Subgroups — C₂×C₄×C₄⋊C₄×C₂₁₁ #41009889

Order 27008.

928 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂³×C₄×C₂₁₁	C₂²
commutator	C₂	C₂²×C₄²×C₂₁₁
Frattini, cosocle	C₂³	C₂⁴×C₂₁₁
socle, prime radical	C₂⁴×C₂₁₁	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄×C₄⋊C₄×C₂₁₁	C₂	direct (N × H)
10. C₂×C₄⋊C₄×C₂₁₁	C₄	direct (N × H)
13. C₂×C₄²×C₂₁₁	C₄	semidirect
32. C₄⋊C₄×C₂₁₁	C₂×C₄	direct (N × H)
34. C₄²×C₂₁₁	C₂×C₄	semidirect
46. C₂×C₄×C₂₁₁	C₄²	semidirect
47. C₂×C₄×C₂₁₁	C₄⋊C₄	direct (N × H)
67. C₄×C₂₁₁	C₂×C₄²	semidirect
68. C₄×C₂₁₁	C₂×C₄⋊C₄	direct (N × H)
69. C₂×C₂₁₁	C₄×C₄⋊C₄	direct (N × H)
75. C₂₁₁	C₂×C₄×C₄⋊C₄	direct (N × H)
76. C₂×C₄×C₄⋊C₄	C₂₁₁	direct (N × H)
80. C₄×C₄⋊C₄	C₂×C₂₁₁	direct (N × H)
86. C₂×C₄⋊C₄	C₄×C₂₁₁	direct (N × H)
89. C₂×C₄²	C₄×C₂₁₁	semidirect
108. C₄⋊C₄	C₂×C₄×C₂₁₁	direct (N × H)
110. C₄²	C₂×C₄×C₂₁₁	semidirect
122. C₂×C₄	C₄²×C₂₁₁	semidirect
123. C₂×C₄	C₄⋊C₄×C₂₁₁	direct (N × H)
143. C₄	C₂×C₄²×C₂₁₁	semidirect
144. C₄	C₂×C₄⋊C₄×C₂₁₁	direct (N × H)
145. C₂	C₄×C₄⋊C₄×C₂₁₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊C₄×C₂₁₁	2
C₂²×C₄²×C₂₁₁	2
C₄×C₄⋊C₄×C₂₁₁	2
C₂×#95×C₂₁₁	2
C₂×C₄×C₄⋊C₄	211

Subgroups — C2×C4×C4⋊C4×C211 #41009889

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₄⋊C₄×C₂₁₁ #41009889