Subgroups — C₂²×C₁₁×#368 #14610605

Order 2816.

662 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₁×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₁×#108	C₂	1	2	yes	maximal
3	C₂×C₁₁×#368	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₁₁×M₁₆	—	2	2	no	—
5	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
6	C₂²×C₁₁×C₁₆	C₄	1	2	yes	—
7	C₂²×C₁₁×C₁₆	C₄	1	6	yes	—
8	C₂²×C₁₁×C₁₆	C₂²	1	6	yes	—
9	C₂×C₁₁×#108	—	2	6	no	—
10	C₂×C₁₁×#108	C₂²	1	12	yes	—
11	C₁₁×#368	C₂²	1	16	yes	direct product factor
12	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
13	C₂×C₁₁×M₁₆	—	2	12	no	—
14	C₂×C₁₁×M₁₆	—	4	6	no	—
15	C₂²×C₈×C₁₁	—	4	2	no	—
16	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₁	C₂×C₄	1	3	yes	—
18	C₂²×C₈×C₁₁	D₄	1	3	yes	—
19	C₂²×C₈×C₁₁	Q₈	1	1	yes	characteristic
20	C₂²×C₈×C₁₁	Q₈	1	3	yes	—
21	C₂²×C₈×C₁₁	C₂³	1	3	yes	—
22	C₂×C₁₁×C₁₆	C₂×C₄	1	12	yes	—
23	C₂×C₁₁×C₁₆	C₂×C₄	1	12	yes	—
24	C₂×C₁₁×C₁₆	C₂³	1	4	yes	—
25	C₁₁×#108	—	2	12	no	—
26	C₁₁×#108	C₂³	1	8	yes	—
27	C₂²×#368	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂⁴×C₁₁	Q₁₆	1	1	yes	characteristic
29	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₁	C₄⋊C₄	1	3	yes	—
31	C₂²×C₄×C₁₁	D₈	1	3	yes	—
32	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
33	C₂²×C₄×C₁₁	Q₁₆	1	3	yes	—
34	C₂²×C₄×C₁₁	C₂×D₄	1	3	yes	—
35	C₁₁×M₁₆	—	2	8	no	—
36	C₁₁×M₁₆	—	4	12	no	—
37	C₂×C₈×C₁₁	—	4	12	no	—
38	C₂×C₈×C₁₁	C₄⋊C₄	1	6	yes	—
39	C₂×C₈×C₁₁	C₄⋊C₄	1	3	yes	—
40	C₂×C₈×C₁₁	C₄⋊C₄	1	3	yes	—
41	C₂×C₈×C₁₁	C₂²×C₄	1	3	yes	—
42	C₂×C₈×C₁₁	C₂²×C₄	1	3	yes	—
43	C₂×C₈×C₁₁	C₂×D₄	1	3	yes	—
44	C₂×C₈×C₁₁	C₂×Q₈	1	3	yes	—
45	C₂×C₈×C₁₁	C₂×Q₈	1	3	yes	—
46	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
47	C₁₁×C₁₆	C₂²×C₄	1	8	yes	—
48	C₂³×C₁₆	C₂×C₁₁	1	1	yes	characteristic
49	C₂²×#108	C₂×C₁₁	1	2	yes	—
50	C₂×#368	C₂×C₁₁	1	12	yes	direct product factor
51	C₂³×C₁₁	—	2	4	no	—
52	C₂³×C₁₁	#107	1	3	yes	—
53	C₂³×C₁₁	#107	1	1	yes	characteristic
54	C₂³×C₁₁	C₂×Q₁₆	1	3	yes	—
55	C₂×C₄×C₁₁	#107	1	6	yes	—
56	C₂×C₄×C₁₁	#107	1	3	yes	—
57	C₂×C₄×C₁₁	#107	1	3	yes	—
58	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
59	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
60	C₂×C₄×C₁₁	C₂×D₈	1	3	yes	—
61	C₂×C₄×C₁₁	C₂×D₈	1	3	yes	—
62	C₂×C₄×C₁₁	C₂×Q₁₆	1	3	yes	—
63	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
64	C₈×C₁₁	—	4	8	no	—
65	C₈×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
66	C₈×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
67	C₈×C₁₁	C₂³×C₄	1	1	yes	characteristic
68	C₈×C₁₁	C₂²×Q₈	1	1	yes	characteristic
69	C₂²×M₁₆	—	2	2	no	—
70	C₂³×C₈	C₂²×C₁₁	1	1	yes	characteristic
71	C₂²×C₁₆	C₄×C₁₁	1	6	yes	—
72	C₂²×C₁₆	C₄×C₁₁	1	2	yes	—
73	C₂²×C₁₆	C₂²×C₁₁	1	6	yes	—
74	C₂×#108	—	2	6	no	—
75	C₂×#108	C₂²×C₁₁	1	12	yes	—
76	#368	C₂²×C₁₁	1	16	yes	direct product factor
77	C₂²×C₁₁	—	2	6	no	—
78	C₂²×C₁₁	—	2	6	no	—
79	C₂²×C₁₁	#368	1	4	yes	direct product factor
80	C₂²×C₁₁	C₂×#107	1	3	yes	—
81	C₂²×C₁₁	C₂×#107	1	3	yes	—
82	C₂²×C₁₁	C₂²×Q₁₆	1	1	yes	characteristic
83	C₄×C₁₁	C₂×#107	1	3	yes	—
84	C₄×C₁₁	C₂×#107	1	3	yes	—
85	C₄×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
86	C₄×C₁₁	C₂²×D₈	1	1	yes	characteristic
87	C₂³×C₄	D₄×C₁₁	1	1	yes	characteristic
88	C₂×M₁₆	—	2	12	no	—
89	C₂×M₁₆	—	4	6	no	—
90	C₂²×C₈	—	4	2	no	—
91	C₂²×C₈	C₂×C₄×C₁₁	1	3	yes	—
92	C₂²×C₈	C₂×C₄×C₁₁	1	1	yes	characteristic
93	C₂²×C₈	D₄×C₁₁	1	3	yes	—
94	C₂²×C₈	Q₈×C₁₁	1	3	yes	—
95	C₂²×C₈	Q₈×C₁₁	1	1	yes	characteristic
96	C₂²×C₈	C₂³×C₁₁	1	3	yes	—
97	C₂×C₁₆	C₂×C₄×C₁₁	1	12	yes	—
98	C₂×C₁₆	C₂×C₄×C₁₁	1	12	yes	—
99	C₂×C₁₆	C₂³×C₁₁	1	4	yes	—
100	#108	—	2	12	no	—
101	#108	C₂³×C₁₁	1	8	yes	—
102	C₂×C₁₁	—	2	1	no	—
103	C₂×C₁₁	—	2	3	no	—
104	C₂×C₁₁	C₂×#368	1	6	yes	direct product factor
105	C₂×C₁₁	C₂²×#107	1	1	yes	characteristic
106	C₂⁴	C₁₁×Q₁₆	1	1	yes	characteristic
107	C₂²×C₄	C₁₁×C₄⋊C₄	1	3	yes	—
108	C₂²×C₄	C₁₁×C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₄	C₁₁×D₈	1	1	yes	characteristic
110	C₂²×C₄	C₁₁×D₈	1	3	yes	—
111	C₂²×C₄	C₁₁×Q₁₆	1	3	yes	—
112	C₂²×C₄	C₂×D₄×C₁₁	1	3	yes	—
113	M₁₆	—	2	8	no	—
114	M₁₆	—	4	12	no	—
115	C₂×C₈	—	4	12	no	—
116	C₂×C₈	C₁₁×C₄⋊C₄	1	6	yes	—
117	C₂×C₈	C₁₁×C₄⋊C₄	1	3	yes	—
118	C₂×C₈	C₁₁×C₄⋊C₄	1	3	yes	—
119	C₂×C₈	C₂²×C₄×C₁₁	1	3	yes	—
120	C₂×C₈	C₂²×C₄×C₁₁	1	3	yes	—
121	C₂×C₈	C₂×D₄×C₁₁	1	3	yes	—
122	C₂×C₈	C₂×Q₈×C₁₁	1	3	yes	—
123	C₂×C₈	C₂×Q₈×C₁₁	1	3	yes	—
124	C₂×C₈	C₂⁴×C₁₁	1	1	yes	characteristic
125	C₁₆	C₂²×C₄×C₁₁	1	8	yes	—
126	C₁₁	C₂²×#368	1	1	yes	characteristic, direct product factor, Sylow
127	C₂³	—	2	4	no	—
128	C₂³	C₁₁×#107	1	3	yes	—
129	C₂³	C₁₁×#107	1	1	yes	characteristic
130	C₂³	C₂×C₁₁×Q₁₆	1	3	yes	—
131	C₂×C₄	C₁₁×#107	1	6	yes	—
132	C₂×C₄	C₁₁×#107	1	3	yes	—
133	C₂×C₄	C₁₁×#107	1	3	yes	—
134	C₂×C₄	C₂×C₁₁×C₄⋊C₄	1	3	yes	—
135	C₂×C₄	C₂×C₁₁×C₄⋊C₄	1	3	yes	—
136	C₂×C₄	C₂×C₁₁×D₈	1	3	yes	—
137	C₂×C₄	C₂×C₁₁×D₈	1	3	yes	—
138	C₂×C₄	C₂×C₁₁×Q₁₆	1	3	yes	—
139	C₂×C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
140	C₈	—	4	8	no	—
141	C₈	C₂×C₁₁×C₄⋊C₄	1	3	yes	—
142	C₈	C₂×C₁₁×C₄⋊C₄	1	3	yes	—
143	C₈	C₂³×C₄×C₁₁	1	1	yes	characteristic
144	C₈	C₂²×Q₈×C₁₁	1	1	yes	characteristic
145	C₂²	—	2	6	no	—
146	C₂²	—	2	6	no	—
147	C₂²	C₁₁×#368	1	4	yes	direct product factor
148	C₂²	C₂×C₁₁×#107	1	3	yes	—
149	C₂²	C₂×C₁₁×#107	1	3	yes	—
150	C₂²	C₂²×C₁₁×Q₁₆	1	1	yes	characteristic
151	C₄	C₂×C₁₁×#107	1	3	yes	—
152	C₄	C₂×C₁₁×#107	1	3	yes	—
153	C₄	C₂²×C₁₁×C₄⋊C₄	1	1	yes	characteristic
154	C₄	C₂²×C₁₁×D₈	1	1	yes	characteristic
155	C₂	—	2	3	no	—
156	C₂	—	2	1	no	—
157	C₂	C₂×C₁₁×#368	1	6	yes	direct product factor
158	C₂	C₂²×C₁₁×#107	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₁	D₈
commutator	C₈	C₂³×C₄×C₁₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₁
socle	C₂³×C₁₁	#107
prime radical	C₂⁴×C₁₁	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₁ ◁ C₂³×C₄×C₁₁ ◁ C₂³×C₈×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₁₁×#368	C₂	direct (N × H)
11. C₁₁×#368	C₂²	direct (N × H)
27. C₂²×#368	C₁₁	direct (N × H)
50. C₂×#368	C₂×C₁₁	direct (N × H)
76. #368	C₂²×C₁₁	direct (N × H)
79. C₂²×C₁₁	#368	direct (N × H)
104. C₂×C₁₁	C₂×#368	direct (N × H)
126. C₁₁	C₂²×#368	direct (N × H)
147. C₂²	C₁₁×#368	direct (N × H)
157. C₂	C₂×C₁₁×#368	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×C₁₁×C₁₆	2
C₂²×C₁₁×#108	2
C₂×C₁₁×#368	2
C₂²×#368	11

Subgroups — C22×C11×#368 #14610605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₁×#368 #14610605