Subgroups — C₂×#391×C₂₁₁ #41009914

Order 27008.

928 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄²×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	#391×C₂₁₁	C₂	1	8	yes	maximal, direct product factor
5	C₂×#95×C₂₁₁	C₂	1	4	yes	maximal
6	C₂²×Q₈×C₂₁₁	—	2	4	no	—
7	C₂²×Q₈×C₂₁₁	—	2	4	no	—
8	C₂²×Q₈×C₂₁₁	C₄	1	8	yes	complemented
9	C₂²×Q₈×C₂₁₁	C₂²	1	4	yes	—
10	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
11	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
12	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
13	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₁₁	—	2	4	no	—
15	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	4	yes	—
16	C₂×C₄²×C₂₁₁	—	2	4	no	—
17	C₂×C₄²×C₂₁₁	C₂²	1	4	yes	—
18	#95×C₂₁₁	C₂²	1	16	yes	—
19	C₂⁴×C₂₁₁	C₂³	1	1	yes	characteristic
20	C₂×Q₈×C₂₁₁	—	2	32	no	—
21	C₂×Q₈×C₂₁₁	—	2	8	no	—
22	C₂×Q₈×C₂₁₁	—	2	8	no	—
23	C₂×Q₈×C₂₁₁	C₂×C₄	1	16	yes	complemented
24	C₂²×C₄×C₂₁₁	—	2	4	no	—
25	C₂²×C₄×C₂₁₁	—	2	4	no	—
26	C₂²×C₄×C₂₁₁	—	2	4	no	—
27	C₂²×C₄×C₂₁₁	—	2	4	no	—
28	C₂²×C₄×C₂₁₁	—	2	4	no	—
29	C₂²×C₄×C₂₁₁	—	2	8	no	—
30	C₂²×C₄×C₂₁₁	C₂×C₄	1	8	yes	—
31	C₂²×C₄×C₂₁₁	C₂×C₄	1	4	yes	—
32	C₂²×C₄×C₂₁₁	D₄	1	4	yes	—
33	C₂²×C₄×C₂₁₁	D₄	1	4	yes	—
34	C₂²×C₄×C₂₁₁	Q₈	1	8	yes	complemented
35	C₂²×C₄×C₂₁₁	C₂³	1	4	yes	—
36	C₂²×C₄×C₂₁₁	C₂³	1	4	yes	—
37	C₂²×C₄×C₂₁₁	C₂³	1	4	yes	—
38	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
39	C₄⋊C₄×C₂₁₁	—	2	8	no	—
40	C₄²×C₂₁₁	—	2	8	no	—
41	C₂³×C₂₁₁	C₂²×C₄	1	2	yes	—
42	C₂³×C₂₁₁	C₂×D₄	1	2	yes	—
43	C₂³×C₂₁₁	C₂×D₄	1	2	yes	—
44	C₂³×C₂₁₁	C₂×Q₈	1	4	yes	—
45	C₂³×C₂₁₁	#41	1	4	yes	—
46	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
47	Q₈×C₂₁₁	—	2	32	no	—
48	C₂×C₄×C₂₁₁	—	2	16	no	—
49	C₂×C₄×C₂₁₁	—	2	16	no	—
50	C₂×C₄×C₂₁₁	—	2	4	no	—
51	C₂×C₄×C₂₁₁	—	2	4	no	—
52	C₂×C₄×C₂₁₁	—	2	4	no	—
53	C₂×C₄×C₂₁₁	—	2	4	no	—
54	C₂×C₄×C₂₁₁	—	2	4	no	—
55	C₂×C₄×C₂₁₁	—	2	8	no	—
56	C₂×C₄×C₂₁₁	—	4	8	no	—
57	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	16	yes	—
58	C₂×C₄×C₂₁₁	C₂²×C₄	1	8	yes	—
59	C₂×C₄×C₂₁₁	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₂₁₁	C₂×D₄	1	4	yes	—
61	C₂×C₄×C₂₁₁	C₂×D₄	1	4	yes	—
62	C₂×C₄×C₂₁₁	C₂×Q₈	1	8	yes	complemented
63	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	4	yes	—
64	C₂²×C₂₁₁	C₄×Q₈	1	8	yes	—
65	C₂²×C₂₁₁	C₂²⋊Q₈	1	8	yes	—
66	C₂²×C₂₁₁	#124	1	4	yes	—
67	C₂²×C₂₁₁	C₄⋊Q₈	1	4	yes	—
68	C₂²×C₂₁₁	C₂³×C₄	1	1	yes	characteristic
69	C₂²×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
70	C₂²×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
71	C₂²×C₂₁₁	C₂²×Q₈	1	2	yes	—
72	C₂²×C₂₁₁	C₂×#41	1	2	yes	—
73	C₄×C₂₁₁	—	2	8	no	—
74	C₄×C₂₁₁	—	2	8	no	—
75	C₄×C₂₁₁	—	4	4	no	—
76	C₄×C₂₁₁	C₂×C₂²⋊C₄	1	8	yes	—
77	C₂×C₂₁₁	#391	1	8	yes	direct product factor
78	C₂×C₂₁₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₂₁₁	C₂×C₄×Q₈	1	2	yes	—
80	C₂×C₂₁₁	C₂×C₂²⋊Q₈	1	2	yes	—
81	C₂×C₂₁₁	C₂×#124	1	1	yes	characteristic
82	C₂×C₂₁₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
83	C₂₁₁	C₂×#391	1	1	yes	characteristic, direct product factor, Sylow
84	C₂×#391	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×Q₈	C₂×C₂₁₁	1	1	yes	characteristic
86	C₂²×C₄⋊C₄	C₂×C₂₁₁	1	1	yes	characteristic
87	C₂²×C₄²	C₂×C₂₁₁	1	1	yes	characteristic
88	#391	C₂×C₂₁₁	1	8	yes	direct product factor
89	C₂×#95	C₂×C₂₁₁	1	4	yes	—
90	C₂²×Q₈	—	2	4	no	—
91	C₂²×Q₈	—	2	4	no	—
92	C₂²×Q₈	C₄×C₂₁₁	1	8	yes	complemented
93	C₂²×Q₈	C₂²×C₂₁₁	1	4	yes	—
94	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
95	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
96	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
97	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
98	C₂×C₄⋊C₄	—	2	4	no	—
99	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	4	yes	—
100	C₂×C₄²	—	2	4	no	—
101	C₂×C₄²	C₂²×C₂₁₁	1	4	yes	—
102	#95	C₂²×C₂₁₁	1	16	yes	—
103	C₂⁴	C₂³×C₂₁₁	1	1	yes	characteristic
104	C₂×Q₈	—	2	32	no	—
105	C₂×Q₈	—	2	8	no	—
106	C₂×Q₈	—	2	8	no	—
107	C₂×Q₈	C₂×C₄×C₂₁₁	1	16	yes	complemented
108	C₂²×C₄	—	2	4	no	—
109	C₂²×C₄	—	2	4	no	—
110	C₂²×C₄	—	2	4	no	—
111	C₂²×C₄	—	2	4	no	—
112	C₂²×C₄	—	2	4	no	—
113	C₂²×C₄	—	2	8	no	—
114	C₂²×C₄	C₂×C₄×C₂₁₁	1	8	yes	—
115	C₂²×C₄	C₂×C₄×C₂₁₁	1	4	yes	—
116	C₂²×C₄	D₄×C₂₁₁	1	4	yes	—
117	C₂²×C₄	D₄×C₂₁₁	1	4	yes	—
118	C₂²×C₄	Q₈×C₂₁₁	1	8	yes	complemented
119	C₂²×C₄	C₂³×C₂₁₁	1	4	yes	—
120	C₂²×C₄	C₂³×C₂₁₁	1	4	yes	—
121	C₂²×C₄	C₂³×C₂₁₁	1	4	yes	—
122	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
123	C₄⋊C₄	—	2	8	no	—
124	C₄²	—	2	8	no	—
125	C₂³	C₂²×C₄×C₂₁₁	1	2	yes	—
126	C₂³	C₂×D₄×C₂₁₁	1	2	yes	—
127	C₂³	C₂×D₄×C₂₁₁	1	2	yes	—
128	C₂³	C₂×Q₈×C₂₁₁	1	4	yes	—
129	C₂³	#41×C₂₁₁	1	4	yes	—
130	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
131	Q₈	—	2	32	no	—
132	C₂×C₄	—	2	16	no	—
133	C₂×C₄	—	2	16	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	4	8	no	—
141	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	16	yes	—
142	C₂×C₄	C₂²×C₄×C₂₁₁	1	8	yes	—
143	C₂×C₄	C₂²×C₄×C₂₁₁	1	4	yes	—
144	C₂×C₄	C₂×D₄×C₂₁₁	1	4	yes	—
145	C₂×C₄	C₂×D₄×C₂₁₁	1	4	yes	—
146	C₂×C₄	C₂×Q₈×C₂₁₁	1	8	yes	complemented
147	C₂²	C₂×C₂²⋊C₄×C₂₁₁	1	4	yes	—
148	C₂²	C₄×Q₈×C₂₁₁	1	8	yes	—
149	C₂²	C₂²⋊Q₈×C₂₁₁	1	8	yes	—
150	C₂²	#124×C₂₁₁	1	4	yes	—
151	C₂²	C₄⋊Q₈×C₂₁₁	1	4	yes	—
152	C₂²	C₂³×C₄×C₂₁₁	1	1	yes	characteristic
153	C₂²	C₂²×D₄×C₂₁₁	1	1	yes	characteristic
154	C₂²	C₂²×D₄×C₂₁₁	1	1	yes	characteristic
155	C₂²	C₂²×Q₈×C₂₁₁	1	2	yes	—
156	C₂²	C₂×#41×C₂₁₁	1	2	yes	—
157	C₄	—	2	8	no	—
158	C₄	—	2	8	no	—
159	C₄	—	4	4	no	—
160	C₄	C₂×C₂²⋊C₄×C₂₁₁	1	8	yes	—
161	C₂	#391×C₂₁₁	1	8	yes	direct product factor
162	C₂	C₂²×C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
163	C₂	C₂×C₄×Q₈×C₂₁₁	1	2	yes	—
164	C₂	C₂×C₂²⋊Q₈×C₂₁₁	1	2	yes	—
165	C₂	C₂×#124×C₂₁₁	1	1	yes	characteristic
166	C₂	C₂×C₄⋊Q₈×C₂₁₁	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. #391×C₂₁₁	C₂	direct (N × H)
8. C₂²×Q₈×C₂₁₁	C₄	semidirect
23. C₂×Q₈×C₂₁₁	C₂×C₄	semidirect
34. C₂²×C₄×C₂₁₁	Q₈	semidirect
62. C₂×C₄×C₂₁₁	C₂×Q₈	semidirect
77. C₂×C₂₁₁	#391	direct (N × H)
83. C₂₁₁	C₂×#391	direct (N × H)
84. C₂×#391	C₂₁₁	direct (N × H)
88. #391	C₂×C₂₁₁	direct (N × H)
92. C₂²×Q₈	C₄×C₂₁₁	semidirect
107. C₂×Q₈	C₂×C₄×C₂₁₁	semidirect
118. C₂²×C₄	Q₈×C₂₁₁	semidirect
146. C₂×C₄	C₂×Q₈×C₂₁₁	semidirect
161. C₂	#391×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
C₂³×Q₈×C₂₁₁	2
C₂²×C₄⋊C₄×C₂₁₁	2
C₂²×C₄²×C₂₁₁	2
#391×C₂₁₁	2
C₂×#95×C₂₁₁	2
C₂×#391	211

Subgroups — C2×#391×C211 #41009914

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#391×C₂₁₁ #41009914