Subgroups — #2368×C₂₁₁ #41009988

Order 27008.

548 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊Q₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	#391×C₂₁₁	C₂	1	2	yes	maximal, complemented
3	#387×C₂₁₁	C₂	1	2	yes	maximal, complemented
4	#387×C₂₁₁	C₂	1	2	yes	maximal, complemented
5	#385×C₂₁₁	C₂	1	2	yes	maximal
6	C₄×C₄⋊C₄×C₂₁₁	C₂	1	2	yes	maximal, complemented
7	C₄×C₂²⋊C₄×C₂₁₁	C₂	1	2	yes	maximal
8	C₄×C₂²⋊C₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic
9	C₂×#95×C₂₁₁	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
12	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₂²⋊Q₈×C₂₁₁	C₄	1	8	yes	complemented
14	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
22	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
23	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
24	#95×C₂₁₁	C₂²	1	1	yes	characteristic
25	#95×C₂₁₁	C₂²	1	1	yes	characteristic
26	#95×C₂₁₁	C₂²	1	2	yes	—
27	#95×C₂₁₁	C₂²	1	2	yes	—
28	#95×C₂₁₁	C₂²	1	2	yes	—
29	#95×C₂₁₁	C₂²	1	2	yes	—
30	#95×C₂₁₁	C₂²	1	2	yes	—
31	#95×C₂₁₁	C₂²	1	2	yes	—
32	C₂⁴×C₂₁₁	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₂₁₁	—	4	2	no	—
34	C₂×Q₈×C₂₁₁	C₂×C₄	1	4	yes	complemented
35	C₂²×C₄×C₂₁₁	—	2	4	no	—
36	C₂²×C₄×C₂₁₁	—	2	4	no	—
37	C₂²×C₄×C₂₁₁	—	4	2	no	—
38	C₂²×C₄×C₂₁₁	—	4	2	no	—
39	C₂²×C₄×C₂₁₁	C₂×C₄	1	4	yes	—
40	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
47	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
48	C₄⋊C₄×C₂₁₁	—	2	4	no	—
49	C₄⋊C₄×C₂₁₁	—	4	2	no	—
50	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	complemented
51	C₄⋊C₄×C₂₁₁	C₂×C₄	1	8	yes	complemented
52	C₂²⋊C₄×C₂₁₁	—	2	2	no	—
53	C₂²⋊C₄×C₂₁₁	C₂×C₄	1	8	yes	—
54	C₂²⋊C₄×C₂₁₁	Q₈	1	4	yes	complemented
55	C₄²×C₂₁₁	—	2	4	no	—
56	C₄²×C₂₁₁	—	4	2	no	—
57	C₄²×C₂₁₁	—	4	2	no	—
58	C₂³×C₂₁₁	—	2	2	no	—
59	C₂³×C₂₁₁	—	2	2	no	—
60	C₂³×C₂₁₁	C₂²×C₄	1	2	yes	—
61	C₂³×C₂₁₁	C₂×Q₈	1	2	yes	—
62	C₂³×C₂₁₁	#41	1	2	yes	—
63	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
64	Q₈×C₂₁₁	—	4	4	no	—
65	C₂×C₄×C₂₁₁	—	2	2	no	—
66	C₂×C₄×C₂₁₁	—	2	2	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	2	no	—
71	C₂×C₄×C₂₁₁	—	2	2	no	—
72	C₂×C₄×C₂₁₁	—	2	2	no	—
73	C₂×C₄×C₂₁₁	—	2	2	no	—
74	C₂×C₄×C₂₁₁	—	2	2	no	—
75	C₂×C₄×C₂₁₁	—	2	2	no	—
76	C₂×C₄×C₂₁₁	—	2	2	no	—
77	C₂×C₄×C₂₁₁	—	2	2	no	—
78	C₂×C₄×C₂₁₁	—	2	2	no	—
79	C₂×C₄×C₂₁₁	—	2	2	no	—
80	C₂×C₄×C₂₁₁	—	2	2	no	—
81	C₂×C₄×C₂₁₁	—	4	4	no	—
82	C₂×C₄×C₂₁₁	—	4	8	no	—
83	C₂×C₄×C₂₁₁	C₂²×C₄	1	4	yes	—
84	C₂×C₄×C₂₁₁	C₂²×C₄	1	4	yes	—
85	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₂₁₁	C₂×Q₈	1	4	yes	—
88	C₂×C₄×C₂₁₁	#41	1	4	yes	—
89	C₂²×C₂₁₁	—	2	2	no	—
90	C₂²×C₂₁₁	—	2	2	no	—
91	C₂²×C₂₁₁	—	2	4	no	—
92	C₂²×C₂₁₁	—	2	4	no	—
93	C₂²×C₂₁₁	C₄×Q₈	1	4	yes	—
94	C₂²×C₂₁₁	C₂³×C₄	1	1	yes	characteristic
95	C₂²×C₂₁₁	C₂²×Q₈	1	1	yes	characteristic
96	C₂²×C₂₁₁	C₂×#41	1	1	yes	characteristic
97	C₂²×C₂₁₁	C₂×#41	1	2	yes	—
98	C₂²×C₂₁₁	H(5,𝔽₂)	1	2	yes	—
99	C₄×C₂₁₁	—	2	4	no	—
100	C₄×C₂₁₁	—	2	2	no	—
101	C₄×C₂₁₁	—	2	2	no	—
102	C₄×C₂₁₁	—	2	4	no	—
103	C₄×C₂₁₁	—	2	4	no	—
104	C₄×C₂₁₁	—	2	4	no	—
105	C₄×C₂₁₁	—	4	2	no	—
106	C₄×C₂₁₁	—	4	2	no	—
107	C₂×C₂₁₁	—	2	4	no	—
108	C₂×C₂₁₁	C₂×C₄×Q₈	1	1	yes	characteristic
109	C₂×C₂₁₁	C₄×#41	1	1	yes	characteristic
110	C₂×C₂₁₁	#518	1	1	yes	characteristic
111	C₂×C₂₁₁	#551	1	2	yes	—
112	C₂×C₂₁₁	#554	1	2	yes	—
113	C₂₁₁	#2368	1	1	yes	characteristic, direct product factor, Sylow
114	#2368	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂×C₂²⋊Q₈	C₂×C₂₁₁	1	1	yes	characteristic
116	#391	C₂×C₂₁₁	1	2	yes	complemented
117	#387	C₂×C₂₁₁	1	2	yes	complemented
118	#387	C₂×C₂₁₁	1	2	yes	complemented
119	#385	C₂×C₂₁₁	1	2	yes	—
120	C₄×C₄⋊C₄	C₂×C₂₁₁	1	2	yes	complemented
121	C₄×C₂²⋊C₄	C₂×C₂₁₁	1	2	yes	—
122	C₄×C₂²⋊C₄	C₂×C₂₁₁	1	1	yes	characteristic
123	C₂×#95	C₂×C₂₁₁	1	1	yes	characteristic
124	C₂²×Q₈	C₂²×C₂₁₁	1	1	yes	characteristic
125	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
126	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
127	C₂²⋊Q₈	C₄×C₂₁₁	1	8	yes	complemented
128	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
129	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
130	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
131	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
132	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
133	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
134	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
135	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
136	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
137	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
138	#95	C₂²×C₂₁₁	1	1	yes	characteristic
139	#95	C₂²×C₂₁₁	1	1	yes	characteristic
140	#95	C₂²×C₂₁₁	1	2	yes	—
141	#95	C₂²×C₂₁₁	1	2	yes	—
142	#95	C₂²×C₂₁₁	1	2	yes	—
143	#95	C₂²×C₂₁₁	1	2	yes	—
144	#95	C₂²×C₂₁₁	1	2	yes	—
145	#95	C₂²×C₂₁₁	1	2	yes	—
146	C₂⁴	C₂³×C₂₁₁	1	1	yes	characteristic
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	C₂×C₄×C₂₁₁	1	4	yes	complemented
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	—	4	2	no	—
153	C₂²×C₄	C₂×C₄×C₂₁₁	1	4	yes	—
154	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
155	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
156	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
157	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
158	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
159	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
160	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
161	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
162	C₄⋊C₄	—	2	4	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	C₂×C₄×C₂₁₁	1	4	yes	complemented
165	C₄⋊C₄	C₂×C₄×C₂₁₁	1	8	yes	complemented
166	C₂²⋊C₄	—	2	2	no	—
167	C₂²⋊C₄	C₂×C₄×C₂₁₁	1	8	yes	—
168	C₂²⋊C₄	Q₈×C₂₁₁	1	4	yes	complemented
169	C₄²	—	2	4	no	—
170	C₄²	—	4	2	no	—
171	C₄²	—	4	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	—	2	2	no	—
174	C₂³	C₂²×C₄×C₂₁₁	1	2	yes	—
175	C₂³	C₂×Q₈×C₂₁₁	1	2	yes	—
176	C₂³	#41×C₂₁₁	1	2	yes	—
177	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
178	Q₈	—	4	4	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	—	4	8	no	—
197	C₂×C₄	C₂²×C₄×C₂₁₁	1	4	yes	—
198	C₂×C₄	C₂²×C₄×C₂₁₁	1	4	yes	—
199	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
200	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
201	C₂×C₄	C₂×Q₈×C₂₁₁	1	4	yes	—
202	C₂×C₄	#41×C₂₁₁	1	4	yes	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	C₄×Q₈×C₂₁₁	1	4	yes	—
208	C₂²	C₂³×C₄×C₂₁₁	1	1	yes	characteristic
209	C₂²	C₂²×Q₈×C₂₁₁	1	1	yes	characteristic
210	C₂²	C₂×#41×C₂₁₁	1	1	yes	characteristic
211	C₂²	C₂×#41×C₂₁₁	1	2	yes	—
212	C₂²	H(5,𝔽₂)×C₂₁₁	1	2	yes	—
213	C₄	—	2	4	no	—
214	C₄	—	2	4	no	—
215	C₄	—	2	4	no	—
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	4	no	—
219	C₄	—	4	2	no	—
220	C₄	—	4	2	no	—
221	C₂	—	2	4	no	—
222	C₂	C₂×C₄×Q₈×C₂₁₁	1	1	yes	characteristic
223	C₂	C₄×#41×C₂₁₁	1	1	yes	characteristic
224	C₂	#518×C₂₁₁	1	1	yes	characteristic
225	C₂	#551×C₂₁₁	1	2	yes	—
226	C₂	#554×C₂₁₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₁₁	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₁₁
Frattini, cosocle	C₂³	C₂⁴×C₂₁₁
prime radical	C₂⁴×C₂₁₁	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #391×C₂₁₁	C₂	semidirect
3. #387×C₂₁₁	C₂	semidirect
4. #387×C₂₁₁	C₂	semidirect
6. C₄×C₄⋊C₄×C₂₁₁	C₂	semidirect
13. C₂²⋊Q₈×C₂₁₁	C₄	semidirect
34. C₂×Q₈×C₂₁₁	C₂×C₄	semidirect
50. C₄⋊C₄×C₂₁₁	C₂×C₄	semidirect
51. C₄⋊C₄×C₂₁₁	C₂×C₄	semidirect
54. C₂²⋊C₄×C₂₁₁	Q₈	semidirect
113. C₂₁₁	#2368	direct (N × H)
114. #2368	C₂₁₁	direct (N × H)
116. #391	C₂×C₂₁₁	semidirect
117. #387	C₂×C₂₁₁	semidirect
118. #387	C₂×C₂₁₁	semidirect
120. C₄×C₄⋊C₄	C₂×C₂₁₁	semidirect
127. C₂²⋊Q₈	C₄×C₂₁₁	semidirect
148. C₂×Q₈	C₂×C₄×C₂₁₁	semidirect
164. C₄⋊C₄	C₂×C₄×C₂₁₁	semidirect
165. C₄⋊C₄	C₂×C₄×C₂₁₁	semidirect
168. C₂²⋊C₄	Q₈×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊Q₈×C₂₁₁	2
#391×C₂₁₁	2
#387×C₂₁₁	2
#385×C₂₁₁	2
C₄×C₄⋊C₄×C₂₁₁	2
C₄×C₂²⋊C₄×C₂₁₁	2
C₂×#95×C₂₁₁	2
#2368	211

Subgroups — #2368×C211 #41009988

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2368×C₂₁₁ #41009988