Subgroups — #16015569

Order 3328.

818 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718801	C₂	1	2	yes	maximal
2	#13116433	C₂	1	6	yes	maximal, complemented
3	C₄×#88052	C₂	1	3	yes	maximal, complemented
4	C₂×#11718053	C₂	1	3	yes	maximal, complemented
5	C₄×C₁₃×#131	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
7	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
8	C₄×C₁₃×#41	C₄	1	1	yes	characteristic
9	C₁₃×#405	C₄	1	3	yes	—
10	C₁₃×#405	C₂²	1	3	yes	—
11	C₄×C₁₃×M₁₆	C₄	1	3	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
13	#11718801	C₄	1	16	yes	complemented
14	C₂×#88052	C₂²	1	6	yes	—
15	C₂²×#6090	C₂²	1	6	yes	—
16	#11718053	C₂²	1	1	yes	characteristic
17	#11718053	C₂²	1	3	yes	—
18	#11718053	C₂²	1	6	yes	—
19	C₄×#6090	C₂²	1	6	yes	—
20	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
21	C₁₃×#131	C₈	1	8	yes	—
22	C₁₃×#131	C₂×C₄	1	8	yes	—
23	C₂×C₁₃×M₁₆	C₂×C₄	1	3	yes	—
24	C₂×C₁₃×M₁₆	C₂×C₄	1	3	yes	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
26	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
27	C₄×Q₈×C₁₃	C₈	1	2	yes	—
28	C₄×D₄×C₁₃	C₈	1	6	yes	—
29	C₁₃×#117	C₂×C₄	1	3	yes	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
31	C₁₃×#97	C₂×C₄	1	6	yes	—
32	C₄×C₈×C₁₃	C₂×C₄	1	3	yes	—
33	C₄×C₈×C₁₃	C₂×C₄	1	3	yes	—
34	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₄×C₈×C₁₃	C₂³	1	3	yes	—
36	#88052	C₂×C₄	1	24	yes	complemented
37	C₂×#6090	C₂×C₄	1	24	yes	complemented
38	C₂×#6090	C₂³	1	2	yes	—
39	C₂×#6090	C₂³	1	6	yes	—
40	#33326	—	13	1	no	maximal, Sylow
41	C₁₃×#41	C₄²	1	4	yes	—
42	C₁₃×#41	C₂×C₈	1	4	yes	—
43	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
44	C₂×D₄×C₁₃	C₂×C₈	1	3	yes	—
45	C₂²×C₄×C₁₃	C₂×C₈	1	6	yes	—
46	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
47	C₁₃×M₁₆	C₄²	1	12	yes	—
48	C₁₃×M₁₆	C₂×C₈	1	12	yes	—
49	C₂×C₈×C₁₃	C₂×C₈	1	12	yes	—
50	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
51	C₂×C₈×C₁₃	C₂²×C₄	1	12	yes	—
52	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
54	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
55	C₁₃×C₄⋊C₄	C₂×C₈	1	6	yes	—
56	C₁₃×C₂²⋊C₄	C₂×C₈	1	6	yes	—
57	C₄²×C₁₃	C₂×C₈	1	6	yes	—
58	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
60	#6090	C₂²×C₄	1	4	yes	—
61	#6090	C₂²×C₄	1	12	yes	—
62	C₂×#504	—	13	2	no	—
63	C₄×#131	D₁₃	1	1	yes	characteristic
64	#2108	—	13	6	no	—
65	C₄×M₃₂	—	13	3	no	—
66	C₂×#346	—	13	3	no	—
67	C₂³×C₁₃	C₂²×C₈	1	3	yes	—
68	Q₈×C₁₃	C₄×C₈	1	4	yes	—
69	D₄×C₁₃	C₄×C₈	1	12	yes	—
70	C₂×C₄×C₁₃	—	4	6	no	—
71	C₂×C₄×C₁₃	C₂×C₄²	1	6	yes	—
72	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
73	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
74	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
75	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
76	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
77	C₈×C₁₃	C₂×C₄²	1	6	yes	—
78	C₈×C₁₃	C₂²×C₈	1	2	yes	—
79	C₈×C₁₃	C₂²×C₈	1	6	yes	—
80	C₈×C₁₃	C₂³×C₄	1	2	yes	—
81	C₂×#131	Dic₁₃	1	1	yes	characteristic
82	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
83	C₄×#41	Dic₁₃	1	1	yes	characteristic
84	#504	—	13	16	no	—
85	C₂×M₃₂	—	13	6	no	—
86	C₂²×C₁₆	—	13	6	no	—
87	#405	Dic₁₃	1	3	yes	—
88	#405	C₂×D₁₃	1	3	yes	—
89	C₄×M₁₆	Dic₁₃	1	3	yes	—
90	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
91	#346	—	13	1	no	—
92	#346	—	13	3	no	—
93	#346	—	13	6	no	—
94	C₄×C₁₆	—	13	6	no	—
95	C₂²×C₁₃	—	2	6	no	—
96	C₂²×C₁₃	C₂×C₄×C₈	1	6	yes	—
97	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
98	C₄×C₁₃	—	2	2	no	—
99	C₄×C₁₃	—	2	6	no	—
100	C₄×C₁₃	C₂×C₄×C₈	1	6	yes	—
101	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
102	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
103	C₂×#41	C₂×Dic₁₃	1	1	yes	characteristic
104	#131	#1055	1	8	yes	—
105	#131	C₄×D₁₃	1	8	yes	—
106	C₂×M₁₆	C₂×Dic₁₃	1	3	yes	—
107	C₂×M₁₆	C₂×Dic₁₃	1	3	yes	—
108	C₂²×C₈	C₂×Dic₁₃	1	3	yes	—
109	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
110	C₄×Q₈	#1055	1	2	yes	—
111	C₄×D₄	#1055	1	6	yes	—
112	#117	C₂×Dic₁₃	1	3	yes	—
113	C₂×C₄²	C₂×Dic₁₃	1	3	yes	—
114	M₃₂	—	13	24	no	—
115	C₂×C₁₆	—	13	24	no	—
116	C₂×C₁₆	—	13	6	no	—
117	C₂×C₁₆	—	13	2	no	—
118	#97	C₂×Dic₁₃	1	6	yes	—
119	C₄×C₈	C₂×Dic₁₃	1	3	yes	—
120	C₄×C₈	C₂×Dic₁₃	1	3	yes	—
121	C₄×C₈	C₂²×D₁₃	1	3	yes	—
122	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
123	C₂×C₁₃	—	2	6	no	—
124	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
125	C₂×C₁₃	#3407	1	2	yes	—
126	#41	C₂×#1055	1	4	yes	—
127	#41	C₄×Dic₁₃	1	4	yes	—
128	C₂×Q₈	C₂×#1055	1	1	yes	characteristic
129	C₂×D₄	C₂×#1055	1	3	yes	—
130	C₂²×C₄	C₂×#1055	1	6	yes	—
131	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
132	M₁₆	C₂×#1055	1	12	yes	—
133	M₁₆	C₄×Dic₁₃	1	12	yes	—
134	C₂×C₈	C₂×#1055	1	12	yes	—
135	C₂×C₈	C₂×C₄×D₁₃	1	12	yes	—
136	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
137	C₂×C₈	C₂²×Dic₁₃	1	3	yes	—
138	C₂×C₈	C₂²×Dic₁₃	1	3	yes	—
139	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
140	C₄⋊C₄	C₂×#1055	1	6	yes	—
141	C₂²⋊C₄	C₂×#1055	1	6	yes	—
142	C₄²	C₂×#1055	1	6	yes	—
143	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
144	C₄²	C₂²×Dic₁₃	1	3	yes	—
145	C₁₆	—	13	4	no	—
146	C₁₆	—	13	12	no	—
147	C₁₃	#33326	1	1	yes	characteristic, complemented, Sylow
148	C₂³	C₂²×#1055	1	3	yes	—
149	Q₈	C₄×#1055	1	4	yes	—
150	D₄	C₄×#1055	1	12	yes	—
151	C₂×C₄	—	4	6	no	—
152	C₂×C₄	C₂²×#1055	1	2	yes	—
153	C₂×C₄	C₂²×#1055	1	6	yes	—
154	C₂×C₄	C₂²×#1055	1	6	yes	—
155	C₂×C₄	C₂²×#1055	1	3	yes	—
156	C₂×C₄	C₂×C₄×Dic₁₃	1	6	yes	—
157	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
158	C₈	C₂²×#1055	1	2	yes	—
159	C₈	C₂²×#1055	1	6	yes	—
160	C₈	C₂×C₄×Dic₁₃	1	6	yes	—
161	C₈	C₂²×C₄×D₁₃	1	2	yes	—
162	C₂²	—	2	6	no	—
163	C₂²	C₂×C₄×#1055	1	6	yes	—
164	C₂²	C₂³×#1055	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	2	6	no	—
167	C₄	C₂×C₄×#1055	1	6	yes	—
168	C₄	C₂³×#1055	1	1	yes	characteristic
169	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
170	C₂	—	2	6	no	—
171	C₂	C₂²×C₄×#1055	1	1	yes	characteristic
172	C₂	#13127939	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator	C₂×C₁₃	C₂²×C₄×C₈
Fitting	C₄×C₁₃×#131	C₂
socle	C₂²×C₁₃	C₂³×C₈
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₄×#131	D₁₃
hypocenter	C₁₃	#33326
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #13116433	C₂	semidirect
3. C₄×#88052	C₂	semidirect
4. C₂×#11718053	C₂	semidirect
13. #11718801	C₄	semidirect
36. #88052	C₂×C₄	semidirect
37. C₂×#6090	C₂×C₄	semidirect
147. C₁₃	#33326	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718801	2
#13116433	2
C₄×#88052	2
C₂×#11718053	2
C₄×C₁₃×#131	2
#33326	13