Subgroups — #16807272

Order 3328.

602 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#6092	C₂	1	2	yes	maximal, complemented
2	#13131215	C₂	1	6	yes	maximal, complemented
3	#13131196	C₂	1	3	yes	maximal, complemented
4	C₄×D₁₃⋊C₁₆	C₂	1	3	yes	maximal, complemented
5	C₈×#6131	C₂	1	1	yes	maximal, characteristic
6	C₄×#6131	C₄	1	1	yes	characteristic
7	C₂×D₁₃⋊C₁₆	C₂²	1	6	yes	—
8	C₈×Q₈×C₁₃	C₄	1	1	yes	characteristic
9	Q₈×#1055	C₂²	1	1	yes	characteristic
10	#11718495	C₂²	1	3	yes	—
11	#11718494	C₄	1	3	yes	—
12	C₈×D₅₂	C₄	1	3	yes	—
13	C₄×C₈×D₁₃	C₂²	1	3	yes	—
14	#11718259	C₂²	1	6	yes	—
15	#11718257	C₂²	1	1	yes	characteristic
16	#11718257	C₂²	1	3	yes	—
17	C₄⋊#6092	C₂²	1	6	yes	—
18	C₄×#6092	C₂²	1	6	yes	—
19	C₂×#6131	C₈	1	1	yes	characteristic
20	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	Q₈×Dic₁₃	C₈	1	1	yes	characteristic
22	C₂×C₈×D₁₃	C₂×C₄	1	3	yes	—
23	C₂×C₈×D₁₃	C₂³	1	3	yes	—
24	#88147	C₈	1	3	yes	—
25	#88146	C₈	1	3	yes	—
26	C₄×D₅₂	C₂×C₄	1	3	yes	—
27	C₄²×D₁₃	C₂×C₄	1	3	yes	—
28	C₂×#6092	C₂³	1	6	yes	—
29	C₂×#6092	C₂³	1	2	yes	—
30	D₁₃⋊C₁₆	—	2	12	no	—
31	C₁₃×C₄⋊C₈	C₂×C₄	1	3	yes	—
32	C₄×C₈×C₁₃	C₂×C₄	1	3	yes	—
33	#88060	C₂×C₄	1	6	yes	—
34	C₈×Dic₁₃	C₂×C₄	1	3	yes	—
35	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
36	C₄⋊#1055	C₂×C₄	1	3	yes	—
37	C₄×#1055	C₂³	1	3	yes	—
38	C₁₆×#41	—	13	1	no	maximal, Sylow
39	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
40	#6131	C₁₆	1	8	yes	complemented
41	C₂×D₅₂	C₂×C₈	1	3	yes	—
42	C₂×C₄×D₁₃	C₂×C₈	1	3	yes	—
43	C₂×C₄×D₁₃	C₂×C₈	1	3	yes	—
44	C₂×C₄×D₁₃	C₂²×C₄	1	3	yes	—
45	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
46	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₁₃×C₄⋊C₄	C₂×C₈	1	3	yes	—
48	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
49	#6103	C₂×C₈	1	6	yes	—
50	C₄⋊Dic₁₃	C₂×C₈	1	3	yes	—
51	C₄×Dic₁₃	C₂×C₈	1	3	yes	—
52	C₄×Dic₁₃	C₂×C₈	1	3	yes	—
53	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×#1055	C₂²×C₄	1	3	yes	—
55	C₂×#1055	C₂⁴	1	1	yes	characteristic
56	C₈×D₁₃	—	2	6	no	—
57	C₈×D₁₃	—	2	6	no	—
58	#6092	—	2	6	no	—
59	#6092	#41	1	4	yes	complemented
60	C₈×#41	—	13	1	no	—
61	Q₈×C₁₆	—	13	2	no	—
62	D₄×C₁₆	—	13	6	no	—
63	#2162	—	13	3	no	—
64	C₂×C₄×C₁₆	—	13	3	no	—
65	C₂²×D₁₃	C₂²×C₈	1	3	yes	—
66	Q₈×C₁₃	C₂×C₁₆	1	4	yes	complemented
67	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
68	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
69	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
70	C₂×Dic₁₃	C₂²×C₈	1	3	yes	—
71	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
72	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
73	D₅₂	C₂×C₁₆	1	12	yes	complemented
74	C₄×D₁₃	—	2	3	no	—
75	C₄×D₁₃	—	2	3	no	—
76	C₄×D₁₃	—	2	6	no	—
77	C₄×D₁₃	C₂×C₁₆	1	12	yes	complemented
78	C₈×C₁₃	—	2	3	no	—
79	C₈×C₁₃	#131	1	2	yes	—
80	#1055	—	2	3	no	—
81	#1055	C₂×#41	1	2	yes	—
82	C₄×#41	—	13	1	no	—
83	C₂²×C₁₆	—	13	6	no	—
84	C₈×Q₈	—	13	1	no	—
85	C₈×Q₈	#261	1	1	yes	characteristic
86	C₈×D₄	—	13	3	no	—
87	C₈×D₄	—	13	3	no	—
88	#431	—	13	3	no	—
89	C₂×C₄×C₈	—	13	3	no	—
90	C₄⋊C₁₆	—	13	6	no	—
91	C₂²⋊C₁₆	—	13	6	no	—
92	C₄×C₁₆	—	13	1	no	—
93	C₄×C₁₆	—	13	6	no	—
94	C₄×C₁₆	—	13	3	no	—
95	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
96	C₂×D₁₃	—	2	3	no	—
97	C₂×D₁₃	—	2	3	no	—
98	C₂×D₁₃	C₂²×C₁₆	1	6	yes	—
99	C₄×C₁₃	—	2	3	no	—
100	C₄×C₁₃	C₂²×C₁₆	1	6	yes	—
101	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
102	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
103	Dic₁₃	—	2	3	no	—
104	Dic₁₃	C₂²×C₁₆	1	2	yes	—
105	Dic₁₃	#504	1	2	yes	—
106	C₂×#41	—	13	1	no	—
107	C₂²×C₈	—	13	3	no	—
108	C₂²×C₈	—	13	3	no	—
109	C₄×Q₈	—	13	1	no	—
110	C₄×Q₈	C₂×#261	1	1	yes	characteristic
111	C₄×D₄	—	13	3	no	—
112	C₄×D₄	—	13	3	no	—
113	#117	—	13	3	no	—
114	C₂×C₄²	—	13	3	no	—
115	C₂×C₁₆	—	13	2	no	—
116	C₂×C₁₆	—	13	6	no	—
117	C₂×C₁₆	—	26	12	no	—
118	C₄⋊C₈	—	13	3	no	—
119	C₄⋊C₈	C₂×#261	1	3	yes	—
120	C₂²⋊C₈	—	13	6	no	—
121	C₄×C₈	—	13	3	no	—
122	C₄×C₈	—	13	1	no	—
123	C₄×C₈	—	13	3	no	—
124	C₄×C₈	C₂×#261	1	3	yes	—
125	C₂×C₁₃	C₈×#41	1	1	yes	characteristic
126	C₂×C₁₃	C₂³×C₁₆	1	1	yes	characteristic
127	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
128	D₁₃	—	2	6	no	—
129	#41	—	13	8	no	—
130	C₂×Q₈	D₁₃⋊C₈	1	1	yes	characteristic
131	C₂×D₄	—	13	3	no	—
132	C₂²×C₄	—	13	3	no	—
133	C₂²×C₄	—	13	3	no	—
134	C₂²×C₄	—	13	3	no	—
135	C₂×C₈	—	13	1	no	—
136	C₂×C₈	—	13	3	no	—
137	C₂×C₈	—	26	6	no	—
138	C₂×C₈	—	26	6	no	—
139	C₂×C₈	C₂²×#261	1	3	yes	—
140	C₂×C₈	C₂²×#261	1	1	yes	characteristic
141	C₄⋊C₄	—	13	3	no	—
142	C₄⋊C₄	D₁₃⋊C₈	1	3	yes	—
143	C₂²⋊C₄	—	13	6	no	—
144	C₄²	—	13	3	no	—
145	C₄²	—	13	3	no	—
146	C₄²	—	13	1	no	—
147	C₄²	C₂²×#261	1	3	yes	—
148	C₁₆	—	13	4	no	—
149	C₁₆	—	26	6	no	—
150	C₁₃	C₁₆×#41	1	1	yes	characteristic, complemented, Sylow
151	C₂³	—	13	3	no	—
152	Q₈	D₁₃⋊C₁₆	1	4	yes	complemented
153	D₄	—	13	12	no	—
154	C₂×C₄	—	13	1	no	—
155	C₂×C₄	—	13	1	no	—
156	C₂×C₄	—	13	12	no	—
157	C₂×C₄	—	13	3	no	—
158	C₂×C₄	—	26	6	no	—
159	C₂×C₄	—	26	3	no	—
160	C₂×C₄	—	26	3	no	—
161	C₂×C₄	C₂×D₁₃⋊C₈	1	3	yes	—
162	C₂×C₄	C₂×D₁₃⋊C₈	1	3	yes	—
163	C₂×C₄	C₂³×#261	1	1	yes	characteristic
164	C₈	—	2	3	no	—
165	C₈	—	13	2	no	—
166	C₈	—	26	3	no	—
167	C₈	#88240	1	2	yes	—
168	C₂²	—	13	6	no	—
169	C₂²	—	26	3	no	—
170	C₂²	—	26	3	no	—
171	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
172	C₄	—	2	3	no	—
173	C₄	—	13	2	no	—
174	C₄	—	13	2	no	—
175	C₄	—	26	3	no	—
176	C₄	C₂×D₁₃⋊C₁₆	1	6	yes	—
177	C₄	C₂×#88240	1	1	yes	characteristic
178	C₄	#41×#261	1	1	yes	characteristic
179	C₂	—	26	6	no	—
180	C₂	#13131642	1	1	yes	characteristic
181	C₂	C₂²×D₁₃⋊C₁₆	1	1	yes	characteristic
182	C₂	D₁₃⋊#504	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×#261
commutator	C₂×C₁₃	C₂³×C₁₆
Frattini	C₂×C₄	C₂³×#261
Fitting	C₈×Q₈×C₁₃	C₄
socle	C₂²×C₁₃	C₂³×C₈
prime radical	C₂×#6131	C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₈×Q₈	#261
hypocenter	C₁₃	C₁₆×#41
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×#6092	C₂	semidirect
2. #13131215	C₂	semidirect
3. #13131196	C₂	semidirect
4. C₄×D₁₃⋊C₁₆	C₂	semidirect
40. #6131	C₁₆	semidirect
59. #6092	#41	semidirect
66. Q₈×C₁₃	C₂×C₁₆	semidirect
73. D₅₂	C₂×C₁₆	semidirect
77. C₄×D₁₃	C₂×C₁₆	semidirect
150. C₁₃	C₁₆×#41	semidirect
152. Q₈	D₁₃⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#6092	2
#13131215	2
#13131196	2
C₄×D₁₃⋊C₁₆	2
C₈×#6131	2
C₁₆×#41	13