Subgroups — #15749788

Order 3328.

242 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718050	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
5	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂²×#6090	C₄	1	2	yes	—
7	C₂×C₄⋊#1055	—	2	1	no	—
8	C₁₃×#335	—	2	1	no	—
9	C₁₃×#335	C₄	1	2	yes	—
10	C₄×#6090	—	2	1	no	—
11	C₄×#6090	—	2	1	no	—
12	#11718050	—	2	1	no	—
13	#11718050	C₄	1	2	yes	—
14	C₂²×C₈×C₁₃	—	4	1	no	—
15	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
18	C₂²×#1055	—	4	1	no	—
19	C₁₃×C₄⋊C₈	—	2	2	no	—
20	C₁₃×C₄⋊C₈	—	4	1	no	—
21	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
25	C₂×#6090	—	2	2	no	—
26	C₂×#6090	—	2	2	no	—
27	C₂×#6090	C₈	1	4	yes	complemented
28	C₄⋊#1055	—	2	2	no	—
29	C₄⋊#1055	—	4	1	no	—
30	#7154	—	13	1	no	maximal, Sylow
31	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	Q₁₆	1	1	yes	characteristic
34	C₂×C₈×C₁₃	—	2	2	no	—
35	C₂×C₈×C₁₃	—	4	2	no	—
36	C₂×C₈×C₁₃	—	8	1	no	—
37	C₂×C₈×C₁₃	—	8	1	no	—
38	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
44	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
47	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
48	C₂×#1055	—	4	2	no	—
49	C₂×#1055	—	8	1	no	—
50	C₂×#1055	—	8	1	no	—
51	#6090	—	2	4	no	—
52	C₂×C₄×C₁₆	—	13	1	no	—
53	C₂×#335	—	13	1	no	—
54	C₂×#335	D₁₃	1	1	yes	characteristic
55	C₂³×C₁₃	#107	1	1	yes	characteristic
56	C₂×C₄×C₁₃	—	2	2	no	—
57	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
58	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
59	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
60	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
61	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
62	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
63	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
64	C₂×C₄×C₁₃	#108	1	2	yes	—
65	C₂×C₄×C₁₃	D₁₆	1	1	yes	characteristic
66	C₂×C₄×C₁₃	SD₃₂	1	1	yes	characteristic
67	C₂×C₄×C₁₃	SD₃₂	1	1	yes	characteristic
68	C₂×C₄×C₁₃	Q₃₂	1	1	yes	characteristic
69	C₈×C₁₃	—	2	1	no	—
70	C₈×C₁₃	—	2	1	no	—
71	C₈×C₁₃	—	8	2	no	—
72	C₈×C₁₃	C₄×C₈	1	1	yes	characteristic
73	C₈×C₁₃	#97	1	1	yes	characteristic
74	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
75	#1055	—	8	2	no	—
76	C₂²×C₁₆	—	13	2	no	—
77	C₂×C₄⋊C₈	—	2	1	no	—
78	C₂×C₄⋊C₈	—	26	1	no	—
79	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
80	C₄×C₁₆	—	26	1	no	—
81	C₄×C₁₆	—	26	1	no	—
82	#335	—	2	1	no	—
83	#335	—	13	2	no	—
84	#335	—	26	1	no	—
85	#335	Dic₁₃	1	2	yes	—
86	C₂²×C₁₃	#335	1	2	yes	—
87	C₂²×C₁₃	#340	1	1	yes	characteristic
88	C₂²×C₁₃	#366	1	1	yes	characteristic
89	C₂²×C₁₃	#367	1	1	yes	characteristic
90	C₂²×C₁₃	#368	1	2	yes	—
91	C₄×C₁₃	—	2	1	no	—
92	C₄×C₁₃	—	2	1	no	—
93	C₄×C₁₃	#334	1	2	yes	—
94	C₄×C₁₃	#336	1	1	yes	characteristic
95	C₄×C₁₃	#341	1	1	yes	characteristic
96	C₄×C₁₃	#357	1	1	yes	characteristic
97	C₄×C₁₃	#357	1	1	yes	characteristic
98	C₄×C₁₃	#358	1	1	yes	characteristic
99	C₄×C₁₃	#358	1	1	yes	characteristic
100	C₂²×C₈	—	4	1	no	—
101	C₂²×C₈	—	52	1	no	—
102	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
103	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
104	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
105	C₂×C₁₆	—	13	4	no	—
106	C₂×C₁₆	—	26	2	no	—
107	C₂×C₁₆	—	26	2	no	—
108	C₄⋊C₈	—	2	2	no	—
109	C₄⋊C₈	—	4	1	no	—
110	C₄⋊C₈	—	26	2	no	—
111	C₄⋊C₈	—	52	1	no	—
112	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
113	C₄×C₈	D₅₂	1	1	yes	characteristic
114	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
115	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
116	C₂×C₁₃	#1277	1	1	yes	characteristic
117	C₂×C₁₃	#1371	1	2	yes	—
118	C₂×C₁₃	#1372	1	2	yes	—
119	C₂×C₁₃	#1380	1	1	yes	characteristic
120	C₂×C₁₃	#1381	1	1	yes	characteristic
121	C₂²×C₄	#6101	1	1	yes	characteristic
122	C₂²×C₄	#6104	1	1	yes	characteristic
123	C₂²×C₄	#6107	1	1	yes	characteristic
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	8	1	no	—
127	C₂×C₈	—	8	1	no	—
128	C₂×C₈	—	52	2	no	—
129	C₂×C₈	—	104	1	no	—
130	C₂×C₈	—	104	1	no	—
131	C₂×C₈	C₈×D₁₃	1	2	yes	—
132	C₂×C₈	#6094	1	2	yes	—
133	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
134	C₂×C₈	#6101	1	1	yes	characteristic
135	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
136	C₂×C₈	#6103	1	1	yes	characteristic
137	C₄²	#6103	1	1	yes	characteristic
138	C₄²	#6105	1	1	yes	characteristic
139	C₄²	#6106	1	1	yes	characteristic
140	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
141	C₁₆	—	26	4	no	—
142	C₁₃	#7154	1	1	yes	characteristic, complemented, Sylow
143	C₂³	#88047	1	1	yes	characteristic
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	#88048	1	1	yes	characteristic
146	C₂×C₄	#88049	1	1	yes	characteristic
147	C₂×C₄	#88050	1	1	yes	characteristic
148	C₂×C₄	#88054	1	2	yes	—
149	C₂×C₄	#88062	1	2	yes	—
150	C₂×C₄	#88066	1	1	yes	characteristic
151	C₂×C₄	#88067	1	1	yes	characteristic
152	C₂×C₄	#88068	1	1	yes	characteristic
153	C₂×C₄	#88069	1	1	yes	characteristic
154	C₂×C₄	#88071	1	1	yes	characteristic
155	C₂×C₄	#88072	1	1	yes	characteristic
156	C₂×C₄	#88075	1	1	yes	characteristic
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	8	2	no	—
160	C₈	—	104	2	no	—
161	C₈	C₈×Dic₁₃	1	1	yes	characteristic
162	C₈	#88055	1	1	yes	characteristic
163	C₈	#88060	1	2	yes	—
164	C₂²	#11718071	1	2	yes	—
165	C₂²	#11718081	1	1	yes	characteristic
166	C₂²	#11718082	1	1	yes	characteristic
167	C₂²	#11718085	1	2	yes	—
168	C₂²	#11718121	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	#11718072	1	2	yes	—
172	C₄	#11718083	1	1	yes	characteristic
173	C₄	#11718084	1	1	yes	characteristic
174	C₄	#11718142	1	1	yes	characteristic
175	C₄	#11718150	1	1	yes	characteristic
176	C₄	#11718154	1	1	yes	characteristic
177	C₄	#11718156	1	1	yes	characteristic
178	C₂	#13112670	1	1	yes	characteristic
179	C₂	#13112907	1	2	yes	—
180	C₂	#13112909	1	2	yes	—
181	C₂	#13112927	1	1	yes	characteristic
182	C₂	#13112929	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#6104
commutator	C₈×C₁₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₁₃
Fitting	C₂×C₁₃×#335	C₂
socle, prime radical	C₂³×C₁₃	#107
cosocle	C₂×C₄×C₈×C₁₃	C₂²
hypercenter	C₂×#335	D₁₃
hypocenter	C₁₃	#7154
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
27. C₂×#6090	C₈	semidirect
142. C₁₃	#7154	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#6090	2
C₂×#11718050	2
C₂×C₁₃×#335	2
#7154	13