Subgroups — #3037×C₂₁₁ #41010657

Order 27008.

484 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#137×C₂₁₁	C₂	1	2	yes	maximal, complemented
2	#532×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Q₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#464×C₂₁₁	C₂	1	4	yes	maximal, complemented
5	#462×C₂₁₁	C₂	1	2	yes	maximal, complemented
6	#460×C₂₁₁	C₂	1	2	yes	maximal, complemented
7	#423×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#417×C₂₁₁	C₂	1	2	yes	maximal, complemented
9	C₂×#41×C₂₁₁	C₂²	1	2	yes	—
10	C₂²×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
11	#137×C₂₁₁	—	2	8	no	—
12	C₂×Q₁₆×C₂₁₁	C₂²	1	4	yes	—
13	C₂×SD₁₆×C₂₁₁	C₂²	1	4	yes	—
14	C₂×M₁₆×C₂₁₁	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
16	#127×C₂₁₁	C₂²	1	1	yes	characteristic
17	#124×C₂₁₁	C₂²	1	2	yes	—
18	#121×C₂₁₁	—	2	1	no	—
19	#121×C₂₁₁	—	2	1	no	—
20	C₄×Q₈×C₂₁₁	—	2	1	no	—
21	C₄×Q₈×C₂₁₁	—	2	1	no	—
22	C₄×Q₈×C₂₁₁	C₂²	1	2	yes	—
23	C₄×Q₈×C₂₁₁	C₂²	1	2	yes	—
24	C₄×D₄×C₂₁₁	—	2	1	no	—
25	C₄×D₄×C₂₁₁	—	2	1	no	—
26	C₂×C₄⋊C₄×C₂₁₁	—	2	1	no	—
27	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₂₁₁	—	2	1	no	—
29	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
30	C₄⋊C₈×C₂₁₁	C₂²	1	2	yes	—
31	C₄⋊C₈×C₂₁₁	C₂²	1	2	yes	—
32	#103×C₂₁₁	C₂²	1	4	yes	—
33	#102×C₂₁₁	C₂²	1	4	yes	—
34	#41×C₂₁₁	—	2	4	no	—
35	#41×C₂₁₁	—	2	4	no	—
36	C₂×Q₈×C₂₁₁	—	2	1	no	—
37	C₂×Q₈×C₂₁₁	—	2	2	no	—
38	C₂×Q₈×C₂₁₁	D₄	1	4	yes	complemented
39	C₂×Q₈×C₂₁₁	C₂³	1	2	yes	—
40	C₂×Q₈×C₂₁₁	C₂³	1	2	yes	—
41	C₂×D₄×C₂₁₁	—	2	2	no	—
42	C₂×D₄×C₂₁₁	—	2	1	no	—
43	C₂×D₄×C₂₁₁	—	2	1	no	—
44	C₂×D₄×C₂₁₁	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₁₁	—	2	1	no	—
46	C₂²×C₄×C₂₁₁	—	2	1	no	—
47	C₂²×C₄×C₂₁₁	—	2	2	no	—
48	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
51	Q₁₆×C₂₁₁	—	2	8	no	—
52	SD₁₆×C₂₁₁	—	2	8	no	—
53	M₁₆×C₂₁₁	—	2	4	no	—
54	C₂×C₈×C₂₁₁	C₂³	1	4	yes	—
55	C₄⋊C₄×C₂₁₁	—	2	2	no	—
56	C₄⋊C₄×C₂₁₁	—	2	2	no	—
57	C₄⋊C₄×C₂₁₁	—	2	1	no	—
58	C₄⋊C₄×C₂₁₁	—	2	1	no	—
59	C₄⋊C₄×C₂₁₁	—	2	1	no	—
60	C₄⋊C₄×C₂₁₁	C₂³	1	2	yes	—
61	C₂²⋊C₄×C₂₁₁	—	2	2	no	—
62	C₂²⋊C₄×C₂₁₁	—	2	2	no	—
63	C₄²×C₂₁₁	—	2	2	no	—
64	C₄²×C₂₁₁	—	2	2	no	—
65	C₄²×C₂₁₁	D₄	1	1	yes	characteristic
66	C₄²×C₂₁₁	D₄	1	1	yes	characteristic
67	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
68	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
69	C₂³×C₂₁₁	—	2	2	no	—
70	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
71	Q₈×C₂₁₁	—	2	4	no	—
72	Q₈×C₂₁₁	—	2	2	no	—
73	Q₈×C₂₁₁	—	2	2	no	—
74	Q₈×C₂₁₁	—	2	2	no	—
75	Q₈×C₂₁₁	C₂×D₄	1	4	yes	—
76	D₄×C₂₁₁	—	2	2	no	—
77	D₄×C₂₁₁	—	2	2	no	—
78	D₄×C₂₁₁	—	4	2	no	—
79	D₄×C₂₁₁	—	4	4	no	—
80	D₄×C₂₁₁	—	4	2	no	—
81	C₂×C₄×C₂₁₁	—	2	4	no	—
82	C₂×C₄×C₂₁₁	—	2	2	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	—	4	1	no	—
87	C₂×C₄×C₂₁₁	—	4	2	no	—
88	C₂×C₄×C₂₁₁	—	4	4	no	—
89	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₂₁₁	#41	1	2	yes	—
96	C₂×C₄×C₂₁₁	C₂⁴	1	1	yes	characteristic
97	C₈×C₂₁₁	—	2	4	no	—
98	C₂²×C₂₁₁	—	2	1	no	—
99	C₂²×C₂₁₁	—	2	1	no	—
100	C₂²×C₂₁₁	—	4	2	no	—
101	C₂²×C₂₁₁	—	4	2	no	—
102	C₂²×C₂₁₁	—	4	2	no	—
103	C₂²×C₂₁₁	#121	1	2	yes	—
104	C₂²×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₂₁₁	—	2	4	no	—
106	C₄×C₂₁₁	—	2	1	no	—
107	C₄×C₂₁₁	—	2	1	no	—
108	C₄×C₂₁₁	—	2	1	no	—
109	C₄×C₂₁₁	—	4	2	no	—
110	C₄×C₂₁₁	—	4	2	no	—
111	C₄×C₂₁₁	#121	1	2	yes	—
112	C₄×C₂₁₁	#137	1	2	yes	complemented
113	C₄×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
114	C₄×C₂₁₁	C₂×#41	1	1	yes	characteristic
115	C₂×C₂₁₁	—	2	2	no	—
116	C₂×C₂₁₁	—	8	2	no	—
117	C₂×C₂₁₁	C₂×#121	1	1	yes	characteristic
118	C₂×C₂₁₁	C₂×#137	1	1	yes	characteristic
119	C₂×C₂₁₁	#575	1	1	yes	characteristic
120	C₂₁₁	#3037	1	1	yes	characteristic, direct product factor, Sylow
121	#3037	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂×#137	C₂×C₂₁₁	1	2	yes	complemented
123	#532	C₂×C₂₁₁	1	1	yes	characteristic
124	C₂×C₄×Q₈	C₂×C₂₁₁	1	1	yes	characteristic, complemented
125	#464	C₂×C₂₁₁	1	4	yes	complemented
126	#462	C₂×C₂₁₁	1	2	yes	complemented
127	#460	C₂×C₂₁₁	1	2	yes	complemented
128	#423	C₂×C₂₁₁	1	1	yes	characteristic, complemented
129	#417	C₂×C₂₁₁	1	2	yes	complemented
130	C₂×#41	C₂²×C₂₁₁	1	2	yes	—
131	C₂²×Q₈	C₂²×C₂₁₁	1	1	yes	characteristic
132	#137	—	2	8	no	—
133	C₂×Q₁₆	C₂²×C₂₁₁	1	4	yes	—
134	C₂×SD₁₆	C₂²×C₂₁₁	1	4	yes	—
135	C₂×M₁₆	C₂²×C₂₁₁	1	2	yes	—
136	C₄⋊Q₈	C₂²×C₂₁₁	1	1	yes	characteristic
137	#127	C₂²×C₂₁₁	1	1	yes	characteristic
138	#124	C₂²×C₂₁₁	1	2	yes	—
139	#121	—	2	1	no	—
140	#121	—	2	1	no	—
141	C₄×Q₈	—	2	1	no	—
142	C₄×Q₈	—	2	1	no	—
143	C₄×Q₈	C₂²×C₂₁₁	1	2	yes	—
144	C₄×Q₈	C₂²×C₂₁₁	1	2	yes	—
145	C₄×D₄	—	2	1	no	—
146	C₄×D₄	—	2	1	no	—
147	C₂×C₄⋊C₄	—	2	1	no	—
148	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
149	C₂×C₄²	—	2	1	no	—
150	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
151	C₄⋊C₈	C₂²×C₂₁₁	1	2	yes	—
152	C₄⋊C₈	C₂²×C₂₁₁	1	2	yes	—
153	#103	C₂²×C₂₁₁	1	4	yes	—
154	#102	C₂²×C₂₁₁	1	4	yes	—
155	#41	—	2	4	no	—
156	#41	—	2	4	no	—
157	C₂×Q₈	—	2	1	no	—
158	C₂×Q₈	—	2	2	no	—
159	C₂×Q₈	D₄×C₂₁₁	1	4	yes	complemented
160	C₂×Q₈	C₂³×C₂₁₁	1	2	yes	—
161	C₂×Q₈	C₂³×C₂₁₁	1	2	yes	—
162	C₂×D₄	—	2	2	no	—
163	C₂×D₄	—	2	1	no	—
164	C₂×D₄	—	2	1	no	—
165	C₂×D₄	C₂³×C₂₁₁	1	2	yes	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	D₄×C₂₁₁	1	1	yes	characteristic
170	C₂²×C₄	D₄×C₂₁₁	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
172	Q₁₆	—	2	8	no	—
173	SD₁₆	—	2	8	no	—
174	M₁₆	—	2	4	no	—
175	C₂×C₈	C₂³×C₂₁₁	1	4	yes	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	C₂³×C₂₁₁	1	2	yes	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	D₄×C₂₁₁	1	1	yes	characteristic
187	C₄²	D₄×C₂₁₁	1	1	yes	characteristic
188	C₄²	C₂³×C₂₁₁	1	1	yes	characteristic
189	C₄²	C₂³×C₂₁₁	1	1	yes	characteristic
190	C₂³	—	2	2	no	—
191	C₂³	C₂×D₄×C₂₁₁	1	1	yes	characteristic
192	Q₈	—	2	4	no	—
193	Q₈	—	2	2	no	—
194	Q₈	—	2	2	no	—
195	Q₈	—	2	2	no	—
196	Q₈	C₂×D₄×C₂₁₁	1	4	yes	—
197	D₄	—	2	2	no	—
198	D₄	—	2	2	no	—
199	D₄	—	4	2	no	—
200	D₄	—	4	4	no	—
201	D₄	—	4	2	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
211	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
212	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
213	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
214	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
215	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
216	C₂×C₄	#41×C₂₁₁	1	2	yes	—
217	C₂×C₄	C₂⁴×C₂₁₁	1	1	yes	characteristic
218	C₈	—	2	4	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	#121×C₂₁₁	1	2	yes	—
225	C₂²	C₂²×D₄×C₂₁₁	1	1	yes	characteristic
226	C₄	—	2	4	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	4	2	no	—
231	C₄	—	4	2	no	—
232	C₄	#121×C₂₁₁	1	2	yes	—
233	C₄	#137×C₂₁₁	1	2	yes	complemented
234	C₄	C₂²×D₄×C₂₁₁	1	1	yes	characteristic
235	C₄	C₂×#41×C₂₁₁	1	1	yes	characteristic
236	C₂	—	2	2	no	—
237	C₂	—	8	2	no	—
238	C₂	C₂×#121×C₂₁₁	1	1	yes	characteristic
239	C₂	C₂×#137×C₂₁₁	1	1	yes	characteristic
240	C₂	#575×C₂₁₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₁₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₁₁
prime radical	#532×C₂₁₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₁₁ ◁ C₂×C₄²×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#137×C₂₁₁	C₂	semidirect
3. C₂×C₄×Q₈×C₂₁₁	C₂	semidirect
4. #464×C₂₁₁	C₂	semidirect
5. #462×C₂₁₁	C₂	semidirect
6. #460×C₂₁₁	C₂	semidirect
7. #423×C₂₁₁	C₂	semidirect
8. #417×C₂₁₁	C₂	semidirect
38. C₂×Q₈×C₂₁₁	D₄	semidirect
112. C₄×C₂₁₁	#137	semidirect
120. C₂₁₁	#3037	direct (N × H)
121. #3037	C₂₁₁	direct (N × H)
122. C₂×#137	C₂×C₂₁₁	semidirect
124. C₂×C₄×Q₈	C₂×C₂₁₁	semidirect
125. #464	C₂×C₂₁₁	semidirect
126. #462	C₂×C₂₁₁	semidirect
127. #460	C₂×C₂₁₁	semidirect
128. #423	C₂×C₂₁₁	semidirect
129. #417	C₂×C₂₁₁	semidirect
159. C₂×Q₈	D₄×C₂₁₁	semidirect
233. C₄	#137×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#137×C₂₁₁	2
#532×C₂₁₁	2
C₂×C₄×Q₈×C₂₁₁	2
#464×C₂₁₁	2
#462×C₂₁₁	2
#460×C₂₁₁	2
#423×C₂₁₁	2
#417×C₂₁₁	2
#3037	211

Subgroups — #3037×C211 #41010657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3037×C₂₁₁ #41010657