Subgroups — #15750104

Order 3328.

262 conjugacy classes of subgroups.

The subgroup lattice has 226 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	C₁₃×#1845	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#107	—	2	1	no	—
5	C₂×C₁₃×#107	C₄	1	2	yes	—
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₁₃×#384	—	2	1	no	—
8	C₂²×#6090	—	2	1	no	—
9	C₂×C₄⋊#1055	—	2	1	no	—
10	C₄⋊#6090	—	2	1	no	—
11	C₄⋊#6090	C₄	1	2	yes	complemented
12	#11718050	—	2	1	no	—
13	#11718050	C₄	1	2	yes	complemented
14	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
16	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
17	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
18	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
19	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
20	C₂²×#1055	—	4	1	no	—
21	C₁₃×#107	—	4	1	no	—
22	C₁₃×#107	—	4	1	no	—
23	C₁₃×#107	—	4	2	no	—
24	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
28	C₂×#6090	—	2	2	no	—
29	C₂×#6090	—	4	1	no	—
30	C₂×#6090	—	4	1	no	—
31	C₄⋊#1055	—	2	2	no	—
32	C₄⋊#1055	—	4	1	no	—
33	#7274	—	13	1	no	maximal, Sylow
34	C₂²×C₄×C₁₃	—	4	2	no	—
35	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
38	C₂×C₈×C₁₃	—	2	1	no	—
39	C₂×C₈×C₁₃	—	2	1	no	—
40	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
45	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
47	C₂×C₈×C₁₃	Q₁₆	1	1	yes	characteristic
48	C₁₃×C₄⋊C₄	—	4	2	no	—
49	C₁₃×C₄⋊C₄	—	4	2	no	—
50	C₁₃×C₄⋊C₄	—	8	2	no	—
51	C₁₃×C₄⋊C₄	—	8	1	no	—
52	C₁₃×C₄⋊C₄	—	8	1	no	—
53	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
55	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
56	C₄²×C₁₃	Q₁₆	1	1	yes	characteristic
57	C₂×#1055	—	4	2	no	—
58	C₂×#1055	—	8	1	no	—
59	C₂×#1055	—	8	1	no	—
60	#6090	—	4	2	no	—
61	C₂×C₄⋊C₁₆	—	13	1	no	—
62	#1845	D₁₃	1	1	yes	characteristic
63	C₂×#335	—	13	1	no	—
64	C₂³×C₁₃	#102	1	1	yes	characteristic
65	C₂×C₄×C₁₃	—	2	1	no	—
66	C₂×C₄×C₁₃	—	2	1	no	—
67	C₂×C₄×C₁₃	—	8	2	no	—
68	C₂×C₄×C₁₃	—	8	2	no	—
69	C₂×C₄×C₁₃	—	8	2	no	—
70	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
71	C₂×C₄×C₁₃	#100	1	1	yes	characteristic
72	C₂×C₄×C₁₃	#101	1	1	yes	characteristic
73	C₂×C₄×C₁₃	#102	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	#103	1	1	yes	characteristic
77	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
78	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
79	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
80	C₂×C₄×C₁₃	D₁₆	1	1	yes	characteristic
81	C₂×C₄×C₁₃	SD₃₂	1	1	yes	characteristic
82	C₂×C₄×C₁₃	SD₃₂	1	1	yes	characteristic
83	C₂×C₄×C₁₃	Q₃₂	1	1	yes	characteristic
84	C₈×C₁₃	—	2	1	no	—
85	C₈×C₁₃	—	2	1	no	—
86	C₈×C₁₃	#102	1	1	yes	characteristic
87	C₈×C₁₃	#103	1	1	yes	characteristic
88	C₈×C₁₃	#106	1	1	yes	characteristic
89	C₈×C₁₃	#107	1	1	yes	characteristic
90	#1055	—	8	2	no	—
91	C₂²×C₁₆	—	26	1	no	—
92	C₂×#107	—	2	1	no	—
93	C₂×#107	Dic₁₃	1	2	yes	—
94	C₂×C₄⋊C₈	—	26	1	no	—
95	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
96	#384	—	2	1	no	—
97	C₄⋊C₁₆	—	13	2	no	—
98	C₄⋊C₁₆	—	26	1	no	—
99	#335	—	13	2	no	—
100	#335	—	26	1	no	—
101	C₂²×C₁₃	#331	1	1	yes	characteristic
102	C₂²×C₁₃	#332	1	1	yes	characteristic
103	C₂²×C₁₃	#340	1	1	yes	characteristic
104	C₂²×C₁₃	#357	1	1	yes	characteristic
105	C₂²×C₁₃	#358	1	1	yes	characteristic
106	C₂²×C₁₃	#361	1	1	yes	characteristic
107	C₂²×C₁₃	#362	1	1	yes	characteristic
108	C₄×C₁₃	—	2	1	no	—
109	C₄×C₁₃	—	2	1	no	—
110	C₄×C₁₃	—	16	2	no	—
111	C₄×C₁₃	#332	1	1	yes	characteristic
112	C₄×C₁₃	#333	1	1	yes	characteristic
113	C₄×C₁₃	#340	1	1	yes	characteristic
114	C₄×C₁₃	#343	1	1	yes	characteristic
115	C₄×C₁₃	#357	1	1	yes	characteristic
116	C₄×C₁₃	#358	1	1	yes	characteristic
117	C₄×C₁₃	#366	1	1	yes	characteristic
118	C₄×C₁₃	#367	1	1	yes	characteristic
119	C₂²×C₈	—	52	1	no	—
120	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
121	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	4	1	no	—
124	C₂×C₄⋊C₄	—	4	1	no	—
125	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
126	C₂×C₁₆	—	26	2	no	—
127	C₂×C₁₆	—	52	1	no	—
128	C₂×C₁₆	—	52	1	no	—
129	#107	—	4	1	no	—
130	#107	—	4	1	no	—
131	#107	—	4	2	no	—
132	C₄⋊C₈	—	26	2	no	—
133	C₄⋊C₈	—	52	1	no	—
134	C₄×C₈	Dic₂₆	1	1	yes	characteristic
135	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
136	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
137	C₄×C₈	D₅₂	1	1	yes	characteristic
138	C₂×C₁₃	#1296	1	1	yes	characteristic
139	C₂×C₁₃	#1361	1	1	yes	characteristic
140	C₂×C₁₃	#1362	1	1	yes	characteristic
141	C₂×C₁₃	#1364	1	1	yes	characteristic
142	C₂×C₁₃	#1365	1	1	yes	characteristic
143	C₂×C₁₃	#1380	1	1	yes	characteristic
144	C₂×C₁₃	#1387	1	1	yes	characteristic
145	C₂²×C₄	—	4	2	no	—
146	C₂²×C₄	#6104	1	1	yes	characteristic
147	C₂²×C₄	#6105	1	1	yes	characteristic
148	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	52	2	no	—
152	C₂×C₈	—	104	1	no	—
153	C₂×C₈	—	104	1	no	—
154	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
155	C₂×C₈	#6101	1	1	yes	characteristic
156	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
157	C₂×C₈	#6103	1	1	yes	characteristic
158	C₂×C₈	#6104	1	1	yes	characteristic
159	C₂×C₈	#6105	1	1	yes	characteristic
160	C₂×C₈	#6106	1	1	yes	characteristic
161	C₂×C₈	#6107	1	1	yes	characteristic
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	8	2	no	—
165	C₄⋊C₄	—	8	1	no	—
166	C₄⋊C₄	—	8	1	no	—
167	C₄²	#6101	1	1	yes	characteristic
168	C₄²	#6103	1	1	yes	characteristic
169	C₄²	#6106	1	1	yes	characteristic
170	C₄²	#6107	1	1	yes	characteristic
171	C₁₆	—	52	2	no	—
172	C₁₃	#7274	1	1	yes	characteristic, complemented, Sylow
173	C₂³	#88072	1	1	yes	characteristic
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	#88047	1	1	yes	characteristic
180	C₂×C₄	#88048	1	1	yes	characteristic
181	C₂×C₄	#88049	1	1	yes	characteristic
182	C₂×C₄	#88050	1	1	yes	characteristic
183	C₂×C₄	#88066	1	1	yes	characteristic
184	C₂×C₄	#88067	1	1	yes	characteristic
185	C₂×C₄	#88068	1	1	yes	characteristic
186	C₂×C₄	#88069	1	1	yes	characteristic
187	C₂×C₄	#88071	1	1	yes	characteristic
188	C₂×C₄	#88072	1	1	yes	characteristic
189	C₂×C₄	#88073	1	1	yes	characteristic
190	C₂×C₄	#88075	1	1	yes	characteristic
191	C₂×C₄	#88075	1	1	yes	characteristic
192	C₂×C₄	#88076	1	1	yes	characteristic
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	104	2	no	—
196	C₈	#88047	1	1	yes	characteristic
197	C₈	#88048	1	1	yes	characteristic
198	C₈	#88049	1	1	yes	characteristic
199	C₈	#88050	1	1	yes	characteristic
200	C₂²	#11718121	1	1	yes	characteristic
201	C₂²	#11718136	1	1	yes	characteristic
202	C₂²	#11718139	1	1	yes	characteristic
203	C₂²	#11718154	1	1	yes	characteristic
204	C₂²	#11718155	1	1	yes	characteristic
205	C₂²	#11718156	1	1	yes	characteristic
206	C₂²	#11718157	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	16	2	no	—
210	C₄	#11718081	1	1	yes	characteristic
211	C₄	#11718082	1	1	yes	characteristic
212	C₄	#11718083	1	1	yes	characteristic
213	C₄	#11718084	1	1	yes	characteristic
214	C₄	#11718121	1	1	yes	characteristic
215	C₄	#11718122	1	1	yes	characteristic
216	C₄	#11718138	1	1	yes	characteristic
217	C₄	#11718139	1	1	yes	characteristic
218	C₂	#13112719	1	1	yes	characteristic
219	C₂	#13112882	1	1	yes	characteristic
220	C₂	#13112885	1	1	yes	characteristic
221	C₂	#13112891	1	1	yes	characteristic
222	C₂	#13112894	1	1	yes	characteristic
223	C₂	#13112927	1	1	yes	characteristic
224	C₂	#13112944	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88072
commutator	C₂×C₈×C₁₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₁₃
Fitting	C₁₃×#1845	C₂
socle, prime radical	C₂³×C₁₃	#102
cosocle	C₂×C₄×C₈×C₁₃	C₂²
hypercenter	#1845	D₁₃
hypocenter	C₁₃	#7274
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
11. C₄⋊#6090	C₄	semidirect
13. #11718050	C₄	semidirect
172. C₁₃	#7274	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
C₁₃×#1845	2
#7274	13