Subgroups — #16016420

Order 3328.

490 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116745	C₂	1	8	yes	maximal, complemented
2	#13116595	C₂	1	2	yes	maximal
3	#13116592	C₂	1	1	yes	maximal, characteristic
4	#13116433	C₂	1	2	yes	maximal
5	C₂×#11718053	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#2960	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#533	C₄	1	1	yes	characteristic
8	C₁₃×#446	C₄	1	4	yes	—
9	C₈×Q₈×C₁₃	C₂²	1	4	yes	—
10	C₁₃×#424	C₄	1	2	yes	—
11	C₁₃×#423	C₄	1	1	yes	characteristic
12	C₁₃×#405	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×#88052	C₂²	1	2	yes	—
15	C₂²×#6090	C₂²	1	1	yes	characteristic
16	C₂²×#6090	C₂²	1	1	yes	characteristic
17	C₄⋊#6090	C₂²	1	4	yes	—
18	C₄⋊#6090	C₂²	1	8	yes	—
19	#11718053	C₂²	1	4	yes	—
20	#11718053	C₂²	1	4	yes	—
21	C₄×#6090	C₂²	1	4	yes	—
22	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
25	C₁₃×C₄⋊Q₈	C₈	1	2	yes	—
26	C₁₃×#125	C₈	1	2	yes	—
27	C₁₃×C₂²⋊Q₈	C₈	1	4	yes	—
28	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
29	C₁₃×#117	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
32	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
33	C₁₃×#97	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₁₃	C₂³	1	2	yes	—
36	C₄×C₈×C₁₃	C₂³	1	4	yes	—
37	#88052	—	2	4	no	—
38	C₂×#6090	—	2	2	no	—
39	C₂×#6090	Q₈	1	4	yes	complemented
40	C₂×#6090	C₂³	1	2	yes	—
41	C₂×#6090	C₂³	1	4	yes	—
42	C₂×#6090	C₂³	1	2	yes	—
43	#33475	—	13	1	no	maximal, Sylow
44	C₂×Q₈×C₁₃	C₂×C₈	1	4	yes	—
45	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₁₃×M₁₆	—	2	4	no	—
48	C₂×C₈×C₁₃	—	2	2	no	—
49	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
50	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
51	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₁₃	#41	1	2	yes	—
54	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
55	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
56	C₁₃×C₄⋊C₄	C₂×C₈	1	8	yes	—
57	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
58	C₁₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
59	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
60	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
61	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
62	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
63	#6090	—	2	2	no	—
64	#6090	—	2	4	no	—
65	#6090	C₂×Q₈	1	4	yes	—
66	#2960	D₁₃	1	1	yes	characteristic
67	#2183	—	13	8	no	—
68	#2151	—	13	2	no	—
69	#2150	—	13	1	no	—
70	#2108	—	13	2	no	—
71	C₂×#346	—	13	1	no	—
72	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
73	Q₈×C₁₃	—	4	4	no	—
74	C₂×C₄×C₁₃	—	4	2	no	—
75	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
76	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
78	C₂×C₄×C₁₃	C₂²×C₈	1	8	yes	—
79	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃	#131	1	2	yes	—
81	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
82	C₈×C₁₃	—	2	2	no	—
83	C₈×C₁₃	—	2	4	no	—
84	C₈×C₁₃	C₄×Q₈	1	2	yes	—
85	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
86	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
87	#533	Dic₁₃	1	1	yes	characteristic
88	C₂×M₃₂	—	13	2	no	—
89	C₂²×C₁₆	—	13	1	no	—
90	C₂²×C₁₆	—	13	1	no	—
91	#446	Dic₁₃	1	4	yes	—
92	C₈×Q₈	C₂×D₁₃	1	4	yes	—
93	#424	Dic₁₃	1	2	yes	—
94	#423	Dic₁₃	1	1	yes	characteristic
95	#405	C₂×D₁₃	1	2	yes	—
96	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
97	C₄⋊C₁₆	—	13	4	no	—
98	C₄⋊C₁₆	—	13	8	no	—
99	#346	—	13	4	no	—
100	#346	—	13	4	no	—
101	C₄×C₁₆	—	13	4	no	—
102	C₂²×C₁₃	—	2	2	no	—
103	C₂²×C₁₃	C₈×Q₈	1	2	yes	—
104	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
105	C₄×C₁₃	—	2	2	no	—
106	C₄×C₁₃	—	2	2	no	—
107	C₄×C₁₃	—	2	8	no	—
108	C₄×C₁₃	C₈×Q₈	1	2	yes	—
109	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
110	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
111	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
112	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
114	C₄⋊Q₈	#1055	1	2	yes	—
115	#125	#1055	1	2	yes	—
116	C₂²⋊Q₈	#1055	1	4	yes	—
117	C₄×Q₈	C₂×Dic₁₃	1	4	yes	—
118	#117	C₂×Dic₁₃	1	2	yes	—
119	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
120	M₃₂	—	26	4	no	—
121	C₂×C₁₆	—	13	4	no	—
122	C₂×C₁₆	—	13	2	no	—
123	C₂×C₁₆	—	13	4	no	—
124	C₂×C₁₆	—	13	2	no	—
125	C₂×C₁₆	—	26	2	no	—
126	C₄⋊C₈	C₂×Dic₁₃	1	4	yes	—
127	C₄⋊C₈	C₂×Dic₁₃	1	8	yes	—
128	#97	C₂×Dic₁₃	1	4	yes	—
129	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
130	C₄×C₈	C₂²×D₁₃	1	4	yes	—
131	C₄×C₈	C₂²×D₁₃	1	2	yes	—
132	C₂×C₁₃	—	2	2	no	—
133	C₂×C₁₃	C₂×C₈×Q₈	1	1	yes	characteristic
134	C₂×C₁₃	#3407	1	2	yes	—
135	C₂×Q₈	C₂×#1055	1	4	yes	—
136	C₂²×C₄	C₂×#1055	1	2	yes	—
137	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
138	M₁₆	—	2	4	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	Q₈×D₁₃	1	2	yes	—
141	C₂×C₈	#6131	1	2	yes	—
142	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
143	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
144	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
145	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
146	C₄⋊C₄	C₂×#1055	1	4	yes	—
147	C₄⋊C₄	C₂×#1055	1	8	yes	—
148	C₄⋊C₄	C₂×#1055	1	4	yes	—
149	C₂²⋊C₄	C₂×#1055	1	4	yes	—
150	C₄²	C₂×#1055	1	1	yes	characteristic
151	C₄²	C₂×#1055	1	1	yes	characteristic
152	C₄²	C₂²×Dic₁₃	1	2	yes	—
153	C₄²	C₂²×Dic₁₃	1	4	yes	—
154	C₁₆	—	13	4	no	—
155	C₁₆	—	26	2	no	—
156	C₁₆	—	26	4	no	—
157	C₁₃	#33475	1	1	yes	characteristic, complemented, Sylow
158	C₂³	C₂²×#1055	1	1	yes	characteristic
159	Q₈	—	4	4	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	C₂²×#1055	1	2	yes	—
162	C₂×C₄	C₂²×#1055	1	2	yes	—
163	C₂×C₄	C₂²×#1055	1	8	yes	—
164	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
165	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
166	C₂×C₄	#88202	1	2	yes	—
167	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	2	4	no	—
170	C₈	Q₈×Dic₁₃	1	2	yes	—
171	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
172	C₈	C₂×#6131	1	1	yes	characteristic
173	C₂²	—	2	2	no	—
174	C₂²	Q₈×#1055	1	2	yes	—
175	C₂²	C₂³×#1055	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	2	8	no	—
179	C₄	Q₈×#1055	1	2	yes	—
180	C₄	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
181	C₄	C₂×#88202	1	1	yes	characteristic
182	C₂	—	2	2	no	—
183	C₂	C₂×Q₈×#1055	1	1	yes	characteristic
184	C₂	#13127939	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#2960	C₂
prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2960	D₁₃
hypocenter	C₁₃	#33475
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
1. #13116745	C₂	semidirect
39. C₂×#6090	Q₈	semidirect
157. C₁₃	#33475	semidirect

Maximal subgroups

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

subgroup	index
#13116745	2
#13116595	2
#13116592	2
#13116433	2
C₂×#11718053	2
C₁₃×#2960	2
#33475	13