Subgroups — #15863332

Order 3328.

1126 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718767	C₂	1	4	yes	maximal
2	#13117243	C₂	1	2	yes	maximal
3	#13115000	C₂	1	4	yes	maximal, complemented
4	#13114996	C₂	1	2	yes	maximal, complemented
5	#13114993	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88180	C₂²	1	4	yes	—
10	C₂×C₄×C₈×C₁₃	C₄	1	8	yes	complemented
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
14	#11718873	—	2	4	no	—
15	#11718767	—	2	16	no	—
16	C₂×#88058	C₂²	1	8	yes	—
17	C₂×#88057	C₂²	1	8	yes	—
18	#11718597	C₂²	1	4	yes	—
19	#11718596	C₂²	1	4	yes	—
20	C₄×C₄⋊Dic₁₃	—	2	4	no	—
21	C₂³×C₄×C₁₃	D₄	1	2	yes	—
22	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
24	C₂²×C₈×C₁₃	D₄	1	8	yes	—
25	C₂²×C₈×C₁₃	Q₈	1	8	yes	—
26	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
27	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
28	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₁₃	D₄	1	2	yes	—
30	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
31	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₂²⋊Dic₁₃	—	2	4	no	—
34	#88180	—	2	16	no	—
35	C₂×C₄⋊Dic₁₃	—	2	8	no	—
36	C₂×C₄⋊Dic₁₃	C₂³	1	8	yes	—
37	C₂×C₄×Dic₁₃	—	2	4	no	—
38	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
39	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	—
41	#88071	—	2	8	no	—
42	#88058	—	2	16	no	—
43	#88057	—	2	16	no	—
44	#20797	—	13	1	no	maximal, Sylow
45	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	—	2	8	no	—
47	C₂²×C₄×C₁₃	C₄⋊C₄	1	8	yes	—
48	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
51	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
54	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
56	C₂²×Dic₁₃	—	2	8	no	—
57	C₂×C₈×C₁₃	C₄⋊C₄	1	32	yes	complemented
58	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
59	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
60	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
61	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
62	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
63	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
64	C₄²×C₁₃	C₄⋊C₄	1	8	yes	—
65	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
66	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
67	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
68	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
69	C₂²⋊Dic₁₃	—	4	8	no	—
70	C₄⋊Dic₁₃	—	2	16	no	—
71	C₄⋊Dic₁₃	—	4	16	no	—
72	C₄×Dic₁₃	—	4	8	no	—
73	C₂×#427	—	13	4	no	—
74	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
75	#2285	—	13	2	no	—
76	#1846	—	13	4	no	—
77	#1845	—	13	2	no	—
78	#1844	—	13	2	no	—
79	C₂³×C₁₃	—	2	4	no	—
80	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
81	C₂³×C₁₃	#127	1	2	yes	—
82	C₂³×C₁₃	C₄⋊Q₈	1	2	yes	—
83	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	—	2	4	no	—
85	C₂×C₄×C₁₃	—	2	16	no	—
86	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
87	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
88	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
89	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
90	C₂×C₄×C₁₃	#127	1	2	yes	—
91	C₂×C₄×C₁₃	C₄⋊Q₈	1	8	yes	—
92	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
93	C₂×C₄×C₁₃	#135	1	8	yes	—
94	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
96	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
97	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
98	C₂×Dic₁₃	—	4	8	no	—
99	C₂×Dic₁₃	—	4	16	no	—
100	C₈×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
101	C₈×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
102	C₂³×C₈	C₂×D₁₃	1	2	yes	—
103	C₂×#117	—	13	4	no	—
104	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
105	#427	—	26	16	no	—
106	C₂×#107	—	13	8	no	—
107	C₂×#106	—	13	8	no	—
108	C₂×C₄×C₈	Dic₁₃	1	8	yes	complemented
109	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
112	#384	—	13	4	no	—
113	#382	—	13	4	no	—
114	#380	—	26	4	no	—
115	C₄×C₄⋊C₄	—	26	4	no	—
116	C₂²×C₁₃	—	2	4	no	—
117	C₂²×C₁₃	—	2	4	no	—
118	C₂²×C₁₃	—	2	4	no	—
119	C₂²×C₁₃	#384	1	4	yes	—
120	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
122	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
123	C₂²×C₁₃	C₂×#135	1	4	yes	—
124	C₄×C₁₃	—	2	4	no	—
125	C₄×C₁₃	#384	1	4	yes	—
126	C₄×C₁₃	#427	1	8	yes	complemented
127	C₄×C₁₃	C₂²×C₄⋊C₄	1	2	yes	—
128	C₄×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
129	Dic₁₃	—	8	8	no	—
130	C₂³×C₄	D₅₂	1	2	yes	—
131	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
132	C₂²×C₈	Dic₂₆	1	8	yes	—
133	C₂²×C₈	D₅₂	1	8	yes	—
134	C₂²×C₈	C₂×Dic₁₃	1	8	yes	—
135	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
136	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
137	#117	—	26	16	no	—
138	C₂×C₄⋊C₄	—	13	8	no	—
139	C₂×C₄⋊C₄	—	26	8	no	—
140	C₂×C₂²⋊C₄	—	26	4	no	—
141	C₂×C₄²	—	26	4	no	—
142	C₂×C₄²	Dic₂₆	1	4	yes	—
143	C₂×C₄²	D₅₂	1	2	yes	—
144	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
145	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
146	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
147	#107	—	26	16	no	—
148	#106	—	26	16	no	—
149	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
150	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
151	C₄×C₈	C₂×Dic₁₃	1	8	yes	—
152	#95	—	26	8	no	—
153	C₂×C₁₃	—	2	4	no	—
154	C₂×C₁₃	C₂×#384	1	1	yes	characteristic
155	C₂×C₁₃	C₂×#427	1	2	yes	—
156	C₂×C₁₃	#3147	1	2	yes	—
157	C₂×C₁₃	#3160	1	2	yes	—
158	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
159	C₂²×C₄	—	2	8	no	—
160	C₂²×C₄	—	26	8	no	—
161	C₂²×C₄	C₄⋊Dic₁₃	1	8	yes	—
162	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
163	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
164	C₂²×C₄	C₂×D₅₂	1	2	yes	—
165	C₂²×C₄	C₂×D₅₂	1	4	yes	—
166	C₂²×C₄	C₂×D₅₂	1	2	yes	—
167	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
168	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
169	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
170	C₂×C₈	C₄⋊Dic₁₃	1	32	yes	complemented
171	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
172	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
173	C₂×C₈	C₂×D₅₂	1	4	yes	—
174	C₂×C₈	C₂×D₅₂	1	4	yes	—
175	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
176	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
177	C₄⋊C₄	—	26	16	no	—
178	C₄⋊C₄	—	52	16	no	—
179	C₂²⋊C₄	—	52	8	no	—
180	C₄²	—	52	8	no	—
181	C₄²	C₄⋊Dic₁₃	1	8	yes	—
182	C₄²	C₂×Dic₂₆	1	2	yes	—
183	C₄²	C₂×Dic₂₆	1	2	yes	—
184	C₄²	C₂²×Dic₁₃	1	2	yes	—
185	C₄²	C₂²×Dic₁₃	1	2	yes	—
186	C₁₃	#20797	1	1	yes	characteristic, complemented, Sylow
187	C₂³	—	2	4	no	—
188	C₂³	#88123	1	2	yes	—
189	C₂³	#88128	1	2	yes	—
190	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
191	C₂³	C₂²×D₅₂	1	1	yes	characteristic
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	16	no	—
194	C₂×C₄	—	52	8	no	—
195	C₂×C₄	—	52	16	no	—
196	C₂×C₄	#88123	1	8	yes	—
197	C₂×C₄	#88123	1	2	yes	—
198	C₂×C₄	#88128	1	2	yes	—
199	C₂×C₄	#88158	1	8	yes	—
200	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
201	C₂×C₄	C₂×C₄⋊Dic₁₃	1	8	yes	—
202	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
203	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
204	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
205	C₂×C₄	C₂²×Dic₂₆	1	2	yes	—
206	C₂×C₄	C₂²×D₅₂	1	2	yes	—
207	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
208	C₈	C₂×C₄⋊Dic₁₃	1	8	yes	—
209	C₈	C₂×C₄⋊Dic₁₃	1	8	yes	—
210	C₂²	—	2	4	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	4	no	—
213	C₂²	#11718596	1	4	yes	—
214	C₂²	C₂×#88123	1	1	yes	characteristic
215	C₂²	C₂×#88128	1	1	yes	characteristic
216	C₂²	C₂×#88158	1	4	yes	—
217	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
218	C₄	—	2	4	no	—
219	C₄	—	104	8	no	—
220	C₄	#11718596	1	4	yes	—
221	C₄	#11718767	1	8	yes	complemented
222	C₄	C₂×#88123	1	2	yes	—
223	C₄	C₂²×C₄⋊Dic₁₃	1	2	yes	—
224	C₂	—	2	4	no	—
225	C₂	C₂×#11718596	1	1	yes	characteristic
226	C₂	C₂×#11718767	1	2	yes	—
227	C₂	#13125264	1	2	yes	—
228	C₂	#13125350	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	#20797
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #13115000	C₂	semidirect
4. #13114996	C₂	semidirect
5. #13114993	C₂	semidirect
10. C₂×C₄×C₈×C₁₃	C₄	semidirect
57. C₂×C₈×C₁₃	C₄⋊C₄	semidirect
108. C₂×C₄×C₈	Dic₁₃	semidirect
126. C₄×C₁₃	#427	semidirect
170. C₂×C₈	C₄⋊Dic₁₃	semidirect
186. C₁₃	#20797	semidirect
221. C₄	#11718767	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718767	2
#13117243	2
#13115000	2
#13114996	2
#13114993	2
C₂²×C₄×C₈×C₁₃	2
#20797	13