Subgroups — #3502×C₂₁₁ #41011122

Order 27008.

964 conjugacy classes of subgroups.

The subgroup lattice has 142 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈²×C₂₁₁	C₂	1	4	yes	maximal, complemented
2	#557×C₂₁₁	C₂	1	8	yes	maximal, complemented
3	#557×C₂₁₁	C₂	1	4	yes	maximal, complemented
4	#544×C₂₁₁	C₂	1	6	yes	maximal
5	#533×C₂₁₁	C₂	1	6	yes	maximal
6	#519×C₂₁₁	C₂	1	2	yes	maximal
7	C₂×C₄×Q₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
9	C₄⋊Q₈×C₂₁₁	C₂²	1	6	yes	—
10	C₄⋊Q₈×C₂₁₁	C₂²	1	6	yes	—
11	C₄⋊Q₈×C₂₁₁	C₂²	1	24	yes	—
12	#125×C₂₁₁	C₂²	1	12	yes	—
13	#125×C₂₁₁	C₂²	1	24	yes	—
14	C₂²⋊Q₈×C₂₁₁	C₂²	1	4	yes	—
15	C₂²⋊Q₈×C₂₁₁	C₂²	1	12	yes	—
16	C₄×Q₈×C₂₁₁	C₂²	1	12	yes	—
17	C₄×Q₈×C₂₁₁	C₂²	1	12	yes	—
18	C₄×Q₈×C₂₁₁	C₂²	1	4	yes	—
19	C₄×Q₈×C₂₁₁	C₂²	1	8	yes	—
20	C₄×Q₈×C₂₁₁	C₂²	1	4	yes	—
21	C₄×Q₈×C₂₁₁	C₂²	1	8	yes	—
22	#117×C₂₁₁	C₂²	1	12	yes	—
23	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	3	yes	—
24	C₂×C₄²×C₂₁₁	C₂²	1	3	yes	—
25	C₂×Q₈×C₂₁₁	Q₈	1	8	yes	complemented
26	C₂×Q₈×C₂₁₁	C₂³	1	4	yes	—
27	C₂×Q₈×C₂₁₁	C₂³	1	12	yes	—
28	C₂×Q₈×C₂₁₁	C₂³	1	4	yes	—
29	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₂₁₁	C₂³	1	3	yes	—
31	C₂²×C₄×C₂₁₁	C₂³	1	3	yes	—
32	C₄⋊C₄×C₂₁₁	C₂³	1	12	yes	—
33	C₄⋊C₄×C₂₁₁	C₂³	1	12	yes	—
34	C₄⋊C₄×C₂₁₁	C₂³	1	12	yes	—
35	C₄⋊C₄×C₂₁₁	C₂³	1	8	yes	—
36	C₄⋊C₄×C₂₁₁	C₂³	1	4	yes	—
37	C₄⋊C₄×C₂₁₁	C₂³	1	12	yes	—
38	C₄⋊C₄×C₂₁₁	C₂³	1	24	yes	—
39	C₂²⋊C₄×C₂₁₁	C₂³	1	8	yes	—
40	C₄²×C₂₁₁	C₂³	1	6	yes	—
41	C₄²×C₂₁₁	C₂³	1	6	yes	—
42	C₄²×C₂₁₁	C₂³	1	24	yes	—
43	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
44	Q₈×C₂₁₁	—	2	8	no	—
45	Q₈×C₂₁₁	—	4	12	no	—
46	Q₈×C₂₁₁	C₂×Q₈	1	16	yes	—
47	C₂×C₄×C₂₁₁	—	2	2	no	—
48	C₂×C₄×C₂₁₁	—	4	3	no	—
49	C₂×C₄×C₂₁₁	C₂×Q₈	1	12	yes	—
50	C₂×C₄×C₂₁₁	C₂⁴	1	1	yes	characteristic
51	C₂×C₄×C₂₁₁	C₂⁴	1	8	yes	—
52	C₂×C₄×C₂₁₁	C₂⁴	1	8	yes	—
53	C₂×C₄×C₂₁₁	C₂⁴	1	6	yes	—
54	C₂×C₄×C₂₁₁	C₂⁴	1	6	yes	—
55	C₂×C₄×C₂₁₁	C₂⁴	1	1	yes	characteristic
56	C₂²×C₂₁₁	—	2	1	no	—
57	C₂²×C₂₁₁	—	2	1	no	—
58	C₂²×C₂₁₁	C₂²×Q₈	1	2	yes	—
59	C₂²×C₂₁₁	C₂⁵	1	1	yes	characteristic
60	C₄×C₂₁₁	—	2	1	no	—
61	C₄×C₂₁₁	—	2	8	no	—
62	C₄×C₂₁₁	—	2	6	no	—
63	C₄×C₂₁₁	—	2	8	no	—
64	C₄×C₂₁₁	C₂²×Q₈	1	12	yes	—
65	C₄×C₂₁₁	2₋¹⁺⁴	1	2	yes	—
66	C₂×C₂₁₁	—	2	2	no	—
67	C₂×C₂₁₁	C₂³×Q₈	1	1	yes	characteristic
68	C₂×C₂₁₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
69	C₂×C₂₁₁	#585	1	1	yes	characteristic
70	C₂₁₁	#3502	1	1	yes	characteristic, direct product factor, Sylow
71	#3502	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	Q₈²	C₂×C₂₁₁	1	4	yes	complemented
73	#557	C₂×C₂₁₁	1	8	yes	complemented
74	#557	C₂×C₂₁₁	1	4	yes	complemented
75	#544	C₂×C₂₁₁	1	6	yes	—
76	#533	C₂×C₂₁₁	1	6	yes	—
77	#519	C₂×C₂₁₁	1	2	yes	—
78	C₂×C₄×Q₈	C₂×C₂₁₁	1	1	yes	characteristic
79	C₂²×Q₈	C₂²×C₂₁₁	1	1	yes	characteristic
80	C₄⋊Q₈	C₂²×C₂₁₁	1	6	yes	—
81	C₄⋊Q₈	C₂²×C₂₁₁	1	6	yes	—
82	C₄⋊Q₈	C₂²×C₂₁₁	1	24	yes	—
83	#125	C₂²×C₂₁₁	1	12	yes	—
84	#125	C₂²×C₂₁₁	1	24	yes	—
85	C₂²⋊Q₈	C₂²×C₂₁₁	1	4	yes	—
86	C₂²⋊Q₈	C₂²×C₂₁₁	1	12	yes	—
87	C₄×Q₈	C₂²×C₂₁₁	1	4	yes	—
88	C₄×Q₈	C₂²×C₂₁₁	1	4	yes	—
89	C₄×Q₈	C₂²×C₂₁₁	1	8	yes	—
90	C₄×Q₈	C₂²×C₂₁₁	1	8	yes	—
91	C₄×Q₈	C₂²×C₂₁₁	1	12	yes	—
92	C₄×Q₈	C₂²×C₂₁₁	1	12	yes	—
93	#117	C₂²×C₂₁₁	1	12	yes	—
94	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	3	yes	—
95	C₂×C₄²	C₂²×C₂₁₁	1	3	yes	—
96	C₂×Q₈	Q₈×C₂₁₁	1	8	yes	complemented
97	C₂×Q₈	C₂³×C₂₁₁	1	4	yes	—
98	C₂×Q₈	C₂³×C₂₁₁	1	12	yes	—
99	C₂×Q₈	C₂³×C₂₁₁	1	4	yes	—
100	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
101	C₂²×C₄	C₂³×C₂₁₁	1	3	yes	—
102	C₂²×C₄	C₂³×C₂₁₁	1	3	yes	—
103	C₄⋊C₄	C₂³×C₂₁₁	1	12	yes	—
104	C₄⋊C₄	C₂³×C₂₁₁	1	12	yes	—
105	C₄⋊C₄	C₂³×C₂₁₁	1	12	yes	—
106	C₄⋊C₄	C₂³×C₂₁₁	1	8	yes	—
107	C₄⋊C₄	C₂³×C₂₁₁	1	4	yes	—
108	C₄⋊C₄	C₂³×C₂₁₁	1	12	yes	—
109	C₄⋊C₄	C₂³×C₂₁₁	1	24	yes	—
110	C₂²⋊C₄	C₂³×C₂₁₁	1	8	yes	—
111	C₄²	C₂³×C₂₁₁	1	6	yes	—
112	C₄²	C₂³×C₂₁₁	1	6	yes	—
113	C₄²	C₂³×C₂₁₁	1	24	yes	—
114	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
115	Q₈	—	2	8	no	—
116	Q₈	—	4	12	no	—
117	Q₈	C₂×Q₈×C₂₁₁	1	16	yes	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	3	no	—
120	C₂×C₄	C₂×Q₈×C₂₁₁	1	12	yes	—
121	C₂×C₄	C₂⁴×C₂₁₁	1	1	yes	characteristic
122	C₂×C₄	C₂⁴×C₂₁₁	1	8	yes	—
123	C₂×C₄	C₂⁴×C₂₁₁	1	8	yes	—
124	C₂×C₄	C₂⁴×C₂₁₁	1	6	yes	—
125	C₂×C₄	C₂⁴×C₂₁₁	1	6	yes	—
126	C₂×C₄	C₂⁴×C₂₁₁	1	1	yes	characteristic
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	C₂²×Q₈×C₂₁₁	1	2	yes	—
130	C₂²	C₂⁵×C₂₁₁	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	8	no	—
133	C₄	—	2	6	no	—
134	C₄	—	2	8	no	—
135	C₄	C₂²×Q₈×C₂₁₁	1	12	yes	—
136	C₄	2₋¹⁺⁴×C₂₁₁	1	2	yes	—
137	C₂	—	2	2	no	—
138	C₂	C₂³×Q₈×C₂₁₁	1	1	yes	characteristic
139	C₂	C₂×2₋¹⁺⁴×C₂₁₁	1	1	yes	characteristic
140	C₂	#585×C₂₁₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₁₁	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₁₁
prime radical	C₂³×C₂₁₁	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈²×C₂₁₁	C₂	semidirect
2. #557×C₂₁₁	C₂	semidirect
3. #557×C₂₁₁	C₂	semidirect
25. C₂×Q₈×C₂₁₁	Q₈	semidirect
70. C₂₁₁	#3502	direct (N × H)
71. #3502	C₂₁₁	direct (N × H)
72. Q₈²	C₂×C₂₁₁	semidirect
73. #557	C₂×C₂₁₁	semidirect
74. #557	C₂×C₂₁₁	semidirect
96. C₂×Q₈	Q₈×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
Q₈²×C₂₁₁	2
#557×C₂₁₁	2
#544×C₂₁₁	2
#533×C₂₁₁	2
#519×C₂₁₁	2
C₂×C₄×Q₈×C₂₁₁	2
#3502	211

Subgroups — #3502×C211 #41011122

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3502×C₂₁₁ #41011122