Subgroups — #15781132

Order 3328.

614 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718356	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718083	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1845	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₁₀₄	—	2	1	no	—
6	C₂²×D₁₀₄	C₂²	1	2	yes	—
7	C₂×#88128	—	2	1	no	—
8	C₂×C₁₃×#107	—	2	1	no	—
9	C₂×C₁₃×#107	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#384	—	2	1	no	—
12	C₂²×#6090	C₂²	1	2	yes	—
13	#11718356	—	2	2	no	—
14	#11718356	C₄	1	4	yes	complemented
15	#11718083	—	2	8	no	—
16	C₄×#6090	—	2	2	no	—
17	C₂²×D₅₂	—	2	2	no	—
18	C₂²×D₅₂	—	4	2	no	—
19	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
22	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
23	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
24	C₂×D₁₀₄	—	2	4	no	—
25	C₂×D₁₀₄	—	2	4	no	—
26	C₂×D₁₀₄	—	2	4	no	—
27	C₂×D₁₀₄	—	2	2	no	—
28	C₂×D₁₀₄	—	4	2	no	—
29	C₂×D₁₀₄	—	4	2	no	—
30	C₂×D₁₀₄	C₂×C₄	1	4	yes	—
31	#88128	—	2	4	no	—
32	#88128	—	4	2	no	—
33	C₁₃×#107	—	2	4	no	—
34	C₁₃×#107	—	4	2	no	—
35	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₁₃	D₄	1	2	yes	—
37	C₂×#6090	—	2	2	no	—
38	C₂×#6090	—	2	2	no	—
39	C₂×#6090	D₄	1	4	yes	complemented
40	#13202	—	13	1	no	maximal, Sylow
41	C₂³×D₁₃	—	4	2	no	—
42	C₂²×C₄×C₁₃	—	4	2	no	—
43	C₂²×C₄×C₁₃	D₈	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂×D₅₂	—	2	4	no	—
46	C₂×D₅₂	—	4	2	no	—
47	C₂×D₅₂	—	4	2	no	—
48	C₂×D₅₂	—	4	2	no	—
49	C₂×D₅₂	—	4	2	no	—
50	C₂×D₅₂	—	4	8	no	—
51	C₂×D₅₂	—	4	2	no	—
52	C₂×D₅₂	—	8	4	no	—
53	C₂×D₅₂	—	8	4	no	—
54	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₁₃	D₈	1	2	yes	—
56	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
57	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃	#41	1	2	yes	—
61	C₁₃×C₄⋊C₄	—	4	2	no	—
62	C₁₃×C₄⋊C₄	—	4	2	no	—
63	C₁₃×C₄⋊C₄	—	8	4	no	—
64	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₁₃	SD₁₆	1	2	yes	—
66	D₁₀₄	—	2	8	no	—
67	D₁₀₄	—	4	8	no	—
68	D₁₀₄	—	4	4	no	—
69	#6090	—	2	4	no	—
70	C₂×#493	—	13	1	no	—
71	C₂×#357	—	13	4	no	—
72	C₂×C₄×C₁₆	—	13	1	no	—
73	#1845	D₁₃	1	1	yes	characteristic, complemented
74	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
75	C₂²×D₁₃	—	4	4	no	—
76	C₂²×D₁₃	—	8	2	no	—
77	C₂²×D₁₃	—	8	2	no	—
78	C₂²×D₁₃	—	8	4	no	—
79	C₂²×D₁₃	—	8	4	no	—
80	C₂×C₄×C₁₃	—	8	2	no	—
81	C₂×C₄×C₁₃	—	8	4	no	—
82	C₂×C₄×C₁₃	#102	1	4	yes	—
83	C₂×C₄×C₁₃	D₁₆	1	4	yes	—
84	C₂×C₄×C₁₃	SD₃₂	1	4	yes	—
85	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
87	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
91	D₅₂	—	4	4	no	—
92	D₅₂	—	4	4	no	—
93	D₅₂	—	8	16	no	—
94	D₅₂	—	8	2	no	—
95	D₅₂	—	8	2	no	—
96	C₈×C₁₃	#102	1	4	yes	—
97	C₈×C₁₃	C₄×D₄	1	2	yes	—
98	C₈×C₁₃	#121	1	2	yes	—
99	C₂²×D₈	—	13	2	no	—
100	C₂²×D₈	—	26	1	no	—
101	C₂×#127	—	26	1	no	—
102	C₂²×C₁₆	—	13	2	no	—
103	#493	—	13	4	no	—
104	#493	—	26	2	no	—
105	C₂×#107	—	2	1	no	—
106	C₂×#107	C₂×D₁₃	1	2	yes	—
107	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
108	#384	—	2	1	no	—
109	#357	—	26	8	no	—
110	C₄×C₁₆	—	26	2	no	—
111	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
112	C₂²×C₁₃	#486	1	1	yes	characteristic
113	C₂²×C₁₃	#493	1	1	yes	characteristic
114	C₂²×C₁₃	C₂×D₁₆	1	2	yes	—
115	C₂²×C₁₃	C₂×SD₃₂	1	2	yes	—
116	C₂×D₁₃	—	8	8	no	—
117	C₂×D₁₃	—	8	4	no	—
118	C₂×D₁₃	—	16	4	no	—
119	C₂×D₁₃	—	16	2	no	—
120	C₂×D₁₃	—	16	2	no	—
121	C₄×C₁₃	—	16	2	no	—
122	C₄×C₁₃	#357	1	8	yes	complemented
123	C₄×C₁₃	#390	1	1	yes	characteristic
124	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
125	C₄×C₁₃	#486	1	1	yes	characteristic
126	C₄×C₁₃	#492	1	1	yes	characteristic
127	C₂²×D₄	—	26	2	no	—
128	C₂²×D₄	—	52	2	no	—
129	C₂×D₈	—	13	4	no	—
130	C₂×D₈	—	26	4	no	—
131	C₂×D₈	—	26	2	no	—
132	C₂×D₈	—	26	4	no	—
133	C₂×D₈	—	26	4	no	—
134	C₂×D₈	—	52	2	no	—
135	C₂×D₈	—	52	2	no	—
136	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
138	#127	—	26	4	no	—
139	#127	—	52	2	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	4	2	no	—
142	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
143	C₂×C₁₆	—	13	4	no	—
144	C₂×C₁₆	—	26	2	no	—
145	C₂×C₁₆	—	26	2	no	—
146	#107	—	2	4	no	—
147	#107	—	4	2	no	—
148	C₄×C₈	C₄×D₁₃	1	2	yes	—
149	C₄×C₈	D₅₂	1	2	yes	—
150	C₂×C₁₃	#1957	1	1	yes	characteristic
151	C₂×C₁₃	C₂×#357	1	2	yes	—
152	C₂×C₁₃	#2240	1	2	yes	—
153	C₂×C₁₃	#2246	1	1	yes	characteristic
154	C₂×C₁₃	#2248	1	1	yes	characteristic
155	D₁₃	—	16	4	no	—
156	C₂⁴	—	52	2	no	—
157	C₂×D₄	—	26	4	no	—
158	C₂×D₄	—	52	2	no	—
159	C₂×D₄	—	52	2	no	—
160	C₂×D₄	—	52	2	no	—
161	C₂×D₄	—	52	2	no	—
162	C₂×D₄	—	52	2	no	—
163	C₂×D₄	—	52	8	no	—
164	C₂×D₄	—	104	4	no	—
165	C₂×D₄	—	104	4	no	—
166	C₂²×C₄	—	4	2	no	—
167	C₂²×C₄	#6104	1	2	yes	—
168	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
169	D₈	—	26	8	no	—
170	D₈	—	52	4	no	—
171	D₈	—	52	8	no	—
172	C₂×C₈	#6103	1	2	yes	—
173	C₂×C₈	#6104	1	2	yes	—
174	C₂×C₈	#6106	1	2	yes	—
175	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
176	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
177	C₂×C₈	D₄×D₁₃	1	2	yes	—
178	C₂×C₈	#6131	1	2	yes	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	8	4	no	—
182	C₄²	#6103	1	2	yes	—
183	C₄²	#6106	1	2	yes	—
184	C₁₆	—	26	4	no	—
185	C₁₃	#13202	1	1	yes	characteristic, complemented, Sylow
186	C₂³	—	52	4	no	—
187	C₂³	—	104	2	no	—
188	C₂³	—	104	2	no	—
189	C₂³	—	104	4	no	—
190	C₂³	—	104	4	no	—
191	C₂³	C₂×#6104	1	1	yes	characteristic
192	D₄	—	52	4	no	—
193	D₄	—	52	4	no	—
194	D₄	—	104	2	no	—
195	D₄	—	104	2	no	—
196	D₄	—	104	16	no	—
197	C₂×C₄	—	8	2	no	—
198	C₂×C₄	—	8	4	no	—
199	C₂×C₄	#88049	1	4	yes	—
200	C₂×C₄	#88066	1	4	yes	—
201	C₂×C₄	#88068	1	4	yes	—
202	C₂×C₄	C₂×#6103	1	1	yes	characteristic
203	C₂×C₄	C₂×#6104	1	1	yes	characteristic
204	C₂×C₄	#88194	1	1	yes	characteristic
205	C₂×C₄	C₂×#6106	1	1	yes	characteristic
206	C₂×C₄	C₂×#6106	1	1	yes	characteristic
207	C₂×C₄	#88201	1	1	yes	characteristic
208	C₈	#88049	1	4	yes	—
209	C₈	#88147	1	2	yes	—
210	C₈	#88149	1	2	yes	—
211	C₂²	—	104	8	no	—
212	C₂²	—	104	4	no	—
213	C₂²	—	208	4	no	—
214	C₂²	—	208	2	no	—
215	C₂²	—	208	2	no	—
216	C₂²	C₂×#88049	1	1	yes	characteristic
217	C₂²	#11718734	1	1	yes	characteristic
218	C₂²	#11718749	1	1	yes	characteristic
219	C₂²	C₂×#88066	1	2	yes	—
220	C₂²	C₂×#88068	1	2	yes	—
221	C₄	—	16	2	no	—
222	C₄	#11718083	1	8	yes	complemented
223	C₄	C₂×#88049	1	1	yes	characteristic
224	C₄	#11718649	1	1	yes	characteristic
225	C₄	#11718746	1	1	yes	characteristic
226	C₄	#11718749	1	1	yes	characteristic
227	C₂	—	208	4	no	—
228	C₂	#13115641	1	1	yes	characteristic
229	C₂	C₂×#11718083	1	2	yes	—
230	C₂	#13117061	1	2	yes	—
231	C₂	#13117085	1	1	yes	characteristic
232	C₂	#13117093	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6104
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1845	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂×#11718356	C₂
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1845	D₁₃
hypocenter	C₁₃	#13202
LCS	G ▷ C₂×C₈×C₁₃ ▷ 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
2. C₂×#11718083	C₂	semidirect
3. C₂×C₄×#6090	C₂	semidirect
4. C₁₃×#1845	C₂	semidirect
14. #11718356	C₄	semidirect
39. C₂×#6090	D₄	semidirect
73. #1845	D₁₃	semidirect
122. C₄×C₁₃	#357	semidirect
185. C₁₃	#13202	semidirect
222. C₄	#11718083	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718356	2
C₂×#11718083	2
C₂×C₄×#6090	2
C₁₃×#1845	2
#13202	13