	—	4	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
216	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
217	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
218	C₂×C₄	#41×C₂₁₁	1	2	yes	—
219	C₂×C₄	C₂⁴×C₂₁₁	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	C₂×D₄×C₂₁₁	1	4	yes	complemented
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	4	4	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	—	8	4	no	—
230	C₂²	#121×C₂₁₁	1	2	yes	—
231	C₂²	AGL(1,ℤ/8)×C₂₁₁	1	2	yes	—
232	C₂²	#137×C₂₁₁	1	2	yes	—
233	C₂²	C₂²×D₄×C₂₁₁	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	4	2	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	2	no	—
238	C₄	#121×C₂₁₁	1	2	yes	—
239	C₄	C₂²×D₄×C₂₁₁	1	1	yes	characteristic
240	C₄	C₂×#41×C₂₁₁	1	1	yes	characteristic
241	C₂	—	4	2	no	—
242	C₂	—	8	2	no	—
243	C₂	#468×C₂₁₁	1	4	yes	direct product factor
244	C₂	C₂×#121×C₂₁₁	1	1	yes	characteristic
245	C₂	C₂×AGL(1,ℤ/8)×C₂₁₁	1	1	yes	characteristic
246	C₂	C₂×#137×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
1. C₂²×SD₁₆×C₂₁₁	C₂	semidirect
2. C₂²×M₁₆×C₂₁₁	C₂	semidirect
3. C₂×C₂²⋊Q₈×C₂₁₁	C₂	semidirect
5. #468×C₂₁₁	C₂	direct (N × H)
6. C₂×#107×C₂₁₁	C₂	semidirect
7. C₂×#103×C₂₁₁	C₂	semidirect
8. C₂×#102×C₂₁₁	C₂	semidirect
14. C₂×SD₁₆×C₂₁₁	C₂²	semidirect
16. C₂×M₁₆×C₂₁₁	C₂²	semidirect
20. C₂²⋊Q₈×C₂₁₁	C₂²	semidirect
28. #107×C₂₁₁	C₂²	semidirect
29. #103×C₂₁₁	C₂²	semidirect
30. #102×C₂₁₁	C₂²	semidirect
55. C₂×C₈×C₂₁₁	D₄	semidirect
97. C₈×C₂₁₁	C₂×D₄	semidirect
119. C₂×C₂₁₁	#468	direct (N × H)
123. C₂₁₁	C₂×#468	direct (N × H)
124. C₂×#468	C₂₁₁	direct (N × H)
125. C₂²×SD₁₆	C₂×C₂₁₁	semidirect
126. C₂²×M₁₆	C₂×C₂₁₁	semidirect
127. C₂×C₂²⋊Q₈	C₂×C₂₁₁	semidirect
129. #468	C₂×C₂₁₁	direct (N × H)
130. C₂×#107	C₂×C₂₁₁	semidirect
131. C₂×#103	C₂×C₂₁₁	semidirect
132. C₂×#102	C₂×C₂₁₁	semidirect
138. C₂×SD₁₆	C₂²×C₂₁₁	semidirect
140. C₂×M₁₆	C₂²×C₂₁₁	semidirect
144. C₂²⋊Q₈	C₂²×C₂₁₁	semidirect
152. #107	C₂²×C₂₁₁	semidirect
153. #103	C₂²×C₂₁₁	semidirect
154. #102	C₂²×C₂₁₁	semidirect
179. C₂×C₈	D₄×C₂₁₁	semidirect
221. C₈	C₂×D₄×C₂₁₁	semidirect
243. C₂	#468×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
C₂²×SD₁₆×C₂₁₁	2
C₂²×M₁₆×C₂₁₁	2
C₂×C₂²⋊Q₈×C₂₁₁	2
C₂×#121×C₂₁₁	2
#468×C₂₁₁	2
C₂×#107×C₂₁₁	2
C₂×#103×C₂₁₁	2
C₂×#102×C₂₁₁	2
C₂×#468	211

Subgroups — C2×#468×C211 #41010671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#468×C₂₁₁ #41010671