Subgroups — #16807368

Order 3328.

494 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131251	C₂	1	6	yes	maximal, complemented
2	Q₈×#6092	C₂	1	2	yes	maximal, complemented
3	#13131197	C₂	1	3	yes	maximal
4	#13131193	C₂	1	3	yes	maximal
5	C₂×Q₈×#1055	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
7	Q₈×#1055	C₄	1	3	yes	—
8	Q₈×#1055	C₄	1	1	yes	characteristic
9	Q₈×#1055	C₂²	1	1	yes	characteristic
10	Q₈×#1055	C₂²	1	3	yes	—
11	C₂×C₄⋊#1055	C₄	1	3	yes	—
12	C₂×C₄×#1055	C₂²	1	3	yes	—
13	C₂²⋊#6092	C₂²	1	1	yes	characteristic
14	C₂²⋊#6092	C₂²	1	3	yes	—
15	#11718231	C₂²	1	6	yes	—
16	C₄⋊#6092	C₂²	1	6	yes	—
17	C₄⋊#6092	C₂²	1	6	yes	—
18	C₄×#6092	C₂²	1	6	yes	—
19	C₂²×Q₈×C₁₃	C₈	1	1	yes	characteristic
20	C₄×Q₈×C₁₃	C₈	1	3	yes	—
21	C₄×Q₈×C₁₃	C₈	1	1	yes	characteristic
22	C₄×Q₈×C₁₃	C₂×C₄	1	3	yes	—
23	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₃×C₄⋊C₄	C₈	1	3	yes	—
25	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
26	C₂²×#1055	C₂×C₄	1	3	yes	—
27	C₂²×#1055	C₂³	1	1	yes	characteristic
28	C₂×#6092	C₂³	1	2	yes	—
29	C₂×#6092	C₂³	1	6	yes	—
30	C₄⋊#1055	C₂×C₄	1	3	yes	—
31	C₄⋊#1055	C₂×C₄	1	6	yes	—
32	C₄⋊#1055	C₂×C₄	1	3	yes	—
33	C₄×#1055	C₂×C₄	1	3	yes	—
34	C₄×#1055	C₂×C₄	1	3	yes	—
35	C₄×#1055	C₂³	1	3	yes	—
36	C₄×#1055	C₂³	1	3	yes	—
37	#33498	—	13	1	no	maximal, Sylow
38	C₂×Q₈×C₁₃	—	2	4	no	—
39	C₂×Q₈×C₁₃	C₂×C₈	1	3	yes	—
40	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
42	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
43	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₁₃×C₄⋊C₄	C₂×C₈	1	6	yes	—
45	C₁₃×C₄⋊C₄	C₂×C₈	1	3	yes	—
46	C₁₃×C₄⋊C₄	C₂×C₈	1	3	yes	—
47	C₄²×C₁₃	C₂×C₈	1	3	yes	—
48	C₄²×C₁₃	C₂×C₈	1	3	yes	—
49	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
50	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
51	C₂×#1055	—	2	2	no	—
52	C₂×#1055	—	4	3	no	—
53	C₂×#1055	C₂²×C₄	1	3	yes	—
54	C₂×#1055	C₂²×C₄	1	3	yes	—
55	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
56	C₂×#1055	C₂⁴	1	1	yes	characteristic
57	#6092	—	2	2	no	—
58	#6092	—	2	6	no	—
59	C₂×C₈×Q₈	—	13	1	no	—
60	#2183	—	13	6	no	—
61	Q₈×C₁₆	—	13	2	no	—
62	#2163	—	13	3	no	—
63	#2162	—	13	3	no	—
64	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
65	Q₈×C₁₃	—	2	6	no	—
66	Q₈×C₁₃	M₃₂	1	4	yes	—
67	C₂×C₄×C₁₃	—	2	6	no	—
68	C₂×C₄×C₁₃	—	2	1	no	—
69	C₂×C₄×C₁₃	—	2	1	no	—
70	C₂×C₄×C₁₃	—	2	6	no	—
71	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
73	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
74	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
75	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
76	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
77	#1055	—	2	1	no	—
78	#1055	—	2	3	no	—
79	#1055	—	2	3	no	—
80	#1055	2₋¹⁺⁴	1	2	yes	—
81	C₂×C₄×Q₈	#261	1	1	yes	characteristic
82	C₈×Q₈	—	13	3	no	—
83	C₈×Q₈	—	13	1	no	—
84	C₈×Q₈	—	13	1	no	—
85	C₈×Q₈	—	13	3	no	—
86	C₂×C₄⋊C₈	—	13	3	no	—
87	C₂×C₄×C₈	—	13	3	no	—
88	C₄⋊C₁₆	—	13	6	no	—
89	C₄⋊C₁₆	—	13	6	no	—
90	C₂²⋊C₁₆	—	13	1	no	—
91	C₂²⋊C₁₆	—	13	3	no	—
92	#346	—	13	6	no	—
93	C₄×C₁₆	—	13	6	no	—
94	C₂²×C₁₃	—	2	1	no	—
95	C₂²×C₁₃	—	2	1	no	—
96	C₂²×C₁₃	#504	1	2	yes	—
97	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
98	C₄×C₁₃	—	2	3	no	—
99	C₄×C₁₃	—	2	3	no	—
100	C₄×C₁₃	—	2	3	no	—
101	C₄×C₁₃	—	2	1	no	—
102	C₄×C₁₃	C₂×M₃₂	1	6	yes	—
103	C₄×C₁₃	#519	1	1	yes	characteristic
104	C₄×C₁₃	#568	1	1	yes	characteristic
105	C₂²×Q₈	#1057	1	1	yes	characteristic
106	C₂²×C₈	—	13	1	no	—
107	C₂²×C₈	—	13	3	no	—
108	C₄×Q₈	#1057	1	3	yes	—
109	C₄×Q₈	#1057	1	1	yes	characteristic
110	C₄×Q₈	C₂×#261	1	3	yes	—
111	C₄×Q₈	C₂×#261	1	1	yes	characteristic
112	C₂×C₄⋊C₄	#1057	1	3	yes	—
113	C₂×C₄²	C₂×#261	1	3	yes	—
114	C₂×C₁₆	—	13	2	no	—
115	C₂×C₁₆	—	13	6	no	—
116	C₄⋊C₈	—	13	3	no	—
117	C₄⋊C₈	—	13	6	no	—
118	C₄⋊C₈	—	13	3	no	—
119	C₄×C₈	—	13	3	no	—
120	C₄×C₈	—	13	3	no	—
121	C₄×C₈	—	13	3	no	—
122	C₄×C₈	—	13	3	no	—
123	C₂×C₁₃	—	2	2	no	—
124	C₂×C₁₃	#2982	1	1	yes	characteristic
125	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
126	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
127	C₂×Q₈	—	2	4	no	—
128	C₂×Q₈	C₂×#1057	1	3	yes	—
129	C₂×Q₈	C₂×#1057	1	1	yes	characteristic
130	C₂²×C₄	C₂×#1057	1	3	yes	—
131	C₂²×C₄	C₂×#1057	1	3	yes	—
132	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
133	C₂×C₈	—	13	1	no	—
134	C₂×C₈	—	13	1	no	—
135	C₂×C₈	—	13	3	no	—
136	C₂×C₈	—	13	3	no	—
137	C₂×C₈	—	26	2	no	—
138	C₂×C₈	—	52	3	no	—
139	C₄⋊C₄	C₂×#1057	1	6	yes	—
140	C₄⋊C₄	C₂×#1057	1	3	yes	—
141	C₄⋊C₄	C₂×#1057	1	3	yes	—
142	C₄²	C₂×#1057	1	3	yes	—
143	C₄²	C₂×#1057	1	3	yes	—
144	C₄²	C₂²×#261	1	3	yes	—
145	C₄²	C₂²×#261	1	3	yes	—
146	C₁₆	—	26	2	no	—
147	C₁₆	—	26	6	no	—
148	C₁₃	#33498	1	1	yes	characteristic, complemented, Sylow
149	C₂³	C₂²×#1057	1	1	yes	characteristic
150	Q₈	—	2	6	no	—
151	Q₈	#88106	1	4	yes	—
152	C₂×C₄	—	2	6	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	6	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
157	C₂×C₄	C₂²×#1057	1	3	yes	—
158	C₂×C₄	C₂²×#1057	1	3	yes	—
159	C₂×C₄	C₂²×#1057	1	3	yes	—
160	C₂×C₄	C₂²×#1057	1	3	yes	—
161	C₂×C₄	C₂³×#261	1	1	yes	characteristic
162	C₈	—	13	2	no	—
163	C₈	—	26	3	no	—
164	C₈	—	26	3	no	—
165	C₈	—	26	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	#11719162	1	2	yes	—
169	C₂²	C₂³×#1057	1	1	yes	characteristic
170	C₄	—	2	3	no	—
171	C₄	—	2	3	no	—
172	C₄	—	2	3	no	—
173	C₄	—	2	1	no	—
174	C₄	C₂×#88106	1	6	yes	—
175	C₄	#11719625	1	1	yes	characteristic
176	C₄	#11719627	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	#13131729	1	1	yes	characteristic
179	C₂	C₂²×#88106	1	1	yes	characteristic
180	C₂	C₂×#11719162	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₂×C₄×Q₈×C₁₃	C₄
prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×C₄×Q₈	#261
hypocenter	C₁₃	#33498
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #13131251	C₂	semidirect
2. Q₈×#6092	C₂	semidirect
148. C₁₃	#33498	semidirect

Maximal subgroups

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

subgroup	index
#13131251	2
Q₈×#6092	2
#13131197	2
#13131193	2
C₂×Q₈×#1055	2
#33498	13