Subgroups — #15756582

Order 3328.

382 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	#13114953	C₂	1	1	yes	maximal, characteristic
2	C₄²×#1055	C₂	1	1	yes	maximal, characteristic
3	#13112670	C₂	1	4	yes	maximal
4	C₁₃×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
6	C₄³×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊#1055	C₂²	1	2	yes	—
8	C₂×C₄×#1055	C₄	1	2	yes	—
9	C₂×C₄×#1055	C₄	1	1	yes	characteristic
10	C₂×C₄×#1055	C₄	1	1	yes	characteristic
11	C₂×C₄×#1055	C₂²	1	2	yes	—
12	C₁₃×#336	—	2	2	no	—
13	#11718116	—	2	2	no	—
14	C₂²×C₈×C₁₃	—	2	2	no	—
15	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₃	D₄	1	2	yes	—
18	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
19	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
20	C₂²×#1055	—	2	2	no	—
21	C₂²×#1055	C₂×C₄	1	2	yes	—
22	C₂²×#1055	C₂×C₄	1	2	yes	—
23	C₁₃×C₄⋊C₈	—	2	4	no	—
24	C₄⋊#1055	—	2	4	no	—
25	C₄×#1055	—	2	2	no	—
26	C₄×#1055	—	2	2	no	—
27	C₄×#1055	—	2	2	no	—
28	C₄×#1055	—	2	4	no	—
29	C₄×#1055	C₈	1	4	yes	complemented
30	#8785	—	13	1	no	maximal, Sylow
31	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
33	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈×C₁₃	—	4	2	no	—
37	C₂×C₈×C₁₃	—	4	4	no	—
38	C₄²×C₁₃	—	2	4	no	—
39	C₄²×C₁₃	—	2	2	no	—
40	C₄²×C₁₃	—	2	4	no	—
41	C₄²×C₁₃	C₂×C₈	1	2	yes	—
42	C₄²×C₁₃	M₁₆	1	2	yes	—
43	C₄²×C₁₃	#41	1	2	yes	—
44	C₄²×C₁₃	#41	1	2	yes	—
45	C₂×#1055	—	2	2	no	—
46	C₂×#1055	—	2	2	no	—
47	C₂×#1055	—	2	2	no	—
48	C₂×#1055	—	2	2	no	—
49	C₂×#1055	—	4	2	no	—
50	C₂×#1055	—	4	4	no	—
51	C₂×#1055	C₂×C₈	1	4	yes	—
52	C₂×#1055	M₁₆	1	4	yes	—
53	#1831	—	13	1	no	—
54	#1831	D₁₃	1	1	yes	characteristic
55	C₄²×C₈	—	13	1	no	—
56	#1277	—	13	4	no	—
57	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃	—	2	2	no	—
59	C₂×C₄×C₁₃	—	2	2	no	—
60	C₂×C₄×C₁₃	—	2	2	no	—
61	C₂×C₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	—	2	2	no	—
63	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
64	C₂×C₄×C₁₃	#108	1	4	yes	—
65	C₂×C₄×C₁₃	#117	1	2	yes	—
66	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
67	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
68	C₂×C₄×C₁₃	#125	1	2	yes	—
69	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
71	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₁₃	#135	1	4	yes	—
73	C₈×C₁₃	—	8	2	no	—
74	#1055	—	2	4	no	—
75	#1055	—	2	4	no	—
76	#1055	—	8	2	no	—
77	C₂×C₄⋊C₈	—	13	2	no	—
78	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
79	C₂×C₄×C₈	—	13	1	no	—
80	C₂×C₄×C₈	—	13	2	no	—
81	C₂×C₄×C₈	—	13	1	no	—
82	C₂×C₄×C₈	—	13	2	no	—
83	C₄³	C₂×D₁₃	1	1	yes	characteristic
84	#336	—	2	2	no	—
85	#336	—	26	2	no	—
86	C₂²×C₁₃	#382	1	1	yes	characteristic
87	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₁₃	#423	1	1	yes	characteristic
89	C₂²×C₁₃	#427	1	2	yes	—
90	C₂²×C₁₃	C₂×#108	1	2	yes	—
91	C₄×C₁₃	—	2	4	no	—
92	C₄×C₁₃	—	2	4	no	—
93	C₄×C₁₃	—	2	2	no	—
94	C₄×C₁₃	#431	1	2	yes	—
95	C₄×C₁₃	#432	1	2	yes	—
96	C₄×C₁₃	#488	1	2	yes	—
97	C₄×C₁₃	#488	1	2	yes	—
98	C₂²×C₈	—	2	2	no	—
99	C₂²×C₈	—	13	2	no	—
100	C₂²×C₈	—	13	2	no	—
101	C₂²×C₈	—	26	2	no	—
102	C₂×C₄²	Dic₂₆	1	2	yes	—
103	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
104	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
105	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
106	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
107	C₄⋊C₈	—	2	4	no	—
108	C₄⋊C₈	—	26	4	no	—
109	C₄×C₈	—	13	4	no	—
110	C₄×C₈	—	26	4	no	—
111	C₄×C₈	—	26	2	no	—
112	C₄×C₈	—	26	2	no	—
113	C₄×C₈	—	26	2	no	—
114	C₂×C₁₃	#1564	1	2	yes	—
115	C₂×C₁₃	#1568	1	2	yes	—
116	C₂×C₁₃	#1831	1	1	yes	characteristic
117	C₂×C₁₃	#1835	1	1	yes	characteristic
118	C₂×C₁₃	#1837	1	1	yes	characteristic
119	C₂²×C₄	#6101	1	2	yes	—
120	C₂²×C₄	#6101	1	2	yes	—
121	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
122	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
123	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	4	4	no	—
126	C₂×C₈	—	13	4	no	—
127	C₂×C₈	—	13	4	no	—
128	C₂×C₈	—	26	2	no	—
129	C₂×C₈	—	26	2	no	—
130	C₂×C₈	—	26	2	no	—
131	C₂×C₈	—	26	2	no	—
132	C₂×C₈	—	52	2	no	—
133	C₂×C₈	—	52	4	no	—
134	C₄²	—	2	2	no	—
135	C₄²	—	2	4	no	—
136	C₄²	—	2	4	no	—
137	C₄²	C₈×D₁₃	1	2	yes	—
138	C₄²	#6094	1	2	yes	—
139	C₄²	#6129	1	2	yes	—
140	C₄²	#6131	1	2	yes	—
141	C₁₃	#8785	1	1	yes	characteristic, complemented, Sylow
142	C₂³	C₂×#6101	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	#88054	1	4	yes	—
149	C₂×C₄	#88062	1	4	yes	—
150	C₂×C₄	#88144	1	2	yes	—
151	C₂×C₄	#88146	1	2	yes	—
152	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
153	C₂×C₄	C₂×#6094	1	1	yes	characteristic
154	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
155	C₂×C₄	#88190	1	1	yes	characteristic
156	C₂×C₄	#88201	1	1	yes	characteristic
157	C₂×C₄	#88204	1	4	yes	—
158	C₈	—	8	2	no	—
159	C₈	—	26	4	no	—
160	C₈	—	26	4	no	—
161	C₈	—	104	2	no	—
162	C₂²	#11718638	1	1	yes	characteristic
163	C₂²	#11718659	1	2	yes	—
164	C₂²	C₂×#88054	1	1	yes	characteristic
165	C₂²	#11718779	1	1	yes	characteristic
166	C₂²	C₂×#88062	1	2	yes	—
167	C₄	—	2	2	no	—
168	C₄	—	2	4	no	—
169	C₄	—	2	4	no	—
170	C₄	#11718494	1	2	yes	—
171	C₄	#11718496	1	2	yes	—
172	C₄	#11718717	1	2	yes	—
173	C₄	#11718729	1	2	yes	—
174	C₂	#13113582	1	2	yes	—
175	C₂	#13113597	1	2	yes	—
176	C₂	#13114952	1	1	yes	characteristic
177	C₂	#13114966	1	1	yes	characteristic
178	C₂	#13114973	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1831	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1831	D₁₃
hypocenter	C₁₃	#8785
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
29. C₄×#1055	C₈	semidirect
141. C₁₃	#8785	semidirect

Maximal subgroups

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

subgroup	index
#13114953	2
C₄²×#1055	2
#13112670	2
C₁₃×#1831	2
#8785	13