Subgroups — #16023016

Order 3328.

2858 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×#11719517	C₂	1	2	yes	maximal
2	#13117757	C₂	1	8	yes	maximal, complemented
3	Q₈×C₄⋊Dic₁₃	C₂	1	8	yes	maximal, complemented
4	#13117454	C₂	1	6	yes	maximal
5	#13117243	C₂	1	6	yes	maximal
6	C₂²×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₁₃	C₄	1	16	yes	complemented
9	C₂×C₄×Q₈×C₁₃	C₂²	1	4	yes	—
10	C₂×C₄×Q₈×C₁₃	C₂²	1	4	yes	—
11	C₂²×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
12	C₂²×C₄²×C₁₃	C₂²	1	3	yes	—
13	#11719517	—	2	16	no	—
14	C₂×Q₈×Dic₁₃	C₂²	1	16	yes	—
15	C₂×#88180	C₂²	1	12	yes	—
16	#11718873	C₂²	1	12	yes	—
17	#11718873	C₂²	1	4	yes	—
18	#11718646	C₂²	1	48	yes	—
19	#11718597	C₂²	1	12	yes	—
20	#11718596	C₂²	1	12	yes	—
21	C₄×C₄⋊Dic₁₃	C₂²	1	12	yes	—
22	C₄×C₄⋊Dic₁₃	C₂²	1	12	yes	—
23	C₂²×Q₈×C₁₃	C₂×C₄	1	8	yes	—
24	C₂²×Q₈×C₁₃	D₄	1	8	yes	—
25	C₂²×Q₈×C₁₃	Q₈	1	8	yes	—
26	C₂²×Q₈×C₁₃	C₂³	1	4	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
29	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
30	C₄×Q₈×C₁₃	C₂×C₄	1	32	yes	—
31	C₄×Q₈×C₁₃	C₂×C₄	1	32	yes	—
32	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	24	yes	—
33	C₂×C₁₃×C₄⋊C₄	C₂³	1	12	yes	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	24	yes	—
35	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
36	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
37	C₂×C₂²⋊Dic₁₃	C₂³	1	8	yes	—
38	Q₈×Dic₁₃	—	2	64	no	—
39	#88180	—	2	48	no	—
40	C₂×C₄⋊Dic₁₃	C₂³	1	24	yes	—
41	C₂×C₄⋊Dic₁₃	C₂³	1	8	yes	—
42	C₂×C₄⋊Dic₁₃	C₂³	1	24	yes	—
43	C₂×C₄×Dic₁₃	C₂³	1	24	yes	—
44	#88071	C₂³	1	8	yes	—
45	#88071	C₂³	1	24	yes	—
46	#34182	—	13	1	no	maximal, Sylow
47	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
48	C₂×Q₈×C₁₃	C₄⋊C₄	1	64	yes	complemented
49	C₂×Q₈×C₁₃	C₂²×C₄	1	16	yes	—
50	C₂×Q₈×C₁₃	C₂×D₄	1	16	yes	—
51	C₂×Q₈×C₁₃	C₂×Q₈	1	16	yes	—
52	C₂²×C₄×C₁₃	—	2	4	no	—
53	C₂²×C₄×C₁₃	—	4	6	no	—
54	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
57	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
58	C₂²×C₄×C₁₃	C₂×Q₈	1	12	yes	—
59	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
61	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
62	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
63	C₂²×Dic₁₃	C₂⁴	1	16	yes	—
64	C₁₃×C₄⋊C₄	C₂²×C₄	1	48	yes	—
65	C₄²×C₁₃	C₂²×C₄	1	24	yes	—
66	C₄²×C₁₃	C₂²×C₄	1	24	yes	—
67	C₂²⋊Dic₁₃	—	2	16	no	—
68	C₄⋊Dic₁₃	—	2	48	no	—
69	C₄⋊Dic₁₃	—	2	16	no	—
70	C₄⋊Dic₁₃	—	4	24	no	—
71	C₄×Dic₁₃	—	2	48	no	—
72	C₂×#519	—	13	2	no	—
73	C₂²×C₄×Q₈	D₁₃	1	1	yes	characteristic
74	#2351	—	13	8	no	—
75	Q₈×C₄⋊C₄	—	13	8	no	—
76	#2316	—	13	6	no	—
77	#2285	—	13	6	no	—
78	C₂³×C₁₃	—	2	2	no	—
79	C₂³×C₁₃	—	2	2	no	—
80	C₂³×C₁₃	C₂³×C₄	1	2	yes	—
81	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
82	C₂³×C₁₃	C₂²×Q₈	1	2	yes	—
83	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
84	Q₈×C₁₃	C₂×C₄⋊C₄	1	64	yes	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	12	no	—
87	C₂×C₄×C₁₃	—	2	8	no	—
88	C₂×C₄×C₁₃	—	4	12	no	—
89	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	48	yes	—
90	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
91	C₂×C₄×C₁₃	C₂³×C₄	1	12	yes	—
92	C₂×C₄×C₁₃	C₂³×C₄	1	12	yes	—
93	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
94	C₂×C₄×C₁₃	C₂²×D₄	1	12	yes	—
95	C₂×C₄×C₁₃	C₂²×Q₈	1	12	yes	—
96	C₂×C₄×C₁₃	2₋¹⁺⁴	1	4	yes	—
97	C₂×Dic₁₃	—	2	16	no	—
98	C₂×Dic₁₃	—	2	16	no	—
99	C₂×Dic₁₃	—	2	16	no	—
100	C₂³×Q₈	C₂×D₁₃	1	1	yes	characteristic
101	#519	—	26	16	no	—
102	C₂×C₄×Q₈	—	13	16	no	—
103	C₂×C₄×Q₈	Dic₁₃	1	16	yes	complemented
104	C₂×C₄×Q₈	C₂×D₁₃	1	4	yes	—
105	C₂×C₄×Q₈	C₂×D₁₃	1	4	yes	—
106	C₂×#117	—	13	12	no	—
107	C₂²×C₄⋊C₄	C₂×D₁₃	1	3	yes	—
108	C₂²×C₄²	C₂×D₁₃	1	3	yes	—
109	#389	—	13	48	no	—
110	#384	—	13	12	no	—
111	#382	—	13	12	no	—
112	#380	—	13	12	no	—
113	#380	—	13	4	no	—
114	C₄×C₄⋊C₄	—	13	12	no	—
115	C₄×C₄⋊C₄	—	13	12	no	—
116	C₂²×C₁₃	—	2	2	no	—
117	C₂²×C₁₃	—	2	2	no	—
118	C₂²×C₁₃	—	2	4	no	—
119	C₂²×C₁₃	—	2	4	no	—
120	C₂²×C₁₃	C₂²×C₄⋊C₄	1	4	yes	—
121	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
122	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
123	C₂²×C₁₃	C₂³×Q₈	1	1	yes	characteristic
124	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	2	yes	—
125	C₂²×C₁₃	#585	1	2	yes	—
126	C₄×C₁₃	—	2	2	no	—
127	C₄×C₁₃	—	2	12	no	—
128	C₄×C₁₃	C₂²×C₄⋊C₄	1	24	yes	—
129	C₄×C₁₃	#519	1	4	yes	complemented
130	Dic₁₃	—	4	16	no	—
131	C₂²×Q₈	Dic₂₆	1	8	yes	—
132	C₂²×Q₈	D₅₂	1	8	yes	—
133	C₂²×Q₈	C₂×Dic₁₃	1	8	yes	—
134	C₂²×Q₈	C₂²×D₁₃	1	4	yes	—
135	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
136	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
137	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
138	C₄×Q₈	—	26	64	no	—
139	C₄×Q₈	C₂×Dic₁₃	1	32	yes	—
140	C₄×Q₈	C₂×Dic₁₃	1	32	yes	—
141	#117	—	26	48	no	—
142	C₂×C₄⋊C₄	—	13	24	no	—
143	C₂×C₄⋊C₄	—	13	8	no	—
144	C₂×C₄⋊C₄	—	13	24	no	—
145	C₂×C₄⋊C₄	C₂×Dic₁₃	1	24	yes	—
146	C₂×C₄⋊C₄	C₂²×D₁₃	1	12	yes	—
147	C₂×C₂²⋊C₄	—	13	8	no	—
148	C₂×C₄²	—	13	24	no	—
149	C₂×C₄²	C₂×Dic₁₃	1	24	yes	—
150	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
151	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
152	#95	—	13	8	no	—
153	#95	—	13	24	no	—
154	C₂×C₁₃	—	2	4	no	—
155	C₂×C₁₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₁₃	C₂×#519	1	1	yes	characteristic
157	C₂×C₁₃	#3428	1	1	yes	characteristic
158	C₂×C₁₃	#3497	1	2	yes	—
159	C₂×C₁₃	#3502	1	2	yes	—
160	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
161	C₂×Q₈	C₄⋊Dic₁₃	1	64	yes	complemented
162	C₂×Q₈	C₂×Dic₂₆	1	16	yes	—
163	C₂×Q₈	C₂×D₅₂	1	16	yes	—
164	C₂×Q₈	C₂²×Dic₁₃	1	16	yes	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	4	6	no	—
167	C₂²×C₄	—	13	16	no	—
168	C₂²×C₄	C₂×Dic₂₆	1	12	yes	—
169	C₂²×C₄	C₂×D₅₂	1	12	yes	—
170	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
171	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
172	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
173	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
174	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
175	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
176	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
177	C₄⋊C₄	—	26	48	no	—
178	C₄⋊C₄	—	26	16	no	—
179	C₄⋊C₄	—	52	24	no	—
180	C₄⋊C₄	C₂²×Dic₁₃	1	48	yes	—
181	C₂²⋊C₄	—	26	16	no	—
182	C₄²	—	26	48	no	—
183	C₄²	C₂²×Dic₁₃	1	24	yes	—
184	C₄²	C₂²×Dic₁₃	1	24	yes	—
185	C₁₃	#34182	1	1	yes	characteristic, complemented, Sylow
186	C₂³	—	2	2	no	—
187	C₂³	—	2	2	no	—
188	C₂³	C₂²×Dic₂₆	1	2	yes	—
189	C₂³	C₂²×D₅₂	1	2	yes	—
190	C₂³	C₂³×Dic₁₃	1	2	yes	—
191	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
192	Q₈	C₂×C₄⋊Dic₁₃	1	64	yes	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	12	no	—
195	C₂×C₄	—	2	8	no	—
196	C₂×C₄	—	4	12	no	—
197	C₂×C₄	—	26	16	no	—
198	C₂×C₄	—	26	16	no	—
199	C₂×C₄	—	26	16	no	—
200	C₂×C₄	C₂×C₄⋊Dic₁₃	1	48	yes	—
201	C₂×C₄	C₂²×Dic₂₆	1	12	yes	—
202	C₂×C₄	C₂²×D₅₂	1	12	yes	—
203	C₂×C₄	#88254	1	4	yes	—
204	C₂×C₄	C₂³×Dic₁₃	1	2	yes	—
205	C₂×C₄	C₂³×Dic₁₃	1	12	yes	—
206	C₂×C₄	C₂³×Dic₁₃	1	12	yes	—
207	C₂×C₄	C₂³×Dic₁₃	1	2	yes	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	2	4	no	—
212	C₂²	C₂²×C₄⋊Dic₁₃	1	4	yes	—
213	C₂²	C₂³×Dic₂₆	1	1	yes	characteristic
214	C₂²	C₂³×D₅₂	1	1	yes	characteristic
215	C₂²	C₂×#88254	1	2	yes	—
216	C₂²	#11719648	1	2	yes	—
217	C₂²	C₂⁴×Dic₁₃	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	12	no	—
220	C₄	—	52	16	no	—
221	C₄	C₂²×C₄⋊Dic₁₃	1	24	yes	—
222	C₄	#11719517	1	4	yes	complemented
223	C₂	—	2	4	no	—
224	C₂	C₂³×C₄⋊Dic₁₃	1	1	yes	characteristic
225	C₂	C₂×#11719517	1	1	yes	characteristic
226	C₂	#13128039	1	1	yes	characteristic
227	C₂	#13128607	1	2	yes	—
228	C₂	#13128646	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₁₃
commutator	C₂²×C₁₃	C₂⁴×C₄
Fitting	C₂²×C₄×Q₈×C₁₃	C₂
socle, cosocle	C₂³×C₁₃	C₂⁵
prime radical	C₂⁴×C₁₃	C₂⁴
hypercenter	C₂²×C₄×Q₈	D₁₃
hypocenter	C₁₃	#34182
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. #13117757	C₂	semidirect
3. Q₈×C₄⋊Dic₁₃	C₂	semidirect
8. C₂×C₄×Q₈×C₁₃	C₄	semidirect
48. C₂×Q₈×C₁₃	C₄⋊C₄	semidirect
103. C₂×C₄×Q₈	Dic₁₃	semidirect
129. C₄×C₁₃	#519	semidirect
161. C₂×Q₈	C₄⋊Dic₁₃	semidirect
185. C₁₃	#34182	semidirect
222. C₄	#11719517	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719517	2
#13117757	2
Q₈×C₄⋊Dic₁₃	2
#13117454	2
#13117243	2
C₂²×C₄×Q₈×C₁₃	2
#34182	13