Subgroups — #2377×C₂₁₁ #41009997

Order 27008.

576 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#121×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	#390×C₂₁₁	C₂	1	2	yes	maximal, complemented
3	#388×C₂₁₁	C₂	1	2	yes	maximal, complemented
4	#387×C₂₁₁	C₂	1	2	yes	maximal, complemented
5	#386×C₂₁₁	C₂	1	2	yes	maximal, complemented
6	#386×C₂₁₁	C₂	1	2	yes	maximal, complemented
7	#385×C₂₁₁	C₂	1	2	yes	maximal, complemented
8	#381×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#380×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₂₁₁	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₂₁₁	C₂²	1	2	yes	—
12	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
13	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
14	#121×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×C₂²⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
25	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
26	#95×C₂₁₁	C₂²	1	2	yes	—
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	C₂⁴×C₂₁₁	C₂³	1	1	yes	characteristic
33	C₂⁴×C₂₁₁	C₂³	1	2	yes	—
34	C₂×D₄×C₂₁₁	—	4	2	no	—
35	C₂×D₄×C₂₁₁	—	4	2	no	—
36	C₂×D₄×C₂₁₁	—	4	2	no	—
37	C₂×D₄×C₂₁₁	C₂×C₄	1	4	yes	—
38	C₂×D₄×C₂₁₁	C₂×C₄	1	8	yes	complemented
39	C₂²×C₄×C₂₁₁	—	4	2	no	—
40	C₂²×C₄×C₂₁₁	—	4	2	no	—
41	C₂²×C₄×C₂₁₁	—	4	2	no	—
42	C₂²×C₄×C₂₁₁	—	4	2	no	—
43	C₂²×C₄×C₂₁₁	C₂×C₄	1	4	yes	complemented
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	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
50	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₂₁₁	—	4	2	no	—
52	C₄⋊C₄×C₂₁₁	—	4	2	no	—
53	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	complemented
54	C₂²⋊C₄×C₂₁₁	—	4	2	no	—
55	C₂²⋊C₄×C₂₁₁	—	4	1	no	—
56	C₂²⋊C₄×C₂₁₁	—	4	2	no	—
57	C₂²⋊C₄×C₂₁₁	—	4	2	no	—
58	C₂²⋊C₄×C₂₁₁	—	4	2	no	—
59	C₂²⋊C₄×C₂₁₁	—	4	1	no	—
60	C₂²⋊C₄×C₂₁₁	C₂×C₄	1	8	yes	complemented
61	C₄²×C₂₁₁	—	4	2	no	—
62	C₂³×C₂₁₁	—	2	1	no	—
63	C₂³×C₂₁₁	—	2	2	no	—
64	C₂³×C₂₁₁	—	2	2	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	—	2	2	no	—
70	C₂³×C₂₁₁	—	2	1	no	—
71	C₂³×C₂₁₁	—	2	2	no	—
72	C₂³×C₂₁₁	C₂²×C₄	1	4	yes	—
73	C₂³×C₂₁₁	C₂²×C₄	1	2	yes	—
74	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
75	D₄×C₂₁₁	—	4	4	no	—
76	D₄×C₂₁₁	—	4	8	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	—	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	—	2	2	no	—
90	C₂×C₄×C₂₁₁	—	2	2	no	—
91	C₂×C₄×C₂₁₁	—	2	2	no	—
92	C₂×C₄×C₂₁₁	—	2	2	no	—
93	C₂×C₄×C₂₁₁	—	4	4	no	—
94	C₂×C₄×C₂₁₁	—	4	8	no	—
95	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₂₁₁	C₂²×C₄	1	4	yes	—
97	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
98	C₂²×C₂₁₁	—	2	2	no	—
99	C₂²×C₂₁₁	—	2	2	no	—
100	C₂²×C₂₁₁	—	2	4	no	—
101	C₂²×C₂₁₁	—	2	4	no	—
102	C₂²×C₂₁₁	—	2	2	no	—
103	C₂²×C₂₁₁	—	2	4	no	—
104	C₂²×C₂₁₁	—	4	2	no	—
105	C₂²×C₂₁₁	—	4	2	no	—
106	C₂²×C₂₁₁	—	4	2	no	—
107	C₂²×C₂₁₁	—	4	2	no	—
108	C₂²×C₂₁₁	—	4	2	no	—
109	C₂²×C₂₁₁	—	4	2	no	—
110	C₂²×C₂₁₁	C₂³×C₄	1	1	yes	characteristic
111	C₂²×C₂₁₁	H(5,𝔽₂)	1	2	yes	—
112	C₂²×C₂₁₁	H(5,𝔽₂)	1	2	yes	—
113	C₂²×C₂₁₁	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂²×C₂₁₁	2₋¹⁺⁴	1	1	yes	characteristic
115	C₄×C₂₁₁	—	2	2	no	—
116	C₄×C₂₁₁	—	2	2	no	—
117	C₄×C₂₁₁	—	2	4	no	—
118	C₄×C₂₁₁	—	4	2	no	—
119	C₄×C₂₁₁	—	4	2	no	—
120	C₄×C₂₁₁	—	4	2	no	—
121	C₄×C₂₁₁	—	4	2	no	—
122	C₂×C₂₁₁	—	4	2	no	—
123	C₂×C₂₁₁	—	4	4	no	—
124	C₂×C₂₁₁	#518	1	1	yes	characteristic
125	C₂×C₂₁₁	#518	1	1	yes	characteristic
126	C₂×C₂₁₁	#520	1	1	yes	characteristic
127	C₂×C₂₁₁	#560	1	2	yes	—
128	C₂×C₂₁₁	#562	1	2	yes	—
129	C₂₁₁	#2377	1	1	yes	characteristic, direct product factor, Sylow
130	#2377	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂×#121	C₂×C₂₁₁	1	1	yes	characteristic
132	#390	C₂×C₂₁₁	1	2	yes	complemented
133	#388	C₂×C₂₁₁	1	2	yes	complemented
134	#387	C₂×C₂₁₁	1	2	yes	complemented
135	#386	C₂×C₂₁₁	1	2	yes	complemented
136	#386	C₂×C₂₁₁	1	2	yes	complemented
137	#385	C₂×C₂₁₁	1	2	yes	complemented
138	#381	C₂×C₂₁₁	1	1	yes	characteristic, complemented
139	#380	C₂×C₂₁₁	1	1	yes	characteristic, complemented
140	C₂²×D₄	C₂²×C₂₁₁	1	1	yes	characteristic
141	C₂²×D₄	C₂²×C₂₁₁	1	2	yes	—
142	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
143	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
144	#121	C₄×C₂₁₁	1	8	yes	complemented
145	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
146	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
147	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
148	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
149	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
150	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
151	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
152	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
153	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
154	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
155	C₂×C₄²	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	1	yes	characteristic
160	#95	C₂²×C₂₁₁	1	2	yes	—
161	#95	C₂²×C₂₁₁	1	2	yes	—
162	C₂⁴	C₂³×C₂₁₁	1	1	yes	characteristic
163	C₂⁴	C₂³×C₂₁₁	1	2	yes	—
164	C₂×D₄	—	4	2	no	—
165	C₂×D₄	—	4	2	no	—
166	C₂×D₄	—	4	2	no	—
167	C₂×D₄	C₂×C₄×C₂₁₁	1	4	yes	—
168	C₂×D₄	C₂×C₄×C₂₁₁	1	8	yes	complemented
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	C₂×C₄×C₂₁₁	1	4	yes	complemented
174	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
175	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
176	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
177	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
178	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
179	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
180	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	C₂×C₄×C₂₁₁	1	4	yes	complemented
184	C₂²⋊C₄	—	4	2	no	—
185	C₂²⋊C₄	—	4	1	no	—
186	C₂²⋊C₄	—	4	2	no	—
187	C₂²⋊C₄	—	4	2	no	—
188	C₂²⋊C₄	—	4	2	no	—
189	C₂²⋊C₄	—	4	1	no	—
190	C₂²⋊C₄	C₂×C₄×C₂₁₁	1	8	yes	complemented
191	C₄²	—	4	2	no	—
192	C₂³	—	2	1	no	—
193	C₂³	—	2	2	no	—
194	C₂³	—	2	2	no	—
195	C₂³	—	2	2	no	—
196	C₂³	—	2	2	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	1	no	—
201	C₂³	—	2	2	no	—
202	C₂³	C₂²×C₄×C₂₁₁	1	4	yes	—
203	C₂³	C₂²×C₄×C₂₁₁	1	2	yes	—
204	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
205	D₄	—	4	4	no	—
206	D₄	—	4	8	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	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	8	no	—
225	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
226	C₂×C₄	C₂²×C₄×C₂₁₁	1	4	yes	—
227	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	4	no	—
231	C₂²	—	2	4	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	4	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	C₂³×C₄×C₂₁₁	1	1	yes	characteristic
241	C₂²	H(5,𝔽₂)×C₂₁₁	1	1	yes	characteristic
242	C₂²	H(5,𝔽₂)×C₂₁₁	1	2	yes	—
243	C₂²	H(5,𝔽₂)×C₂₁₁	1	2	yes	—
244	C₂²	2₋¹⁺⁴×C₂₁₁	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	4	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	2	no	—
251	C₄	—	4	2	no	—
252	C₂	—	4	2	no	—
253	C₂	—	4	4	no	—
254	C₂	#518×C₂₁₁	1	1	yes	characteristic
255	C₂	#518×C₂₁₁	1	1	yes	characteristic
256	C₂	#520×C₂₁₁	1	1	yes	characteristic
257	C₂	#560×C₂₁₁	1	2	yes	—
258	C₂	#562×C₂₁₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₁₁	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₁₁
Frattini, cosocle	C₂³	C₂⁴×C₂₁₁
prime radical	C₂×#121×C₂₁₁	C₂
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. #390×C₂₁₁	C₂	semidirect
3. #388×C₂₁₁	C₂	semidirect
4. #387×C₂₁₁	C₂	semidirect
5. #386×C₂₁₁	C₂	semidirect
6. #386×C₂₁₁	C₂	semidirect
7. #385×C₂₁₁	C₂	semidirect
8. #381×C₂₁₁	C₂	semidirect
9. #380×C₂₁₁	C₂	semidirect
14. #121×C₂₁₁	C₄	semidirect
38. C₂×D₄×C₂₁₁	C₂×C₄	semidirect
43. C₂²×C₄×C₂₁₁	C₂×C₄	semidirect
53. C₄⋊C₄×C₂₁₁	C₂×C₄	semidirect
60. C₂²⋊C₄×C₂₁₁	C₂×C₄	semidirect
129. C₂₁₁	#2377	direct (N × H)
130. #2377	C₂₁₁	direct (N × H)
132. #390	C₂×C₂₁₁	semidirect
133. #388	C₂×C₂₁₁	semidirect
134. #387	C₂×C₂₁₁	semidirect
135. #386	C₂×C₂₁₁	semidirect
136. #386	C₂×C₂₁₁	semidirect
137. #385	C₂×C₂₁₁	semidirect
138. #381	C₂×C₂₁₁	semidirect
139. #380	C₂×C₂₁₁	semidirect
144. #121	C₄×C₂₁₁	semidirect
168. C₂×D₄	C₂×C₄×C₂₁₁	semidirect
173. C₂²×C₄	C₂×C₄×C₂₁₁	semidirect
183. C₄⋊C₄	C₂×C₄×C₂₁₁	semidirect
190. C₂²⋊C₄	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×#121×C₂₁₁	2
#390×C₂₁₁	2
#388×C₂₁₁	2
#387×C₂₁₁	2
#386×C₂₁₁	2
#385×C₂₁₁	2
#381×C₂₁₁	2
#380×C₂₁₁	2
#2377	211

Subgroups — #2377×C211 #41009997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2377×C₂₁₁ #41009997