Subgroups — #16016414

Order 3328.

506 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116745	C₂	1	4	yes	maximal, complemented
2	Q₈×#6090	C₂	1	4	yes	maximal, complemented
3	#13116595	C₂	1	2	yes	maximal
4	#13116592	C₂	1	1	yes	maximal, characteristic
5	#13116433	C₂	1	2	yes	maximal
6	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#2960	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#533	C₄	1	1	yes	characteristic
9	C₁₃×#446	C₄	1	4	yes	—
10	C₈×Q₈×C₁₃	C₂²	1	4	yes	—
11	C₁₃×#424	C₄	1	2	yes	—
12	C₁₃×#423	C₄	1	1	yes	characteristic
13	C₁₃×#405	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88052	C₂²	1	2	yes	—
16	C₂²×#6090	C₂²	1	1	yes	characteristic
17	C₂²×#6090	C₂²	1	1	yes	characteristic
18	C₄⋊#6090	C₂²	1	4	yes	—
19	C₄⋊#6090	C₂²	1	8	yes	—
20	#11718053	C₂²	1	4	yes	—
21	C₄×#6090	C₂²	1	2	yes	—
22	C₄×#6090	C₂²	1	2	yes	—
23	C₄×#6090	C₂²	1	4	yes	—
24	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	C₁₃×C₄⋊Q₈	C₈	1	2	yes	—
28	C₁₃×#125	C₈	1	2	yes	—
29	C₁₃×C₂²⋊Q₈	C₈	1	4	yes	—
30	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
31	C₁₃×#117	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
34	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
35	C₁₃×#97	C₂×C₄	1	4	yes	—
36	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₁₃	C₂³	1	2	yes	—
38	C₄×C₈×C₁₃	C₂³	1	4	yes	—
39	#88052	—	2	4	no	—
40	C₂×#6090	—	2	2	no	—
41	C₂×#6090	Q₈	1	4	yes	complemented
42	C₂×#6090	C₂³	1	2	yes	—
43	C₂×#6090	C₂³	1	4	yes	—
44	C₂×#6090	C₂³	1	1	yes	characteristic
45	C₂×#6090	C₂³	1	1	yes	characteristic
46	#33474	—	13	1	no	maximal, Sylow
47	C₂×Q₈×C₁₃	C₂×C₈	1	4	yes	—
48	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₁₃×M₁₆	—	2	4	no	—
51	C₂×C₈×C₁₃	—	2	2	no	—
52	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
53	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
54	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
56	C₂×C₈×C₁₃	#41	1	2	yes	—
57	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
58	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
59	C₁₃×C₄⋊C₄	C₂×C₈	1	8	yes	—
60	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
61	C₁₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
62	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
63	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
64	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
65	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
66	#6090	—	2	2	no	—
67	#6090	—	2	4	no	—
68	#6090	C₂×Q₈	1	2	yes	—
69	#6090	C₂×Q₈	1	2	yes	—
70	#2960	D₁₃	1	1	yes	characteristic
71	#2183	—	13	4	no	—
72	Q₈×C₁₆	—	13	4	no	—
73	#2151	—	13	2	no	—
74	#2150	—	13	1	no	—
75	#2108	—	13	2	no	—
76	C₂×C₄×C₁₆	—	13	1	no	—
77	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
78	Q₈×C₁₃	—	2	8	no	—
79	C₂×C₄×C₁₃	—	2	4	no	—
80	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
81	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
82	C₂×C₄×C₁₃	C₂²×C₈	1	8	yes	—
83	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
85	C₂×C₄×C₁₃	#131	1	2	yes	—
86	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
87	C₈×C₁₃	—	2	2	no	—
88	C₈×C₁₃	—	2	4	no	—
89	C₈×C₁₃	C₄×Q₈	1	2	yes	—
90	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
91	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
92	#533	Dic₁₃	1	1	yes	characteristic
93	C₂×M₃₂	—	13	2	no	—
94	C₂²×C₁₆	—	13	1	no	—
95	C₂²×C₁₆	—	13	1	no	—
96	#446	Dic₁₃	1	4	yes	—
97	C₈×Q₈	C₂×D₁₃	1	4	yes	—
98	#424	Dic₁₃	1	2	yes	—
99	#423	Dic₁₃	1	1	yes	characteristic
100	#405	C₂×D₁₃	1	2	yes	—
101	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
102	C₄⋊C₁₆	—	13	4	no	—
103	C₄⋊C₁₆	—	13	8	no	—
104	#346	—	13	4	no	—
105	C₄×C₁₆	—	13	2	no	—
106	C₄×C₁₆	—	13	2	no	—
107	C₄×C₁₆	—	13	4	no	—
108	C₂²×C₁₃	—	2	2	no	—
109	C₂²×C₁₃	C₈×Q₈	1	2	yes	—
110	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
111	C₄×C₁₃	—	2	2	no	—
112	C₄×C₁₃	—	2	8	no	—
113	C₄×C₁₃	C₈×Q₈	1	2	yes	—
114	C₄×C₁₃	#504	1	4	yes	—
115	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
116	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
117	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
118	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
120	C₄⋊Q₈	#1055	1	2	yes	—
121	#125	#1055	1	2	yes	—
122	C₂²⋊Q₈	#1055	1	4	yes	—
123	C₄×Q₈	C₂×Dic₁₃	1	4	yes	—
124	#117	C₂×Dic₁₃	1	2	yes	—
125	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
126	M₃₂	—	26	4	no	—
127	C₂×C₁₆	—	13	4	no	—
128	C₂×C₁₆	—	13	1	no	—
129	C₂×C₁₆	—	13	1	no	—
130	C₂×C₁₆	—	13	2	no	—
131	C₂×C₁₆	—	13	4	no	—
132	C₂×C₁₆	—	26	2	no	—
133	C₄⋊C₈	C₂×Dic₁₃	1	4	yes	—
134	C₄⋊C₈	C₂×Dic₁₃	1	8	yes	—
135	#97	C₂×Dic₁₃	1	4	yes	—
136	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
137	C₄×C₈	C₂²×D₁₃	1	2	yes	—
138	C₄×C₈	C₂²×D₁₃	1	4	yes	—
139	C₂×C₁₃	—	2	2	no	—
140	C₂×C₁₃	C₂×C₈×Q₈	1	1	yes	characteristic
141	C₂×C₁₃	C₂×#504	1	2	yes	—
142	C₂×Q₈	C₂×#1055	1	4	yes	—
143	C₂²×C₄	C₂×#1055	1	2	yes	—
144	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
145	M₁₆	—	2	4	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	Q₈×D₁₃	1	2	yes	—
148	C₂×C₈	#6131	1	2	yes	—
149	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
150	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
151	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
152	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
153	C₄⋊C₄	C₂×#1055	1	4	yes	—
154	C₄⋊C₄	C₂×#1055	1	8	yes	—
155	C₄⋊C₄	C₂×#1055	1	4	yes	—
156	C₂²⋊C₄	C₂×#1055	1	4	yes	—
157	C₄²	C₂×#1055	1	1	yes	characteristic
158	C₄²	C₂×#1055	1	1	yes	characteristic
159	C₄²	C₂²×Dic₁₃	1	2	yes	—
160	C₄²	C₂²×Dic₁₃	1	4	yes	—
161	C₁₆	—	13	2	no	—
162	C₁₆	—	13	2	no	—
163	C₁₆	—	26	2	no	—
164	C₁₆	—	26	4	no	—
165	C₁₃	#33474	1	1	yes	characteristic, complemented, Sylow
166	C₂³	C₂²×#1055	1	1	yes	characteristic
167	Q₈	—	2	8	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	C₂²×#1055	1	2	yes	—
170	C₂×C₄	C₂²×#1055	1	8	yes	—
171	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
172	C₂×C₄	C₂²×#1055	1	2	yes	—
173	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
174	C₂×C₄	#88202	1	2	yes	—
175	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	2	4	no	—
178	C₈	Q₈×Dic₁₃	1	2	yes	—
179	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
180	C₈	C₂×#6131	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	Q₈×#1055	1	2	yes	—
183	C₂²	C₂³×#1055	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	2	8	no	—
186	C₄	Q₈×#1055	1	2	yes	—
187	C₄	#11718801	1	4	yes	—
188	C₄	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
189	C₄	C₂×#88202	1	1	yes	characteristic
190	C₂	—	2	2	no	—
191	C₂	C₂×Q₈×#1055	1	1	yes	characteristic
192	C₂	C₂×#11718801	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	#33474
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
2. Q₈×#6090	C₂	semidirect
41. C₂×#6090	Q₈	semidirect
165. C₁₃	#33474	semidirect

Maximal subgroups

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

subgroup	index
#13116745	2
Q₈×#6090	2
#13116595	2
#13116592	2
#13116433	2
C₂×C₄×#6090	2
C₁₃×#2960	2
#33474	13