Subgroups — #15778313

Order 3328.

310 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718098	C₂	1	1	yes	maximal, characteristic
2	#13116620	C₂	1	1	yes	maximal, characteristic
3	#13114715	C₂	1	2	yes	maximal
4	#13112932	C₂	1	2	yes	maximal
5	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88059	C₂²	1	2	yes	—
10	#11718767	C₂²	1	2	yes	—
11	#11718762	—	2	2	no	—
12	C₁₃×#346	C₄	1	4	yes	complemented
13	#11718115	—	2	2	no	—
14	#11718098	C₄	1	4	yes	complemented
15	#11718097	C₄	1	2	yes	complemented
16	#11718096	C₄	1	2	yes	complemented
17	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
20	#88180	—	2	2	no	—
21	C₂×#6099	—	2	2	no	—
22	C₂×C₁₃×C₁₆	—	2	1	no	—
23	C₂×C₁₃×C₁₆	—	2	1	no	—
24	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₁₃	Q₈	1	2	yes	—
31	#88059	—	2	2	no	—
32	#88059	C₂×C₄	1	4	yes	—
33	#88058	C₂×C₄	1	4	yes	—
34	#88057	—	2	2	no	—
35	#88055	—	4	2	no	—
36	C₈×Dic₁₃	—	4	2	no	—
37	#12671	—	13	1	no	maximal, Sylow
38	C₂²×C₄×C₁₃	—	2	1	no	—
39	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₁₃	D₈	1	2	yes	—
42	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
43	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
48	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
49	C₄²×C₁₃	—	2	1	no	—
50	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
51	C₂²⋊Dic₁₃	—	4	2	no	—
52	C₄⋊Dic₁₃	—	2	4	no	—
53	C₄×Dic₁₃	—	4	2	no	—
54	#6099	—	2	4	no	—
55	#6099	—	4	2	no	—
56	C₂×#1055	—	4	2	no	—
57	C₁₃×C₁₆	—	2	2	no	—
58	C₁₃×C₁₆	C₄²	1	2	yes	—
59	C₁₃×C₁₆	C₄²	1	2	yes	—
60	C₂×#368	—	13	1	no	—
61	#2158	—	13	1	no	—
62	C₂×#346	D₁₃	1	1	yes	characteristic
63	#1772	—	13	2	no	—
64	#1382	—	13	2	no	—
65	C₂³×C₁₃	#135	1	1	yes	characteristic
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	—	4	2	no	—
68	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
73	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
74	C₂×Dic₁₃	—	4	4	no	—
75	C₈×C₁₃	#107	1	4	yes	—
76	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
77	C₈×C₁₃	#117	1	1	yes	characteristic
78	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
79	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
80	#1055	—	4	4	no	—
81	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
82	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
83	C₂×#108	—	13	2	no	—
84	#427	—	13	2	no	—
85	#405	—	26	2	no	—
86	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
87	#368	—	13	4	no	—
88	#367	—	13	2	no	—
89	#366	—	13	2	no	—
90	#346	Dic₁₃	1	4	yes	complemented
91	#339	—	26	2	no	—
92	C₂²×C₁₃	—	2	2	no	—
93	C₂²×C₁₃	#427	1	1	yes	characteristic
94	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
95	C₂²×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
96	C₄×C₁₃	—	4	2	no	—
97	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
98	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
99	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
100	C₄×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
101	Dic₁₃	—	8	4	no	—
102	C₂×M₁₆	—	26	2	no	—
103	C₂²×C₈	D₅₂	1	1	yes	characteristic
104	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
105	#117	—	26	2	no	—
106	C₂×C₄²	D₅₂	1	1	yes	characteristic
107	C₂×C₁₆	—	2	1	no	—
108	C₂×C₁₆	—	2	1	no	—
109	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
110	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
111	C₂×C₁₆	C₄×D₁₃	1	2	yes	—
112	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
113	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
114	#108	—	13	4	no	—
115	#108	—	26	2	no	—
116	#107	—	13	4	no	—
117	#106	—	26	2	no	—
118	#97	—	52	2	no	—
119	C₄×C₈	—	52	2	no	—
120	C₄×C₈	Dic₂₆	1	2	yes	—
121	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
122	C₂×C₁₃	—	2	1	no	—
123	C₂×C₁₃	—	2	1	no	—
124	C₂×C₁₃	C₄×#107	1	1	yes	characteristic
125	C₂×C₁₃	#2179	1	2	yes	—
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
128	M₁₆	—	26	4	no	—
129	M₁₆	—	52	2	no	—
130	C₂×C₈	—	52	2	no	—
131	C₂×C₈	D₁₀₄	1	2	yes	—
132	C₂×C₈	Dic₅₂	1	2	yes	—
133	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
134	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
135	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
137	C₂×C₈	#6127	1	1	yes	characteristic
138	C₂×C₈	#6127	1	1	yes	characteristic
139	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
140	C₄⋊C₄	—	26	4	no	—
141	C₂²⋊C₄	—	52	2	no	—
142	C₄²	—	2	1	no	—
143	C₄²	—	52	2	no	—
144	C₄²	C₄⋊Dic₁₃	1	2	yes	—
145	C₁₆	—	2	2	no	—
146	C₁₆	C₄×Dic₁₃	1	2	yes	—
147	C₁₆	C₄×Dic₁₃	1	2	yes	—
148	C₁₃	#12671	1	1	yes	characteristic, complemented, Sylow
149	C₂³	#88158	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	52	4	no	—
153	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
154	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
155	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
156	C₂×C₄	#88158	1	1	yes	characteristic
157	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
159	C₈	—	52	4	no	—
160	C₈	#88058	1	4	yes	—
161	C₈	C₄×Dic₂₆	1	1	yes	characteristic
162	C₈	C₄×Dic₂₆	1	1	yes	characteristic
163	C₈	C₂×C₄×Dic₁₃	1	1	yes	characteristic
164	C₈	#88180	1	1	yes	characteristic
165	C₂²	—	2	2	no	—
166	C₂²	C₄×D₁₀₄	1	1	yes	characteristic
167	C₂²	C₄×Dic₅₂	1	1	yes	characteristic
168	C₂²	#11718767	1	1	yes	characteristic
169	C₄	—	4	2	no	—
170	C₄	—	104	4	no	—
171	C₄	C₄×D₁₀₄	1	1	yes	characteristic
172	C₄	C₄×Dic₅₂	1	1	yes	characteristic
173	C₄	C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
174	C₄	C₂×#88058	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	C₄×#88058	1	1	yes	characteristic
178	C₂	#13116723	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₀₄
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#346	C₂
socle	C₂²×C₁₃	C₄×D₈
prime radical	C₂³×C₁₃	#135
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#346	D₁₃
hypocenter	C₁₃	#12671
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
12. C₁₃×#346	C₄	semidirect
14. #11718098	C₄	semidirect
15. #11718097	C₄	semidirect
16. #11718096	C₄	semidirect
90. #346	Dic₁₃	semidirect
148. C₁₃	#12671	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718098	2
#13116620	2
#13114715	2
#13112932	2
C₂×C₁₃×#346	2
#12671	13