Subgroups — #2728×C₂₁₁ #41010348

Order 27008.

512 conjugacy classes of subgroups.

The subgroup lattice has 240 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	2	yes	maximal, complemented
2	#400×C₂₁₁	C₂	1	2	yes	maximal, complemented
3	#398×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#397×C₂₁₁	C₂	1	2	yes	maximal, complemented
5	#394×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	#391×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#388×C₂₁₁	C₂	1	2	yes	maximal, complemented
8	#386×C₂₁₁	C₂	1	2	yes	maximal
9	#385×C₂₁₁	C₂	1	2	yes	maximal, complemented
10	C₂²×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₂₁₁	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
13	C₂²⋊Q₈×C₂₁₁	—	2	8	no	—
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×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
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×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
31	#95×C₂₁₁	C₂²	1	1	yes	characteristic
32	C₂⁴×C₂₁₁	C₂³	1	2	yes	—
33	C₂×Q₈×C₂₁₁	—	4	1	no	—
34	C₂×Q₈×C₂₁₁	—	4	1	no	—
35	C₂×Q₈×C₂₁₁	D₄	1	4	yes	complemented
36	C₂×D₄×C₂₁₁	—	2	4	no	—
37	C₂×D₄×C₂₁₁	—	4	1	no	—
38	C₂²×C₄×C₂₁₁	—	2	4	no	—
39	C₂²×C₄×C₂₁₁	—	4	2	no	—
40	C₂²×C₄×C₂₁₁	—	4	2	no	—
41	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
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	2	yes	—
47	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
48	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
49	C₄⋊C₄×C₂₁₁	—	2	4	no	—
50	C₄⋊C₄×C₂₁₁	—	2	4	no	—
51	C₄⋊C₄×C₂₁₁	—	2	4	no	—
52	C₄⋊C₄×C₂₁₁	—	4	1	no	—
53	C₄⋊C₄×C₂₁₁	—	4	1	no	—
54	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
55	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
56	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
57	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
58	C₂²⋊C₄×C₂₁₁	—	4	2	no	—
59	C₄²×C₂₁₁	—	4	1	no	—
60	C₂³×C₂₁₁	—	2	2	no	—
61	C₂³×C₂₁₁	—	2	2	no	—
62	C₂³×C₂₁₁	—	2	2	no	—
63	C₂³×C₂₁₁	—	2	2	no	—
64	C₂³×C₂₁₁	—	2	2	no	—
65	C₂³×C₂₁₁	—	2	2	no	—
66	C₂³×C₂₁₁	#41	1	4	yes	—
67	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
68	Q₈×C₂₁₁	—	4	4	no	—
69	D₄×C₂₁₁	—	4	4	no	—
70	C₂×C₄×C₂₁₁	—	2	1	no	—
71	C₂×C₄×C₂₁₁	—	2	1	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	—	2	2	no	—
82	C₂×C₄×C₂₁₁	—	2	2	no	—
83	C₂×C₄×C₂₁₁	—	2	2	no	—
84	C₂×C₄×C₂₁₁	—	2	1	no	—
85	C₂×C₄×C₂₁₁	—	2	2	no	—
86	C₂×C₄×C₂₁₁	—	2	1	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	—	4	8	no	—
91	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₂₁₁	C₂×D₄	1	4	yes	—
93	C₂²×C₂₁₁	—	2	4	no	—
94	C₂²×C₂₁₁	—	2	4	no	—
95	C₂²×C₂₁₁	—	2	4	no	—
96	C₂²×C₂₁₁	—	4	2	no	—
97	C₂²×C₂₁₁	—	4	2	no	—
98	C₂²×C₂₁₁	—	4	2	no	—
99	C₂²×C₂₁₁	—	4	2	no	—
100	C₂²×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
101	C₂²×C₂₁₁	C₂×#41	1	2	yes	—
102	C₂²×C₂₁₁	H(5,𝔽₂)	1	1	yes	characteristic
103	C₂²×C₂₁₁	H(5,𝔽₂)	1	1	yes	characteristic
104	C₂²×C₂₁₁	2₋¹⁺⁴	1	2	yes	—
105	C₄×C₂₁₁	—	2	4	no	—
106	C₄×C₂₁₁	—	2	2	no	—
107	C₄×C₂₁₁	—	4	2	no	—
108	C₄×C₂₁₁	—	4	2	no	—
109	C₄×C₂₁₁	—	4	2	no	—
110	C₄×C₂₁₁	—	4	2	no	—
111	C₄×C₂₁₁	—	4	1	no	—
112	C₄×C₂₁₁	—	4	1	no	—
113	C₂×C₂₁₁	—	4	4	no	—
114	C₂×C₂₁₁	#534	1	1	yes	characteristic
115	C₂×C₂₁₁	#539	1	2	yes	—
116	C₂×C₂₁₁	#548	1	2	yes	—
117	C₂×C₂₁₁	#551	1	1	yes	characteristic
118	C₂×C₂₁₁	#563	1	1	yes	characteristic
119	C₂₁₁	#2728	1	1	yes	characteristic, direct product factor, Sylow
120	#2728	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂×C₂²⋊Q₈	C₂×C₂₁₁	1	2	yes	complemented
122	#400	C₂×C₂₁₁	1	2	yes	complemented
123	#398	C₂×C₂₁₁	1	1	yes	characteristic, complemented
124	#397	C₂×C₂₁₁	1	2	yes	complemented
125	#394	C₂×C₂₁₁	1	1	yes	characteristic
126	#391	C₂×C₂₁₁	1	1	yes	characteristic, complemented
127	#388	C₂×C₂₁₁	1	2	yes	complemented
128	#386	C₂×C₂₁₁	1	2	yes	—
129	#385	C₂×C₂₁₁	1	2	yes	complemented
130	C₂²×Q₈	C₂²×C₂₁₁	1	1	yes	characteristic
131	C₂²×D₄	C₂²×C₂₁₁	1	1	yes	characteristic
132	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
133	C₂²⋊Q₈	—	2	8	no	—
134	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
135	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
136	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
137	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
138	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
139	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
140	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
141	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
142	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
143	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
144	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
145	#95	C₂²×C₂₁₁	1	2	yes	—
146	#95	C₂²×C₂₁₁	1	2	yes	—
147	#95	C₂²×C₂₁₁	1	2	yes	—
148	#95	C₂²×C₂₁₁	1	2	yes	—
149	#95	C₂²×C₂₁₁	1	2	yes	—
150	#95	C₂²×C₂₁₁	1	1	yes	characteristic
151	#95	C₂²×C₂₁₁	1	1	yes	characteristic
152	C₂⁴	C₂³×C₂₁₁	1	2	yes	—
153	C₂×Q₈	—	4	1	no	—
154	C₂×Q₈	—	4	1	no	—
155	C₂×Q₈	D₄×C₂₁₁	1	4	yes	complemented
156	C₂×D₄	—	2	4	no	—
157	C₂×D₄	—	4	1	no	—
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	—	4	2	no	—
161	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
162	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
163	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
164	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
165	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
166	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
167	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
168	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	4	1	no	—
173	C₄⋊C₄	—	4	1	no	—
174	C₂²⋊C₄	—	2	4	no	—
175	C₂²⋊C₄	—	2	4	no	—
176	C₂²⋊C₄	—	2	4	no	—
177	C₂²⋊C₄	—	2	4	no	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₄²	—	4	1	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	2	no	—
186	C₂³	#41×C₂₁₁	1	4	yes	—
187	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
188	Q₈	—	4	4	no	—
189	D₄	—	4	4	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	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	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	8	no	—
211	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
212	C₂×C₄	C₂×D₄×C₂₁₁	1	4	yes	—
213	C₂²	—	2	4	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	C₂²×D₄×C₂₁₁	1	1	yes	characteristic
221	C₂²	C₂×#41×C₂₁₁	1	2	yes	—
222	C₂²	H(5,𝔽₂)×C₂₁₁	1	1	yes	characteristic
223	C₂²	H(5,𝔽₂)×C₂₁₁	1	1	yes	characteristic
224	C₂²	2₋¹⁺⁴×C₂₁₁	1	2	yes	—
225	C₄	—	2	4	no	—
226	C₄	—	2	2	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	2	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	2	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	1	no	—
233	C₂	—	4	4	no	—
234	C₂	#534×C₂₁₁	1	1	yes	characteristic
235	C₂	#539×C₂₁₁	1	2	yes	—
236	C₂	#548×C₂₁₁	1	2	yes	—
237	C₂	#551×C₂₁₁	1	1	yes	characteristic
238	C₂	#563×C₂₁₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₁₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₁₁
prime radical	C₂²×D₄×C₂₁₁	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊Q₈×C₂₁₁	C₂	semidirect
2. #400×C₂₁₁	C₂	semidirect
3. #398×C₂₁₁	C₂	semidirect
4. #397×C₂₁₁	C₂	semidirect
6. #391×C₂₁₁	C₂	semidirect
7. #388×C₂₁₁	C₂	semidirect
9. #385×C₂₁₁	C₂	semidirect
35. C₂×Q₈×C₂₁₁	D₄	semidirect
119. C₂₁₁	#2728	direct (N × H)
120. #2728	C₂₁₁	direct (N × H)
121. C₂×C₂²⋊Q₈	C₂×C₂₁₁	semidirect
122. #400	C₂×C₂₁₁	semidirect
123. #398	C₂×C₂₁₁	semidirect
124. #397	C₂×C₂₁₁	semidirect
126. #391	C₂×C₂₁₁	semidirect
127. #388	C₂×C₂₁₁	semidirect
129. #385	C₂×C₂₁₁	semidirect
155. C₂×Q₈	D₄×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
#400×C₂₁₁	2
#398×C₂₁₁	2
#397×C₂₁₁	2
#394×C₂₁₁	2
#391×C₂₁₁	2
#388×C₂₁₁	2
#386×C₂₁₁	2
#385×C₂₁₁	2
#2728	211

Subgroups — #2728×C211 #41010348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2728×C₂₁₁ #41010348