Subgroups — #2547×C₂₁₁ #41010167

Order 27008.

604 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#124×C₂₁₁	C₂	1	2	yes	maximal
3	C₂×C₄×D₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	#397×C₂₁₁	C₂	1	4	yes	maximal, complemented
5	#391×C₂₁₁	C₂	1	2	yes	maximal, complemented
6	#380×C₂₁₁	C₂	1	4	yes	maximal, complemented
7	C₄×C₄⋊C₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₈×C₂₁₁	C₂²	1	2	yes	—
9	C₂²×D₄×C₂₁₁	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₂₁₁	—	2	4	no	—
12	#124×C₂₁₁	—	2	8	no	—
13	C₄×D₄×C₂₁₁	—	2	4	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×C₂²⋊C₄×C₂₁₁	C₂²	1	8	yes	—
19	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
20	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
22	#95×C₂₁₁	C₂²	1	2	yes	—
23	#95×C₂₁₁	C₂²	1	8	yes	—
24	C₂⁴×C₂₁₁	C₂³	1	2	yes	—
25	C₂×Q₈×C₂₁₁	—	2	4	no	—
26	C₂×Q₈×C₂₁₁	—	2	4	no	—
27	C₂×Q₈×C₂₁₁	—	2	4	no	—
28	C₂×D₄×C₂₁₁	—	2	2	no	—
29	C₂×D₄×C₂₁₁	—	2	2	no	—
30	C₂×D₄×C₂₁₁	D₄	1	4	yes	—
31	C₂²×C₄×C₂₁₁	—	2	4	no	—
32	C₂²×C₄×C₂₁₁	—	2	8	no	—
33	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
34	C₂²×C₄×C₂₁₁	C₂³	1	4	yes	—
35	C₂²×C₄×C₂₁₁	C₂³	1	4	yes	—
36	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₂₁₁	—	2	2	no	—
40	C₄⋊C₄×C₂₁₁	—	2	4	no	—
41	C₄⋊C₄×C₂₁₁	—	2	4	no	—
42	C₄⋊C₄×C₂₁₁	—	2	4	no	—
43	C₂²⋊C₄×C₂₁₁	—	2	16	no	—
44	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
45	C₄²×C₂₁₁	—	2	4	no	—
46	C₄²×C₂₁₁	D₄	1	4	yes	—
47	C₂³×C₂₁₁	—	2	2	no	—
48	C₂³×C₂₁₁	—	2	4	no	—
49	C₂³×C₂₁₁	—	2	2	no	—
50	C₂³×C₂₁₁	—	2	4	no	—
51	C₂³×C₂₁₁	C₂×D₄	1	4	yes	—
52	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
53	Q₈×C₂₁₁	—	4	8	no	—
54	D₄×C₂₁₁	—	2	8	no	—
55	C₂×C₄×C₂₁₁	—	2	2	no	—
56	C₂×C₄×C₂₁₁	—	2	4	no	—
57	C₂×C₄×C₂₁₁	—	2	4	no	—
58	C₂×C₄×C₂₁₁	—	2	2	no	—
59	C₂×C₄×C₂₁₁	—	2	4	no	—
60	C₂×C₄×C₂₁₁	—	2	4	no	—
61	C₂×C₄×C₂₁₁	—	2	4	no	—
62	C₂×C₄×C₂₁₁	—	2	4	no	—
63	C₂×C₄×C₂₁₁	—	2	4	no	—
64	C₂×C₄×C₂₁₁	—	4	8	no	—
65	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₂₁₁	#41	1	2	yes	—
70	C₂×C₄×C₂₁₁	#41	1	4	yes	—
71	C₂²×C₂₁₁	—	2	4	no	—
72	C₂²×C₂₁₁	—	2	8	no	—
73	C₂²×C₂₁₁	—	4	2	no	—
74	C₂²×C₂₁₁	—	4	2	no	—
75	C₂²×C₂₁₁	—	4	4	no	—
76	C₂²×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
77	C₂²×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
78	C₂²×C₂₁₁	C₂×#41	1	1	yes	characteristic
79	C₂²×C₂₁₁	C₂×#41	1	2	yes	—
80	C₂²×C₂₁₁	2₋¹⁺⁴	1	2	yes	—
81	C₄×C₂₁₁	—	2	2	no	—
82	C₄×C₂₁₁	—	4	4	no	—
83	C₄×C₂₁₁	—	4	2	no	—
84	C₄×C₂₁₁	—	4	4	no	—
85	C₄×C₂₁₁	#121	1	4	yes	—
86	C₄×C₂₁₁	#124	1	4	yes	complemented
87	C₂×C₂₁₁	—	4	4	no	—
88	C₂×C₂₁₁	C₂×#121	1	1	yes	characteristic
89	C₂×C₂₁₁	C₂×#124	1	1	yes	characteristic
90	C₂×C₂₁₁	#536	1	1	yes	characteristic
91	C₂×C₂₁₁	#547	1	2	yes	—
92	C₂×C₂₁₁	#556	1	2	yes	—
93	C₂₁₁	#2547	1	1	yes	characteristic, direct product factor, Sylow
94	#2547	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂×C₄⋊Q₈	C₂×C₂₁₁	1	1	yes	characteristic, complemented
96	C₂×#124	C₂×C₂₁₁	1	2	yes	—
97	C₂×C₄×D₄	C₂×C₂₁₁	1	1	yes	characteristic
98	#397	C₂×C₂₁₁	1	4	yes	complemented
99	#391	C₂×C₂₁₁	1	2	yes	complemented
100	#380	C₂×C₂₁₁	1	4	yes	complemented
101	C₄×C₄⋊C₄	C₂×C₂₁₁	1	1	yes	characteristic, complemented
102	C₂²×Q₈	C₂²×C₂₁₁	1	2	yes	—
103	C₂²×D₄	C₂²×C₂₁₁	1	1	yes	characteristic
104	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
105	C₄⋊Q₈	—	2	4	no	—
106	#124	—	2	8	no	—
107	C₄×D₄	—	2	4	no	—
108	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
109	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
110	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
111	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
112	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
113	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	8	yes	—
114	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
115	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
116	#95	C₂²×C₂₁₁	1	2	yes	—
117	#95	C₂²×C₂₁₁	1	8	yes	—
118	C₂⁴	C₂³×C₂₁₁	1	2	yes	—
119	C₂×Q₈	—	2	4	no	—
120	C₂×Q₈	—	2	4	no	—
121	C₂×Q₈	—	2	4	no	—
122	C₂×D₄	—	2	2	no	—
123	C₂×D₄	—	2	2	no	—
124	C₂×D₄	D₄×C₂₁₁	1	4	yes	—
125	C₂²×C₄	—	2	4	no	—
126	C₂²×C₄	—	2	8	no	—
127	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
128	C₂²×C₄	C₂³×C₂₁₁	1	4	yes	—
129	C₂²×C₄	C₂³×C₂₁₁	1	4	yes	—
130	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
131	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
132	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	—	2	4	no	—
135	C₄⋊C₄	—	2	4	no	—
136	C₄⋊C₄	—	2	4	no	—
137	C₂²⋊C₄	—	2	16	no	—
138	C₂²⋊C₄	—	2	4	no	—
139	C₄²	—	2	4	no	—
140	C₄²	D₄×C₂₁₁	1	4	yes	—
141	C₂³	—	2	2	no	—
142	C₂³	—	2	4	no	—
143	C₂³	—	2	2	no	—
144	C₂³	—	2	4	no	—
145	C₂³	C₂×D₄×C₂₁₁	1	4	yes	—
146	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
147	Q₈	—	4	8	no	—
148	D₄	—	2	8	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	4	8	no	—
159	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
160	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
161	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
162	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
163	C₂×C₄	#41×C₂₁₁	1	2	yes	—
164	C₂×C₄	#41×C₂₁₁	1	4	yes	—
165	C₂²	—	2	4	no	—
166	C₂²	—	2	8	no	—
167	C₂²	—	4	2	no	—
168	C₂²	—	4	2	no	—
169	C₂²	—	4	4	no	—
170	C₂²	C₂²×D₄×C₂₁₁	1	1	yes	characteristic
171	C₂²	C₂²×D₄×C₂₁₁	1	1	yes	characteristic
172	C₂²	C₂×#41×C₂₁₁	1	1	yes	characteristic
173	C₂²	C₂×#41×C₂₁₁	1	2	yes	—
174	C₂²	2₋¹⁺⁴×C₂₁₁	1	2	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	4	4	no	—
177	C₄	—	4	2	no	—
178	C₄	—	4	4	no	—
179	C₄	#121×C₂₁₁	1	4	yes	—
180	C₄	#124×C₂₁₁	1	4	yes	complemented
181	C₂	—	4	4	no	—
182	C₂	C₂×#121×C₂₁₁	1	1	yes	characteristic
183	C₂	C₂×#124×C₂₁₁	1	1	yes	characteristic
184	C₂	#536×C₂₁₁	1	1	yes	characteristic
185	C₂	#547×C₂₁₁	1	2	yes	—
186	C₂	#556×C₂₁₁	1	2	yes	—

	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×C₄⋊Q₈×C₂₁₁	C₂	semidirect
4. #397×C₂₁₁	C₂	semidirect
5. #391×C₂₁₁	C₂	semidirect
6. #380×C₂₁₁	C₂	semidirect
7. C₄×C₄⋊C₄×C₂₁₁	C₂	semidirect
86. C₄×C₂₁₁	#124	semidirect
93. C₂₁₁	#2547	direct (N × H)
94. #2547	C₂₁₁	direct (N × H)
95. C₂×C₄⋊Q₈	C₂×C₂₁₁	semidirect
98. #397	C₂×C₂₁₁	semidirect
99. #391	C₂×C₂₁₁	semidirect
100. #380	C₂×C₂₁₁	semidirect
101. C₄×C₄⋊C₄	C₂×C₂₁₁	semidirect
180. C₄	#124×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Q₈×C₂₁₁	2
C₂×#124×C₂₁₁	2
C₂×C₄×D₄×C₂₁₁	2
#397×C₂₁₁	2
#391×C₂₁₁	2
#380×C₂₁₁	2
C₄×C₄⋊C₄×C₂₁₁	2
#2547	211

Subgroups — #2547×C211 #41010167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2547×C₂₁₁ #41010167