Subgroups — #2329×C₂₁₁ #41009949

Order 27008.

528 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×C₂²⋊Q₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	#391×C₂₁₁	C₂	1	2	yes	maximal, complemented
3	#389×C₂₁₁	C₂	1	2	yes	maximal, complemented
4	#387×C₂₁₁	C₂	1	2	yes	maximal, complemented
5	#387×C₂₁₁	C₂	1	2	yes	maximal, complemented
6	#385×C₂₁₁	C₂	1	2	yes	maximal
7	#380×C₂₁₁	C₂	1	1	yes	maximal, characteristic
8	C₄×C₂²⋊C₄×C₂₁₁	C₂	1	2	yes	maximal
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	1	yes	characteristic
12	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂²⋊Q₈×C₂₁₁	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊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×C₄²×C₂₁₁	C₂²	1	2	yes	—
24	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
25	#95×C₂₁₁	C₂²	1	2	yes	—
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	1	no	—
34	C₂×Q₈×C₂₁₁	—	4	1	no	—
35	C₂×Q₈×C₂₁₁	C₂×C₄	1	4	yes	complemented
36	C₂²×C₄×C₂₁₁	—	2	2	no	—
37	C₂²×C₄×C₂₁₁	—	4	1	no	—
38	C₂²×C₄×C₂₁₁	—	4	1	no	—
39	C₂²×C₄×C₂₁₁	—	4	1	no	—
40	C₂²×C₄×C₂₁₁	—	4	1	no	—
41	C₂²×C₄×C₂₁₁	—	4	1	no	—
42	C₂²×C₄×C₂₁₁	—	4	1	no	—
43	C₂²×C₄×C₂₁₁	C₂×C₄	1	4	yes	—
44	C₂²×C₄×C₂₁₁	Q₈	1	4	yes	complemented
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	2	yes	—
51	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
53	C₄⋊C₄×C₂₁₁	—	4	2	no	—
54	C₄⋊C₄×C₂₁₁	—	4	1	no	—
55	C₄⋊C₄×C₂₁₁	—	4	1	no	—
56	C₄⋊C₄×C₂₁₁	—	4	2	no	—
57	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	complemented
58	C₄⋊C₄×C₂₁₁	C₂×C₄	1	8	yes	complemented
59	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
60	C₂²⋊C₄×C₂₁₁	C₂×C₄	1	8	yes	—
61	C₄²×C₂₁₁	—	4	2	no	—
62	C₄²×C₂₁₁	—	4	2	no	—
63	C₂³×C₂₁₁	—	2	1	no	—
64	C₂³×C₂₁₁	—	2	1	no	—
65	C₂³×C₂₁₁	—	2	1	no	—
66	C₂³×C₂₁₁	—	2	1	no	—
67	C₂³×C₂₁₁	C₂²×C₄	1	2	yes	—
68	C₂³×C₂₁₁	C₂×Q₈	1	2	yes	—
69	C₂³×C₂₁₁	#41	1	2	yes	—
70	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
71	Q₈×C₂₁₁	—	4	4	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	1	no	—
77	C₂×C₄×C₂₁₁	—	2	1	no	—
78	C₂×C₄×C₂₁₁	—	2	2	no	—
79	C₂×C₄×C₂₁₁	—	2	1	no	—
80	C₂×C₄×C₂₁₁	—	2	1	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	—	4	4	no	—
92	C₂×C₄×C₂₁₁	—	4	4	no	—
93	C₂×C₄×C₂₁₁	—	4	4	no	—
94	C₂×C₄×C₂₁₁	C₂²×C₄	1	4	yes	—
95	C₂×C₄×C₂₁₁	C₂²×C₄	1	4	yes	—
96	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₂₁₁	C₂×Q₈	1	4	yes	—
99	C₂²×C₂₁₁	—	2	2	no	—
100	C₂²×C₂₁₁	—	2	2	no	—
101	C₂²×C₂₁₁	—	2	2	no	—
102	C₂²×C₂₁₁	—	2	2	no	—
103	C₂²×C₂₁₁	—	2	2	no	—
104	C₂²×C₂₁₁	—	2	2	no	—
105	C₂²×C₂₁₁	C₄×Q₈	1	4	yes	—
106	C₂²×C₂₁₁	C₂³×C₄	1	1	yes	characteristic
107	C₂²×C₂₁₁	C₂²×Q₈	1	1	yes	characteristic
108	C₂²×C₂₁₁	C₂×#41	1	1	yes	characteristic
109	C₂²×C₂₁₁	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂²×C₂₁₁	H(5,𝔽₂)	1	1	yes	characteristic
111	C₂²×C₂₁₁	H(5,𝔽₂)	1	1	yes	characteristic
112	C₂²×C₂₁₁	2₋¹⁺⁴	1	1	yes	characteristic
113	C₄×C₂₁₁	—	2	2	no	—
114	C₄×C₂₁₁	—	2	2	no	—
115	C₄×C₂₁₁	—	2	4	no	—
116	C₄×C₂₁₁	—	2	4	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	C₂×C₂₁₁	—	2	4	no	—
122	C₂×C₂₁₁	C₂×C₄×Q₈	1	1	yes	characteristic
123	C₂×C₂₁₁	#518	1	1	yes	characteristic
124	C₂×C₂₁₁	#520	1	1	yes	characteristic
125	C₂×C₂₁₁	#538	1	1	yes	characteristic
126	C₂×C₂₁₁	#539	1	1	yes	characteristic
127	C₂×C₂₁₁	#543	1	1	yes	characteristic
128	C₂×C₂₁₁	#544	1	1	yes	characteristic
129	C₂₁₁	#2329	1	1	yes	characteristic, direct product factor, Sylow
130	#2329	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂×C₂²⋊Q₈	C₂×C₂₁₁	1	1	yes	characteristic
132	#391	C₂×C₂₁₁	1	2	yes	complemented
133	#389	C₂×C₂₁₁	1	2	yes	complemented
134	#387	C₂×C₂₁₁	1	2	yes	complemented
135	#387	C₂×C₂₁₁	1	2	yes	complemented
136	#385	C₂×C₂₁₁	1	2	yes	—
137	#380	C₂×C₂₁₁	1	1	yes	characteristic
138	C₄×C₂²⋊C₄	C₂×C₂₁₁	1	2	yes	—
139	C₂×#95	C₂×C₂₁₁	1	1	yes	characteristic
140	C₂²×Q₈	C₂²×C₂₁₁	1	1	yes	characteristic
141	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
142	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
143	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
144	C₂²⋊Q₈	C₄×C₂₁₁	1	8	yes	complemented
145	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
146	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
147	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
148	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
149	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
150	C₂×C₄⋊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×C₄²	C₂²×C₂₁₁	1	2	yes	—
154	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
155	#95	C₂²×C₂₁₁	1	2	yes	—
156	#95	C₂²×C₂₁₁	1	2	yes	—
157	#95	C₂²×C₂₁₁	1	2	yes	—
158	#95	C₂²×C₂₁₁	1	2	yes	—
159	#95	C₂²×C₂₁₁	1	2	yes	—
160	#95	C₂²×C₂₁₁	1	2	yes	—
161	#95	C₂²×C₂₁₁	1	2	yes	—
162	C₂⁴	C₂³×C₂₁₁	1	1	yes	characteristic
163	C₂×Q₈	—	4	1	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂×Q₈	C₂×C₄×C₂₁₁	1	4	yes	complemented
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	4	1	no	—
168	C₂²×C₄	—	4	1	no	—
169	C₂²×C₄	—	4	1	no	—
170	C₂²×C₄	—	4	1	no	—
171	C₂²×C₄	—	4	1	no	—
172	C₂²×C₄	—	4	1	no	—
173	C₂²×C₄	C₂×C₄×C₂₁₁	1	4	yes	—
174	C₂²×C₄	Q₈×C₂₁₁	1	4	yes	complemented
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	2	yes	—
181	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
182	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄⋊C₄	—	4	1	no	—
185	C₄⋊C₄	—	4	1	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	C₂×C₄×C₂₁₁	1	4	yes	complemented
188	C₄⋊C₄	C₂×C₄×C₂₁₁	1	8	yes	complemented
189	C₂²⋊C₄	—	2	4	no	—
190	C₂²⋊C₄	C₂×C₄×C₂₁₁	1	8	yes	—
191	C₄²	—	4	2	no	—
192	C₄²	—	4	2	no	—
193	C₂³	—	2	1	no	—
194	C₂³	—	2	1	no	—
195	C₂³	—	2	1	no	—
196	C₂³	—	2	1	no	—
197	C₂³	C₂²×C₄×C₂₁₁	1	2	yes	—
198	C₂³	C₂×Q₈×C₂₁₁	1	2	yes	—
199	C₂³	#41×C₂₁₁	1	2	yes	—
200	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
201	Q₈	—	4	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	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	—	4	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	C₂²×C₄×C₂₁₁	1	4	yes	—
225	C₂×C₄	C₂²×C₄×C₂₁₁	1	4	yes	—
226	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
227	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
228	C₂×C₄	C₂×Q₈×C₂₁₁	1	4	yes	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	C₄×Q₈×C₂₁₁	1	4	yes	—
236	C₂²	C₂³×C₄×C₂₁₁	1	1	yes	characteristic
237	C₂²	C₂²×Q₈×C₂₁₁	1	1	yes	characteristic
238	C₂²	C₂×#41×C₂₁₁	1	1	yes	characteristic
239	C₂²	H(5,𝔽₂)×C₂₁₁	1	1	yes	characteristic
240	C₂²	H(5,𝔽₂)×C₂₁₁	1	1	yes	characteristic
241	C₂²	H(5,𝔽₂)×C₂₁₁	1	1	yes	characteristic
242	C₂²	2₋¹⁺⁴×C₂₁₁	1	1	yes	characteristic
243	C₄	—	2	4	no	—
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	4	no	—
247	C₄	—	2	4	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	2	no	—
251	C₂	—	2	4	no	—
252	C₂	C₂×C₄×Q₈×C₂₁₁	1	1	yes	characteristic
253	C₂	#518×C₂₁₁	1	1	yes	characteristic
254	C₂	#520×C₂₁₁	1	1	yes	characteristic
255	C₂	#538×C₂₁₁	1	1	yes	characteristic
256	C₂	#539×C₂₁₁	1	1	yes	characteristic
257	C₂	#543×C₂₁₁	1	1	yes	characteristic
258	C₂	#544×C₂₁₁	1	1	yes	characteristic

	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. #389×C₂₁₁	C₂	semidirect
4. #387×C₂₁₁	C₂	semidirect
5. #387×C₂₁₁	C₂	semidirect
14. C₂²⋊Q₈×C₂₁₁	C₄	semidirect
35. C₂×Q₈×C₂₁₁	C₂×C₄	semidirect
44. C₂²×C₄×C₂₁₁	Q₈	semidirect
57. C₄⋊C₄×C₂₁₁	C₂×C₄	semidirect
58. C₄⋊C₄×C₂₁₁	C₂×C₄	semidirect
129. C₂₁₁	#2329	direct (N × H)
130. #2329	C₂₁₁	direct (N × H)
132. #391	C₂×C₂₁₁	semidirect
133. #389	C₂×C₂₁₁	semidirect
134. #387	C₂×C₂₁₁	semidirect
135. #387	C₂×C₂₁₁	semidirect
144. C₂²⋊Q₈	C₄×C₂₁₁	semidirect
165. C₂×Q₈	C₂×C₄×C₂₁₁	semidirect
174. C₂²×C₄	Q₈×C₂₁₁	semidirect
187. C₄⋊C₄	C₂×C₄×C₂₁₁	semidirect
188. C₄⋊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×C₂²⋊Q₈×C₂₁₁	2
#391×C₂₁₁	2
#389×C₂₁₁	2
#387×C₂₁₁	2
#385×C₂₁₁	2
#380×C₂₁₁	2
C₄×C₂²⋊C₄×C₂₁₁	2
C₂×#95×C₂₁₁	2
#2329	211

Subgroups — #2329×C211 #41009949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2329×C₂₁₁ #41009949