Subgroups — #15749859

Order 3328.

274 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718055	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1772	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#108	C₄	1	1	yes	characteristic
5	C₁₃×#427	C₄	1	1	yes	characteristic
6	C₁₃×#405	—	2	1	no	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88052	—	2	1	no	—
9	C₂×C₄×#1055	—	2	1	no	—
10	C₁₃×#339	—	2	1	no	—
11	#11718055	—	2	1	no	—
12	#11718055	C₄	1	2	yes	complemented
13	C₈×#1055	—	2	1	no	—
14	C₈×#1055	C₄	1	2	yes	complemented
15	C₂×C₁₃×M₁₆	—	2	1	no	—
16	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
18	C₁₃×#117	—	2	1	no	—
19	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
20	C₂²×#1055	—	2	2	no	—
21	C₁₃×#108	—	2	1	no	—
22	C₁₃×#108	C₈	1	2	yes	complemented
23	C₁₃×#107	C₈	1	2	yes	complemented
24	C₁₃×#106	—	2	1	no	—
25	C₁₃×#97	—	4	1	no	—
26	C₄×C₈×C₁₃	—	4	1	no	—
27	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
31	#88052	—	2	2	no	—
32	#88052	—	4	1	no	—
33	C₂×#6090	—	4	1	no	—
34	C₄×#1055	—	2	2	no	—
35	C₄×#1055	—	2	2	no	—
36	#7179	—	13	1	no	maximal, Sylow
37	C₂²×C₄×C₁₃	—	2	1	no	—
38	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₁₃×M₁₆	—	2	2	no	—
40	C₁₃×M₁₆	—	4	1	no	—
41	C₂×C₈×C₁₃	—	4	1	no	—
42	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
49	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
50	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
51	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
52	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₁₃	Q₁₆	1	1	yes	characteristic
54	C₁₃×C₄⋊C₄	—	2	2	no	—
55	C₁₃×C₂²⋊C₄	—	4	1	no	—
56	C₄²×C₁₃	—	2	1	no	—
57	C₄²×C₁₃	—	4	1	no	—
58	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂×#1055	—	2	8	no	—
61	C₂×#1055	—	2	4	no	—
62	#6090	—	4	2	no	—
63	C₂×#364	—	13	1	no	—
64	#1772	D₁₃	1	1	yes	characteristic
65	C₂×C₈²	—	13	1	no	—
66	C₂³×C₁₃	#104	1	1	yes	characteristic
67	C₂×C₄×C₁₃	—	2	4	no	—
68	C₂×C₄×C₁₃	—	2	1	no	—
69	C₂×C₄×C₁₃	—	2	1	no	—
70	C₂×C₄×C₁₃	—	4	2	no	—
71	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
75	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
76	C₂×C₄×C₁₃	#104	1	1	yes	characteristic
77	C₈×C₁₃	—	4	2	no	—
78	C₈×C₁₃	C₄×C₈	1	1	yes	characteristic
79	C₈×C₁₃	#97	1	1	yes	characteristic
80	C₈×C₁₃	#102	1	1	yes	characteristic
81	C₈×C₁₃	#103	1	1	yes	characteristic
82	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
83	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
84	C₈×C₁₃	#106	1	1	yes	characteristic
85	C₈×C₁₃	#107	1	1	yes	characteristic
86	#1055	—	2	8	no	—
87	C₂×M₃₂	—	26	1	no	—
88	C₂×#108	Dic₁₃	1	1	yes	characteristic
89	#427	Dic₁₃	1	1	yes	characteristic
90	#405	—	2	1	no	—
91	C₂×C₄×C₈	—	26	1	no	—
92	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
93	#364	—	13	2	no	—
94	#364	—	26	1	no	—
95	#339	—	2	1	no	—
96	C₈²	—	13	2	no	—
97	C₈²	—	26	1	no	—
98	C₂²×C₁₃	—	2	2	no	—
99	C₂²×C₁₃	D₄⋊C₈	1	1	yes	characteristic
100	C₂²×C₁₃	Q₈⋊C₈	1	1	yes	characteristic
101	C₂²×C₁₃	#339	1	1	yes	characteristic
102	C₄×C₁₃	—	2	2	no	—
103	C₄×C₁₃	—	2	2	no	—
104	C₄×C₁₃	—	8	2	no	—
105	C₄×C₁₃	D₄⋊C₈	1	1	yes	characteristic
106	C₄×C₁₃	Q₈⋊C₈	1	1	yes	characteristic
107	C₄×C₁₃	#336	1	1	yes	characteristic
108	C₄×C₁₃	#340	1	1	yes	characteristic
109	C₂×M₁₆	—	2	1	no	—
110	C₂²×C₈	—	26	2	no	—
111	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
112	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
113	#117	—	2	1	no	—
114	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
115	M₃₂	—	26	2	no	—
116	M₃₂	—	52	1	no	—
117	C₂×C₁₆	—	52	1	no	—
118	#108	—	2	1	no	—
119	#108	#1055	1	2	yes	complemented
120	#107	#1055	1	2	yes	complemented
121	#106	—	2	1	no	—
122	#97	—	4	1	no	—
123	C₄×C₈	—	4	1	no	—
124	C₄×C₈	—	26	2	no	—
125	C₄×C₈	—	26	2	no	—
126	C₄×C₈	Dic₂₆	1	1	yes	characteristic
127	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
128	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
129	C₄×C₈	D₅₂	1	1	yes	characteristic
130	C₂×C₁₃	—	2	1	no	—
131	C₂×C₁₃	—	2	1	no	—
132	C₂×C₁₃	#1279	1	1	yes	characteristic
133	C₂×C₁₃	#1335	1	2	yes	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
136	M₁₆	—	2	2	no	—
137	M₁₆	—	4	1	no	—
138	C₂×C₈	—	4	1	no	—
139	C₂×C₈	—	26	4	no	—
140	C₂×C₈	—	26	8	no	—
141	C₂×C₈	C₂×#1055	1	1	yes	characteristic
142	C₂×C₈	C₂×#1055	1	1	yes	characteristic
143	C₂×C₈	#6099	1	1	yes	characteristic
144	C₂×C₈	#6099	1	1	yes	characteristic
145	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
146	C₂×C₈	#6101	1	1	yes	characteristic
147	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
148	C₂×C₈	#6103	1	1	yes	characteristic
149	C₂×C₈	#6104	1	1	yes	characteristic
150	C₂×C₈	#6105	1	1	yes	characteristic
151	C₂×C₈	#6106	1	1	yes	characteristic
152	C₂×C₈	#6107	1	1	yes	characteristic
153	C₄⋊C₄	—	2	2	no	—
154	C₂²⋊C₄	—	4	1	no	—
155	C₄²	—	2	1	no	—
156	C₄²	—	4	1	no	—
157	C₄²	#6101	1	1	yes	characteristic
158	C₄²	#6103	1	1	yes	characteristic
159	C₁₆	—	52	2	no	—
160	C₁₃	#7179	1	1	yes	characteristic, complemented, Sylow
161	C₂³	#88077	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	C₂²⋊#1055	1	1	yes	characteristic
167	C₂×C₄	C₂²⋊#1055	1	1	yes	characteristic
168	C₂×C₄	#88071	1	1	yes	characteristic
169	C₂×C₄	#88072	1	1	yes	characteristic
170	C₂×C₄	#88075	1	1	yes	characteristic
171	C₂×C₄	#88077	1	1	yes	characteristic
172	C₈	—	4	2	no	—
173	C₈	—	26	8	no	—
174	C₈	C₄×#1055	1	1	yes	characteristic
175	C₈	#88043	1	1	yes	characteristic
176	C₈	C₄⋊#1055	1	1	yes	characteristic
177	C₈	C₄⋊#1055	1	1	yes	characteristic
178	C₈	#88047	1	1	yes	characteristic
179	C₈	#88048	1	1	yes	characteristic
180	C₈	#88049	1	1	yes	characteristic
181	C₈	#88050	1	1	yes	characteristic
182	C₂²	—	2	2	no	—
183	C₂²	#11718124	1	1	yes	characteristic
184	C₂²	D₄⋊#1055	1	1	yes	characteristic
185	C₂²	Q₈⋊#1055	1	1	yes	characteristic
186	C₄	—	2	2	no	—
187	C₄	—	2	2	no	—
188	C₄	—	8	2	no	—
189	C₄	#11718116	1	1	yes	characteristic
190	C₄	#11718121	1	1	yes	characteristic
191	C₄	D₄⋊#1055	1	1	yes	characteristic
192	C₄	Q₈⋊#1055	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#13112677	1	1	yes	characteristic
196	C₂	#13112809	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	#6104
commutator	C₈×C₁₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₁₃
Fitting	C₁₃×#1772	C₂
socle	C₂²×C₁₃	D₄⋊C₈
prime radical	C₂³×C₁₃	#104
cosocle	C₂×C₄×C₈×C₁₃	C₂²
hypercenter	#1772	D₁₃
hypocenter	C₁₃	#7179
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
12. #11718055	C₄	semidirect
14. C₈×#1055	C₄	semidirect
22. C₁₃×#108	C₈	semidirect
23. C₁₃×#107	C₈	semidirect
119. #108	#1055	semidirect
120. #107	#1055	semidirect
160. C₁₃	#7179	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718055	2
C₂×C₈×#1055	2
C₁₃×#1772	2
#7179	13