Subgroups — #2343×C₂₁₁ #41009963

Order 27008.

532 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	C₂×#117×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	#387×C₂₁₁	C₂	1	2	yes	maximal, complemented
3	#387×C₂₁₁	C₂	1	2	yes	maximal, complemented
4	#385×C₂₁₁	C₂	1	2	yes	maximal
5	#383×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#382×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#381×C₂₁₁	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₂²⋊C₄×C₂₁₁	C₂	1	2	yes	maximal
10	#376×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#95×C₂₁₁	C₂	1	1	yes	maximal, characteristic
12	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
13	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
14	#117×C₂₁₁	C₄	1	8	yes	complemented
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×C₄⋊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×C₂²⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
23	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
24	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
26	#95×C₂₁₁	C₂²	1	1	yes	characteristic
27	#95×C₂₁₁	C₂²	1	1	yes	characteristic
28	#95×C₂₁₁	C₂²	1	2	yes	—
29	#95×C₂₁₁	C₂²	1	1	yes	characteristic
30	#95×C₂₁₁	C₂²	1	2	yes	—
31	#95×C₂₁₁	C₂²	1	2	yes	—
32	#95×C₂₁₁	C₂²	1	2	yes	—
33	#95×C₂₁₁	C₂²	1	1	yes	characteristic
34	#95×C₂₁₁	C₂²	1	2	yes	—
35	#95×C₂₁₁	C₂²	1	2	yes	—
36	C₂⁴×C₂₁₁	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₂₁₁	—	2	4	no	—
38	C₂²×C₄×C₂₁₁	—	2	4	no	—
39	C₂²×C₄×C₂₁₁	—	2	4	no	—
40	C₂²×C₄×C₂₁₁	—	4	2	no	—
41	C₂²×C₄×C₂₁₁	C₂×C₄	1	4	yes	—
42	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
44	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
52	C₄⋊C₄×C₂₁₁	—	2	4	no	—
53	C₄⋊C₄×C₂₁₁	—	4	2	no	—
54	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	complemented
55	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	complemented
56	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
57	C₂²⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	—
58	C₂²⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	—
59	C₄²×C₂₁₁	—	2	4	no	—
60	C₄²×C₂₁₁	—	4	2	no	—
61	C₄²×C₂₁₁	C₂×C₄	1	4	yes	complemented
62	C₄²×C₂₁₁	C₂×C₄	1	4	yes	complemented
63	C₂³×C₂₁₁	—	2	2	no	—
64	C₂³×C₂₁₁	—	2	1	no	—
65	C₂³×C₂₁₁	—	2	1	no	—
66	C₂³×C₂₁₁	C₂²×C₄	1	2	yes	—
67	C₂³×C₂₁₁	#41	1	4	yes	—
68	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
69	C₂×C₄×C₂₁₁	—	2	8	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	1	no	—
76	C₂×C₄×C₂₁₁	—	2	2	no	—
77	C₂×C₄×C₂₁₁	—	2	1	no	—
78	C₂×C₄×C₂₁₁	—	2	1	no	—
79	C₂×C₄×C₂₁₁	—	2	1	no	—
80	C₂×C₄×C₂₁₁	—	2	2	no	—
81	C₂×C₄×C₂₁₁	—	2	2	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	—	2	2	no	—
87	C₂×C₄×C₂₁₁	—	2	2	no	—
88	C₂×C₄×C₂₁₁	—	2	2	no	—
89	C₂×C₄×C₂₁₁	—	4	8	no	—
90	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₂₁₁	#41	1	4	yes	—
97	C₂²×C₂₁₁	—	2	2	no	—
98	C₂²×C₂₁₁	—	2	2	no	—
99	C₂²×C₂₁₁	—	2	4	no	—
100	C₂²×C₂₁₁	—	2	4	no	—
101	C₂²×C₂₁₁	#117	1	4	yes	—
102	C₂²×C₂₁₁	C₂³×C₄	1	1	yes	characteristic
103	C₂²×C₂₁₁	C₂×#41	1	2	yes	—
104	C₂²×C₂₁₁	C₂×#41	1	2	yes	—
105	C₂²×C₂₁₁	H(5,𝔽₂)	1	1	yes	characteristic
106	C₂²×C₂₁₁	2₋¹⁺⁴	1	1	yes	characteristic
107	C₄×C₂₁₁	—	2	2	no	—
108	C₄×C₂₁₁	—	2	2	no	—
109	C₄×C₂₁₁	—	2	2	no	—
110	C₄×C₂₁₁	—	2	2	no	—
111	C₄×C₂₁₁	—	2	2	no	—
112	C₄×C₂₁₁	—	2	2	no	—
113	C₄×C₂₁₁	—	2	4	no	—
114	C₄×C₂₁₁	—	4	2	no	—
115	C₄×C₂₁₁	—	4	2	no	—
116	C₄×C₂₁₁	—	4	2	no	—
117	C₂×C₂₁₁	—	2	4	no	—
118	C₂×C₂₁₁	C₂×#117	1	1	yes	characteristic
119	C₂×C₂₁₁	C₄×#41	1	1	yes	characteristic
120	C₂×C₂₁₁	#520	1	1	yes	characteristic
121	C₂×C₂₁₁	#551	1	2	yes	—
122	C₂×C₂₁₁	#552	1	2	yes	—
123	C₂₁₁	#2343	1	1	yes	characteristic, direct product factor, Sylow
124	#2343	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂×#117	C₂×C₂₁₁	1	1	yes	characteristic
126	#387	C₂×C₂₁₁	1	2	yes	complemented
127	#387	C₂×C₂₁₁	1	2	yes	complemented
128	#385	C₂×C₂₁₁	1	2	yes	—
129	#383	C₂×C₂₁₁	1	1	yes	characteristic, complemented
130	#382	C₂×C₂₁₁	1	1	yes	characteristic, complemented
131	#381	C₂×C₂₁₁	1	1	yes	characteristic
132	C₄×C₄⋊C₄	C₂×C₂₁₁	1	1	yes	characteristic, complemented
133	C₄×C₂²⋊C₄	C₂×C₂₁₁	1	2	yes	—
134	#376	C₂×C₂₁₁	1	1	yes	characteristic, complemented
135	C₂×#95	C₂×C₂₁₁	1	1	yes	characteristic
136	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
137	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
138	#117	C₄×C₂₁₁	1	8	yes	complemented
139	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
140	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
141	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
142	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
143	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
144	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
145	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
146	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
147	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
148	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
149	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
150	#95	C₂²×C₂₁₁	1	1	yes	characteristic
151	#95	C₂²×C₂₁₁	1	1	yes	characteristic
152	#95	C₂²×C₂₁₁	1	2	yes	—
153	#95	C₂²×C₂₁₁	1	1	yes	characteristic
154	#95	C₂²×C₂₁₁	1	2	yes	—
155	#95	C₂²×C₂₁₁	1	2	yes	—
156	#95	C₂²×C₂₁₁	1	2	yes	—
157	#95	C₂²×C₂₁₁	1	1	yes	characteristic
158	#95	C₂²×C₂₁₁	1	2	yes	—
159	#95	C₂²×C₂₁₁	1	2	yes	—
160	C₂⁴	C₂³×C₂₁₁	1	1	yes	characteristic
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	C₂×C₄×C₂₁₁	1	4	yes	—
166	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
167	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
168	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
169	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
172	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
173	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
174	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
175	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	C₂×C₄×C₂₁₁	1	4	yes	complemented
179	C₄⋊C₄	C₂×C₄×C₂₁₁	1	4	yes	complemented
180	C₂²⋊C₄	—	2	4	no	—
181	C₂²⋊C₄	C₂×C₄×C₂₁₁	1	4	yes	—
182	C₂²⋊C₄	C₂×C₄×C₂₁₁	1	4	yes	—
183	C₄²	—	2	4	no	—
184	C₄²	—	4	2	no	—
185	C₄²	C₂×C₄×C₂₁₁	1	4	yes	complemented
186	C₄²	C₂×C₄×C₂₁₁	1	4	yes	complemented
187	C₂³	—	2	2	no	—
188	C₂³	—	2	1	no	—
189	C₂³	—	2	1	no	—
190	C₂³	C₂²×C₄×C₂₁₁	1	2	yes	—
191	C₂³	#41×C₂₁₁	1	4	yes	—
192	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
193	C₂×C₄	—	2	8	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	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	—	2	2	no	—
213	C₂×C₄	—	4	8	no	—
214	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
215	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
216	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
217	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
218	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
219	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
220	C₂×C₄	#41×C₂₁₁	1	4	yes	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	4	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	#117×C₂₁₁	1	4	yes	—
226	C₂²	C₂³×C₄×C₂₁₁	1	1	yes	characteristic
227	C₂²	C₂×#41×C₂₁₁	1	2	yes	—
228	C₂²	C₂×#41×C₂₁₁	1	2	yes	—
229	C₂²	H(5,𝔽₂)×C₂₁₁	1	1	yes	characteristic
230	C₂²	2₋¹⁺⁴×C₂₁₁	1	1	yes	characteristic
231	C₄	—	2	4	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	2	no	—
237	C₄	—	2	2	no	—
238	C₄	—	4	2	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	2	no	—
241	C₂	—	2	4	no	—
242	C₂	C₂×#117×C₂₁₁	1	1	yes	characteristic
243	C₂	C₄×#41×C₂₁₁	1	1	yes	characteristic
244	C₂	#520×C₂₁₁	1	1	yes	characteristic
245	C₂	#551×C₂₁₁	1	2	yes	—
246	C₂	#552×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. #387×C₂₁₁	C₂	semidirect
3. #387×C₂₁₁	C₂	semidirect
5. #383×C₂₁₁	C₂	semidirect
6. #382×C₂₁₁	C₂	semidirect
8. C₄×C₄⋊C₄×C₂₁₁	C₂	semidirect
10. #376×C₂₁₁	C₂	semidirect
14. #117×C₂₁₁	C₄	semidirect
54. C₄⋊C₄×C₂₁₁	C₂×C₄	semidirect
55. C₄⋊C₄×C₂₁₁	C₂×C₄	semidirect
61. C₄²×C₂₁₁	C₂×C₄	semidirect
62. C₄²×C₂₁₁	C₂×C₄	semidirect
123. C₂₁₁	#2343	direct (N × H)
124. #2343	C₂₁₁	direct (N × H)
126. #387	C₂×C₂₁₁	semidirect
127. #387	C₂×C₂₁₁	semidirect
129. #383	C₂×C₂₁₁	semidirect
130. #382	C₂×C₂₁₁	semidirect
132. C₄×C₄⋊C₄	C₂×C₂₁₁	semidirect
134. #376	C₂×C₂₁₁	semidirect
138. #117	C₄×C₂₁₁	semidirect
178. C₄⋊C₄	C₂×C₄×C₂₁₁	semidirect
179. C₄⋊C₄	C₂×C₄×C₂₁₁	semidirect
185. C₄²	C₂×C₄×C₂₁₁	semidirect
186. C₄²	C₂×C₄×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#117×C₂₁₁	2
#387×C₂₁₁	2
#385×C₂₁₁	2
#383×C₂₁₁	2
#382×C₂₁₁	2
#381×C₂₁₁	2
C₄×C₄⋊C₄×C₂₁₁	2
C₄×C₂²⋊C₄×C₂₁₁	2
#376×C₂₁₁	2
C₂×#95×C₂₁₁	2
#2343	211

Subgroups — #2343×C211 #41009963

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2343×C₂₁₁ #41009963