Subgroups — #2296×C₂₁₁ #41009916

Order 27008.

656 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#117×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#117×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄²×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	#389×C₂₁₁	C₂	1	4	yes	maximal, complemented
5	#385×C₂₁₁	C₂	1	4	yes	maximal
6	C₄×C₄⋊C₄×C₂₁₁	C₂	1	4	yes	maximal, complemented
7	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
8	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
9	#117×C₂₁₁	—	2	4	no	—
10	#117×C₂₁₁	C₄	1	8	yes	complemented
11	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	8	yes	—
12	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
14	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
16	C₂×C₄²×C₂₁₁	—	2	4	no	—
17	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
18	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
19	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
20	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
21	#95×C₂₁₁	C₂²	1	8	yes	—
22	C₂⁴×C₂₁₁	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₂₁₁	—	2	4	no	—
24	C₂²×C₄×C₂₁₁	—	2	8	no	—
25	C₂²×C₄×C₂₁₁	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₂₁₁	D₄	1	4	yes	—
27	C₂²×C₄×C₂₁₁	Q₈	1	4	yes	—
28	C₂²×C₄×C₂₁₁	C₂³	1	4	yes	—
29	C₂²×C₄×C₂₁₁	C₂³	1	4	yes	—
30	C₂²×C₄×C₂₁₁	C₂³	1	4	yes	—
31	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₂₁₁	—	2	4	no	—
34	C₄⋊C₄×C₂₁₁	—	2	16	no	—
35	C₄⋊C₄×C₂₁₁	C₂×C₄	1	8	yes	complemented
36	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
37	C₂²⋊C₄×C₂₁₁	C₂×C₄	1	8	yes	—
38	C₄²×C₂₁₁	—	2	4	no	—
39	C₄²×C₂₁₁	—	2	4	no	—
40	C₄²×C₂₁₁	—	2	4	no	—
41	C₄²×C₂₁₁	C₂×C₄	1	8	yes	complemented
42	C₂³×C₂₁₁	—	2	4	no	—
43	C₂³×C₂₁₁	C₂²×C₄	1	2	yes	—
44	C₂³×C₂₁₁	C₂×D₄	1	2	yes	—
45	C₂³×C₂₁₁	C₂×Q₈	1	2	yes	—
46	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
47	C₂×C₄×C₂₁₁	—	2	16	no	—
48	C₂×C₄×C₂₁₁	—	2	4	no	—
49	C₂×C₄×C₂₁₁	—	2	8	no	—
50	C₂×C₄×C₂₁₁	—	2	4	no	—
51	C₂×C₄×C₂₁₁	—	2	4	no	—
52	C₂×C₄×C₂₁₁	—	2	8	no	—
53	C₂×C₄×C₂₁₁	C₄⋊C₄	1	16	yes	complemented
54	C₂×C₄×C₂₁₁	C₂²×C₄	1	8	yes	—
55	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
60	C₂×C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₂₁₁	#41	1	8	yes	—
62	C₂²×C₂₁₁	—	2	4	no	—
63	C₂²×C₂₁₁	—	2	4	no	—
64	C₂²×C₂₁₁	—	2	4	no	—
65	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	4	yes	—
66	C₂²×C₂₁₁	C₂³×C₄	1	1	yes	characteristic
67	C₂²×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
68	C₂²×C₂₁₁	C₂²×Q₈	1	1	yes	characteristic
69	C₂²×C₂₁₁	C₂×#41	1	4	yes	—
70	C₄×C₂₁₁	—	2	8	no	—
71	C₄×C₂₁₁	—	2	8	no	—
72	C₄×C₂₁₁	—	4	4	no	—
73	C₄×C₂₁₁	C₂×C₄⋊C₄	1	4	yes	—
74	C₄×C₂₁₁	C₂×C₄⋊C₄	1	4	yes	—
75	C₂×C₂₁₁	—	2	4	no	—
76	C₂×C₂₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₂₁₁	C₄×#41	1	2	yes	—
78	C₂×C₂₁₁	#525	1	2	yes	—
79	C₂×C₂₁₁	#533	1	2	yes	—
80	C₂₁₁	#2296	1	1	yes	characteristic, direct product factor, Sylow
81	#2296	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×#117	C₂×C₂₁₁	1	1	yes	characteristic
83	C₂×#117	C₂×C₂₁₁	1	1	yes	characteristic
84	C₂²×C₄²	C₂×C₂₁₁	1	1	yes	characteristic
85	#389	C₂×C₂₁₁	1	4	yes	complemented
86	#385	C₂×C₂₁₁	1	4	yes	—
87	C₄×C₄⋊C₄	C₂×C₂₁₁	1	4	yes	complemented
88	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
89	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
90	#117	—	2	4	no	—
91	#117	C₄×C₂₁₁	1	8	yes	complemented
92	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
93	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	8	yes	—
94	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
95	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
96	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
97	C₂×C₄²	—	2	4	no	—
98	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
99	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
100	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
101	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
102	#95	C₂²×C₂₁₁	1	8	yes	—
103	C₂⁴	C₂³×C₂₁₁	1	1	yes	characteristic
104	C₂²×C₄	—	2	4	no	—
105	C₂²×C₄	—	2	8	no	—
106	C₂²×C₄	C₂×C₄×C₂₁₁	1	4	yes	—
107	C₂²×C₄	D₄×C₂₁₁	1	4	yes	—
108	C₂²×C₄	Q₈×C₂₁₁	1	4	yes	—
109	C₂²×C₄	C₂³×C₂₁₁	1	4	yes	—
110	C₂²×C₄	C₂³×C₂₁₁	1	4	yes	—
111	C₂²×C₄	C₂³×C₂₁₁	1	4	yes	—
112	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
113	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	16	no	—
116	C₄⋊C₄	C₂×C₄×C₂₁₁	1	8	yes	complemented
117	C₂²⋊C₄	—	2	4	no	—
118	C₂²⋊C₄	C₂×C₄×C₂₁₁	1	8	yes	—
119	C₄²	—	2	4	no	—
120	C₄²	—	2	4	no	—
121	C₄²	—	2	4	no	—
122	C₄²	C₂×C₄×C₂₁₁	1	8	yes	complemented
123	C₂³	—	2	4	no	—
124	C₂³	C₂²×C₄×C₂₁₁	1	2	yes	—
125	C₂³	C₂×D₄×C₂₁₁	1	2	yes	—
126	C₂³	C₂×Q₈×C₂₁₁	1	2	yes	—
127	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
128	C₂×C₄	—	2	16	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	8	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	2	8	no	—
134	C₂×C₄	C₄⋊C₄×C₂₁₁	1	16	yes	complemented
135	C₂×C₄	C₂²×C₄×C₂₁₁	1	8	yes	—
136	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
137	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
138	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
139	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
140	C₂×C₄	C₂×Q₈×C₂₁₁	1	2	yes	—
141	C₂×C₄	C₂×Q₈×C₂₁₁	1	2	yes	—
142	C₂×C₄	#41×C₂₁₁	1	8	yes	—
143	C₂²	—	2	4	no	—
144	C₂²	—	2	4	no	—
145	C₂²	—	2	4	no	—
146	C₂²	C₂×C₄⋊C₄×C₂₁₁	1	4	yes	—
147	C₂²	C₂³×C₄×C₂₁₁	1	1	yes	characteristic
148	C₂²	C₂²×D₄×C₂₁₁	1	1	yes	characteristic
149	C₂²	C₂²×Q₈×C₂₁₁	1	1	yes	characteristic
150	C₂²	C₂×#41×C₂₁₁	1	4	yes	—
151	C₄	—	2	8	no	—
152	C₄	—	2	8	no	—
153	C₄	—	4	4	no	—
154	C₄	C₂×C₄⋊C₄×C₂₁₁	1	4	yes	—
155	C₄	C₂×C₄⋊C₄×C₂₁₁	1	4	yes	—
156	C₂	—	2	4	no	—
157	C₂	C₂²×C₄⋊C₄×C₂₁₁	1	1	yes	characteristic
158	C₂	C₄×#41×C₂₁₁	1	2	yes	—
159	C₂	#525×C₂₁₁	1	2	yes	—
160	C₂	#533×C₂₁₁	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₁₁	C₂³
commutator	C₂²	C₂³×C₄×C₂₁₁
Frattini, cosocle	C₂³	C₂⁴×C₂₁₁
socle	C₂³×C₂₁₁	C₂⁴
prime radical	C₂⁴×C₂₁₁	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. #389×C₂₁₁	C₂	semidirect
6. C₄×C₄⋊C₄×C₂₁₁	C₂	semidirect
10. #117×C₂₁₁	C₄	semidirect
35. C₄⋊C₄×C₂₁₁	C₂×C₄	semidirect
41. C₄²×C₂₁₁	C₂×C₄	semidirect
53. C₂×C₄×C₂₁₁	C₄⋊C₄	semidirect
80. C₂₁₁	#2296	direct (N × H)
81. #2296	C₂₁₁	direct (N × H)
85. #389	C₂×C₂₁₁	semidirect
87. C₄×C₄⋊C₄	C₂×C₂₁₁	semidirect
91. #117	C₄×C₂₁₁	semidirect
116. C₄⋊C₄	C₂×C₄×C₂₁₁	semidirect
122. C₄²	C₂×C₄×C₂₁₁	semidirect
134. C₂×C₄	C₄⋊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×#117×C₂₁₁	2
C₂²×C₄²×C₂₁₁	2
#389×C₂₁₁	2
#385×C₂₁₁	2
C₄×C₄⋊C₄×C₂₁₁	2
#2296	211

Subgroups — #2296×C211 #41009916

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2296×C₂₁₁ #41009916