Subgroups — #14670162

Order 2816.

282 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	C₂×C₄×C₁₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#13068788	C₂	1	1	yes	maximal, characteristic
3	#13066709	C₂	1	4	yes	maximal
4	C₁₁×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₁×#108	—	2	1	no	—
6	C₂×C₁₁×#108	C₂²	1	2	yes	—
7	C₁₁×#423	—	2	1	no	—
8	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
9	C₂×#66883	—	2	1	no	—
10	C₂×#66883	C₂²	1	2	yes	—
11	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
12	#10622537	—	2	1	no	—
13	C₄×C₁₁⋊C₁₆	C₄	1	2	yes	—
14	C₄×C₁₁⋊C₁₆	C₄	1	2	yes	—
15	C₂×C₁₁×M₁₆	—	2	2	no	—
16	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
19	C₂×#4318	—	2	2	no	—
20	C₁₁×#108	—	2	4	no	—
21	C₁₁×#108	—	4	2	no	—
22	C₁₁×C₄⋊C₈	—	4	2	no	—
23	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₁	Q₈	1	1	yes	characteristic
26	C₄×C₈×C₁₁	Q₈	1	1	yes	characteristic
27	#66883	—	2	4	no	—
28	#66883	—	4	2	no	—
29	C₂×C₁₁⋊C₁₆	—	2	2	no	—
30	C₂×C₁₁⋊C₁₆	—	2	2	no	—
31	C₂×C₁₁⋊C₁₆	C₂×C₄	1	4	yes	—
32	C₄⋊(C₁₁⋊C₈)	—	4	2	no	—
33	#12891	—	11	1	no	maximal, Sylow
34	C₂²×C₄×C₁₁	Q₁₆	1	2	yes	—
35	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
36	C₁₁×M₁₆	—	4	2	no	—
37	C₁₁×M₁₆	—	4	2	no	—
38	C₂×C₈×C₁₁	—	4	2	no	—
39	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₁	SD₁₆	1	2	yes	—
42	C₂×C₈×C₁₁	SD₁₆	1	2	yes	—
43	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₁₁	#41	1	2	yes	—
46	C₂×C₈×C₁₁	#41	1	2	yes	—
47	C₄²×C₁₁	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₁₁	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₁₁	D₈	1	2	yes	—
50	#4318	—	4	2	no	—
51	#4318	—	4	2	no	—
52	C₂×C₁₁⋊C₈	—	4	2	no	—
53	C₁₁⋊C₁₆	—	2	4	no	—
54	C₂×C₄×C₁₆	—	11	1	no	—
55	#1848	—	11	1	no	—
56	#1848	D₁₁	1	1	yes	characteristic
57	#1381	—	11	4	no	—
58	C₂³×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×C₄×C₁₁	—	2	4	no	—
60	C₂×C₄×C₁₁	#107	1	2	yes	—
61	C₂×C₄×C₁₁	#107	1	2	yes	—
62	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
64	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
65	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
68	C₈×C₁₁	—	8	2	no	—
69	C₈×C₁₁	#106	1	2	yes	—
70	C₈×C₁₁	#106	1	2	yes	—
71	C₈×C₁₁	#117	1	2	yes	—
72	C₈×C₁₁	#125	1	2	yes	—
73	C₁₁⋊C₈	—	8	2	no	—
74	C₂²×C₁₆	—	11	2	no	—
75	C₂×#108	—	2	1	no	—
76	C₂×#108	—	11	2	no	—
77	C₂×#108	—	22	1	no	—
78	C₂×#108	C₂×D₁₁	1	2	yes	—
79	#423	—	2	1	no	—
80	#423	—	22	1	no	—
81	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
82	C₄×C₁₆	—	11	2	no	—
83	C₄×C₁₆	—	11	2	no	—
84	C₂²×C₁₁	—	2	2	no	—
85	C₂²×C₁₁	C₂×#107	1	1	yes	characteristic
86	C₂²×C₁₁	#487	1	1	yes	characteristic
87	C₂²×C₁₁	#487	1	1	yes	characteristic
88	C₄×C₁₁	—	2	2	no	—
89	C₄×C₁₁	#368	1	4	yes	—
90	C₄×C₁₁	#382	1	1	yes	characteristic
91	C₄×C₁₁	C₂×#106	1	1	yes	characteristic
92	C₄×C₁₁	#486	1	1	yes	characteristic
93	C₄×C₁₁	#486	1	1	yes	characteristic
94	C₂×M₁₆	—	2	2	no	—
95	C₂×M₁₆	—	22	2	no	—
96	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
97	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
98	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
99	C₂×C₁₆	—	11	4	no	—
100	C₂×C₁₆	—	22	2	no	—
101	C₂×C₁₆	—	22	2	no	—
102	#108	—	2	4	no	—
103	#108	—	4	2	no	—
104	#108	—	22	4	no	—
105	#108	—	44	2	no	—
106	C₄⋊C₈	—	4	2	no	—
107	C₄⋊C₈	—	44	2	no	—
108	C₄×C₈	Dic₂₂	1	1	yes	characteristic
109	C₄×C₈	Dic₂₂	1	1	yes	characteristic
110	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
111	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
112	C₂×C₁₁	—	2	1	no	—
113	C₂×C₁₁	—	2	1	no	—
114	C₂×C₁₁	#1834	1	1	yes	characteristic
115	C₂×C₁₁	C₂×#368	1	2	yes	—
116	C₂²×C₄	#4326	1	2	yes	—
117	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
118	M₁₆	—	4	2	no	—
119	M₁₆	—	4	2	no	—
120	M₁₆	—	44	2	no	—
121	M₁₆	—	44	2	no	—
122	C₂×C₈	—	4	2	no	—
123	C₂×C₈	—	44	2	no	—
124	C₂×C₈	Dic₁₁⋊C₄	1	1	yes	characteristic
125	C₂×C₈	Dic₁₁⋊C₄	1	1	yes	characteristic
126	C₂×C₈	#4324	1	2	yes	—
127	C₂×C₈	#4325	1	2	yes	—
128	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
129	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
130	C₂×C₈	#4341	1	2	yes	—
131	C₂×C₈	#4343	1	2	yes	—
132	C₄²	Dic₁₁⋊C₄	1	1	yes	characteristic
133	C₄²	Dic₁₁⋊C₄	1	1	yes	characteristic
134	C₄²	#4323	1	2	yes	—
135	C₁₆	—	22	4	no	—
136	C₁₁	#12891	1	1	yes	characteristic, complemented, Sylow
137	C₂³	C₂×#4326	1	1	yes	characteristic
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	#66871	1	2	yes	—
140	C₂×C₄	#66871	1	2	yes	—
141	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
142	C₂×C₄	C₂×#4323	1	1	yes	characteristic
143	C₂×C₄	C₂×#4324	1	1	yes	characteristic
144	C₂×C₄	#66989	1	1	yes	characteristic
145	C₂×C₄	C₂×#4325	1	1	yes	characteristic
146	C₂×C₄	#67000	1	1	yes	characteristic
147	C₈	—	8	2	no	—
148	C₈	—	88	2	no	—
149	C₈	#66872	1	2	yes	—
150	C₈	#66872	1	2	yes	—
151	C₈	#66943	1	2	yes	—
152	C₈	#66945	1	2	yes	—
153	C₂²	—	2	2	no	—
154	C₂²	C₂×#66871	1	1	yes	characteristic
155	C₂²	#10622694	1	1	yes	characteristic
156	C₂²	#10622706	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	#10622131	1	4	yes	—
159	C₄	C₂×#66872	1	1	yes	characteristic
160	C₄	#10622590	1	1	yes	characteristic
161	C₄	#10622683	1	1	yes	characteristic
162	C₄	#10622701	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	#13068743	1	1	yes	characteristic
166	C₂	C₂×#10622131	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#4323
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₁₁×#1848	C₂
socle	C₂²×C₁₁	C₂×#107
prime radical	C₂³×C₁₁	C₂×Q₁₆
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	#1848	D₁₁
hypocenter	C₁₁	#12891
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
136. C₁₁	#12891	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₁₁⋊C₁₆	2
#13068788	2
#13066709	2
C₁₁×#1848	2
#12891	11