Subgroups — #15749829

Order 3328.

266 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116592	C₂	1	1	yes	maximal, characteristic
2	C₄×#88052	C₂	1	1	yes	maximal, characteristic
3	C₈×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
5	C₂×C₄×C₈×C₁₃	—	2	1	no	—
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₄×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
8	C₂×#88052	—	2	1	no	—
9	C₂×#88052	—	2	1	no	—
10	C₁₃×#336	—	2	1	no	—
11	C₄⋊#6090	—	2	1	no	—
12	C₄⋊#6090	C₄	1	2	yes	complemented
13	#11718053	C₄	1	2	yes	complemented
14	C₄×#6090	—	2	1	no	—
15	C₂²×C₈×C₁₃	—	2	1	no	—
16	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
18	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
19	C₂×C₄²×C₁₃	—	2	1	no	—
20	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₁₃×C₄⋊C₈	—	2	1	no	—
22	C₁₃×C₄⋊C₈	C₈	1	2	yes	—
23	C₄×C₈×C₁₃	—	4	1	no	—
24	C₄×C₈×C₁₃	—	4	1	no	—
25	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
29	C₁₃×#95	C₈	1	2	yes	—
30	#88052	—	2	2	no	—
31	#88052	—	2	2	no	—
32	#88052	—	4	1	no	—
33	#88052	—	4	1	no	—
34	C₂×#6090	—	2	2	no	—
35	C₂×#6090	—	2	2	no	—
36	#7169	—	13	1	no	maximal, Sylow
37	C₂²×C₄×C₁₃	—	2	2	no	—
38	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
41	C₂×C₈×C₁₃	—	2	1	no	—
42	C₂×C₈×C₁₃	—	2	1	no	—
43	C₂×C₈×C₁₃	—	2	2	no	—
44	C₂×C₈×C₁₃	—	4	1	no	—
45	C₂×C₈×C₁₃	—	4	1	no	—
46	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₄⋊C₄	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	4	no	—
55	C₄²×C₁₃	—	4	1	no	—
56	C₄²×C₁₃	—	4	1	no	—
57	C₄²×C₁₃	C₄²	1	1	yes	characteristic
58	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
59	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
60	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
61	#6090	—	2	4	no	—
62	#6090	—	4	2	no	—
63	#2150	—	13	1	no	—
64	C₄×M₃₂	—	13	1	no	—
65	C₈×C₄⋊C₄	D₁₃	1	1	yes	characteristic
66	C₂³×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	4	no	—
69	C₂×C₄×C₁₃	—	4	1	no	—
70	C₂×C₄×C₁₃	—	4	2	no	—
71	C₂×C₄×C₁₃	—	4	2	no	—
72	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
76	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
77	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
78	C₂×C₄×C₁₃	#104	1	1	yes	characteristic
79	C₂×C₄×C₁₃	#104	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
82	C₈×C₁₃	—	2	1	no	—
83	C₈×C₁₃	—	2	1	no	—
84	C₈×C₁₃	—	4	2	no	—
85	C₈×C₁₃	#102	1	1	yes	characteristic
86	C₈×C₁₃	#103	1	1	yes	characteristic
87	C₈×C₁₃	#106	1	1	yes	characteristic
88	C₈×C₁₃	#107	1	1	yes	characteristic
89	C₂×M₃₂	—	26	1	no	—
90	C₂×M₃₂	—	26	1	no	—
91	C₂×C₄⋊C₈	Dic₁₃	1	1	yes	characteristic
92	C₂×C₄×C₈	—	2	1	no	—
93	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
94	C₄×C₄⋊C₄	Dic₁₃	1	1	yes	characteristic
95	C₄⋊C₁₆	—	13	2	no	—
96	C₄⋊C₁₆	—	26	1	no	—
97	#346	—	13	2	no	—
98	C₄×C₁₆	—	26	1	no	—
99	#336	—	2	1	no	—
100	C₂²×C₁₃	—	2	1	no	—
101	C₂²×C₁₃	—	2	1	no	—
102	C₂²×C₁₃	D₄⋊C₈	1	1	yes	characteristic
103	C₂²×C₁₃	Q₈⋊C₈	1	1	yes	characteristic
104	C₂²×C₁₃	#336	1	1	yes	characteristic
105	C₄×C₁₃	—	2	1	no	—
106	C₄×C₁₃	—	2	1	no	—
107	C₄×C₁₃	—	2	1	no	—
108	C₄×C₁₃	—	4	4	no	—
109	C₄×C₁₃	D₄⋊C₈	1	1	yes	characteristic
110	C₄×C₁₃	Q₈⋊C₈	1	1	yes	characteristic
111	C₄×C₁₃	#339	1	1	yes	characteristic
112	C₄×C₁₃	#340	1	1	yes	characteristic
113	C₄×C₁₃	#364	1	2	yes	—
114	C₂²×C₈	—	2	1	no	—
115	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
116	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	2	1	no	—
118	C₂×C₄²	—	2	1	no	—
119	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
120	M₃₂	—	26	2	no	—
121	M₃₂	—	26	2	no	—
122	M₃₂	—	52	1	no	—
123	M₃₂	—	52	1	no	—
124	C₂×C₁₆	—	26	2	no	—
125	C₂×C₁₆	—	26	2	no	—
126	C₄⋊C₈	—	2	1	no	—
127	C₄⋊C₈	#1055	1	2	yes	—
128	C₄×C₈	—	4	1	no	—
129	C₄×C₈	—	4	1	no	—
130	C₄×C₈	Dic₂₆	1	1	yes	characteristic
131	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
132	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
133	C₄×C₈	D₅₂	1	1	yes	characteristic
134	#95	#1055	1	2	yes	—
135	C₂×C₁₃	—	2	1	no	—
136	C₂×C₁₃	—	2	1	no	—
137	C₂×C₁₃	#1279	1	1	yes	characteristic
138	C₂×C₁₃	#1376	1	1	yes	characteristic
139	C₂×C₁₃	#1378	1	1	yes	characteristic
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
142	C₂²×C₄	#6099	1	1	yes	characteristic
143	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	#6101	1	1	yes	characteristic
150	C₂×C₈	#6101	1	1	yes	characteristic
151	C₂×C₈	#6103	1	1	yes	characteristic
152	C₂×C₈	#6103	1	1	yes	characteristic
153	C₂×C₈	#6104	1	1	yes	characteristic
154	C₂×C₈	#6105	1	1	yes	characteristic
155	C₂×C₈	#6106	1	1	yes	characteristic
156	C₂×C₈	#6107	1	1	yes	characteristic
157	C₄⋊C₄	—	2	4	no	—
158	C₄²	—	4	1	no	—
159	C₄²	—	4	1	no	—
160	C₄²	C₂×#1055	1	1	yes	characteristic
161	C₄²	#6099	1	1	yes	characteristic
162	C₄²	C₄×Dic₁₃	1	1	yes	characteristic
163	C₄²	C₄⋊Dic₁₃	1	1	yes	characteristic
164	C₁₆	—	26	4	no	—
165	C₁₆	—	52	2	no	—
166	C₁₃	#7169	1	1	yes	characteristic, complemented, Sylow
167	C₂³	C₂²⋊#1055	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	C₄×#1055	1	1	yes	characteristic
174	C₂×C₄	#88043	1	1	yes	characteristic
175	C₂×C₄	C₄⋊#1055	1	1	yes	characteristic
176	C₂×C₄	C₄⋊#1055	1	1	yes	characteristic
177	C₂×C₄	C₂²⋊#1055	1	1	yes	characteristic
178	C₂×C₄	#88071	1	1	yes	characteristic
179	C₂×C₄	#88072	1	1	yes	characteristic
180	C₂×C₄	#88075	1	1	yes	characteristic
181	C₂×C₄	#88077	1	1	yes	characteristic
182	C₂×C₄	#88077	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	1	no	—
185	C₈	—	4	2	no	—
186	C₈	#88047	1	1	yes	characteristic
187	C₈	#88048	1	1	yes	characteristic
188	C₈	#88049	1	1	yes	characteristic
189	C₈	#88050	1	1	yes	characteristic
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	#11718116	1	1	yes	characteristic
193	C₂²	D₄⋊#1055	1	1	yes	characteristic
194	C₂²	Q₈⋊#1055	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	4	4	no	—
199	C₄	#11718055	1	2	yes	—
200	C₄	#11718121	1	1	yes	characteristic
201	C₄	#11718124	1	1	yes	characteristic
202	C₄	D₄⋊#1055	1	1	yes	characteristic
203	C₄	Q₈⋊#1055	1	1	yes	characteristic
204	C₂	—	2	1	no	—
205	C₂	—	2	1	no	—
206	C₂	#13112677	1	1	yes	characteristic
207	C₂	#13112918	1	1	yes	characteristic
208	C₂	#13112922	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#6103
commutator	C₂×C₄×C₁₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₁₃
Fitting	C₈×C₁₃×C₄⋊C₄	C₂
socle	C₂²×C₁₃	#336
prime radical	C₂³×C₁₃	C₂²⋊C₈
cosocle	C₂×C₄×C₈×C₁₃	C₂²
hypercenter	C₈×C₄⋊C₄	D₁₃
hypocenter	C₁₃	#7169
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
12. C₄⋊#6090	C₄	semidirect
13. #11718053	C₄	semidirect
166. C₁₃	#7169	semidirect

Maximal subgroups

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

subgroup	index
#13116592	2
C₄×#88052	2
C₈×C₁₃×C₄⋊C₄	2
#7169	13