Subgroups — #16022949

Order 3328.

5318 conjugacy classes of subgroups.

The subgroup lattice has 216 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, complemented
2	#13117677	C₂	1	16	yes	maximal, complemented
3	#13117418	C₂	1	6	yes	maximal, complemented
4	#13117219	C₂	1	6	yes	maximal, complemented
5	C₂³×C₁₃×#41	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×#41	—	2	16	no	—
7	C₂²×C₁₃×#41	C₄	1	16	yes	complemented
8	C₂²×C₁₃×#41	C₂²	1	8	yes	—
9	C₂³×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂³×D₄×C₁₃	C₂²	1	3	yes	—
11	C₂⁴×C₄×C₁₃	C₂²	1	3	yes	—
12	#11719517	—	2	16	no	—
13	C₂×Q₈×Dic₁₃	C₂²	1	16	yes	complemented
14	C₂×#88180	C₂²	1	12	yes	—
15	#11718908	C₂²	1	12	yes	—
16	#11718882	C₂²	1	48	yes	complemented
17	#11718873	C₂²	1	24	yes	complemented
18	#11718872	C₂²	1	4	yes	—
19	C₄×C₂²⋊Dic₁₃	C₂²	1	24	yes	complemented
20	C₂⁵×C₁₃	C₂³	1	3	yes	—
21	C₂×C₁₃×#41	—	2	128	no	—
22	C₂×C₁₃×#41	—	2	64	no	—
23	C₂×C₁₃×#41	C₂×C₄	1	64	yes	—
24	C₂²×Q₈×C₁₃	C₂×C₄	1	8	yes	—
25	C₂²×Q₈×C₁₃	D₄	1	16	yes	—
26	C₂²×Q₈×C₁₃	C₂³	1	4	yes	—
27	C₂²×D₄×C₁₃	—	2	24	no	—
28	C₂²×D₄×C₁₃	C₂×C₄	1	24	yes	—
29	C₂²×D₄×C₁₃	C₂³	1	12	yes	—
30	C₂³×C₄×C₁₃	—	2	24	no	—
31	C₂³×C₄×C₁₃	C₂×C₄	1	24	yes	—
32	C₂³×C₄×C₁₃	C₂³	1	12	yes	—
33	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
35	C₂×C₂²⋊Dic₁₃	C₂³	1	8	yes	—
36	C₂×C₂²⋊Dic₁₃	C₂³	1	48	yes	—
37	Q₈×Dic₁₃	—	2	64	no	—
38	#88180	—	2	48	no	—
39	C₂×C₄⋊Dic₁₃	C₂³	1	24	yes	—
40	C₂×C₄×Dic₁₃	C₂³	1	24	yes	—
41	#88071	C₂³	1	16	yes	—
42	#34174	—	13	1	no	maximal, Sylow
43	C₂⁴×C₁₃	—	2	12	no	—
44	C₂⁴×C₁₃	—	4	12	no	—
45	C₂⁴×C₁₃	C₂²×C₄	1	12	yes	—
46	C₂⁴×C₁₃	C₂⁴	1	6	yes	—
47	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
48	C₁₃×#41	—	2	256	no	—
49	C₂×Q₈×C₁₃	C₂²⋊C₄	1	64	yes	complemented
50	C₂×Q₈×C₁₃	C₂²×C₄	1	16	yes	—
51	C₂×Q₈×C₁₃	C₂×D₄	1	32	yes	—
52	C₂×D₄×C₁₃	—	2	96	no	—
53	C₂×D₄×C₁₃	—	2	48	no	—
54	C₂×D₄×C₁₃	C₂²×C₄	1	48	yes	—
55	C₂²×C₄×C₁₃	—	2	4	no	—
56	C₂²×C₄×C₁₃	—	2	96	no	—
57	C₂²×C₄×C₁₃	—	2	48	no	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	48	yes	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	24	yes	—
62	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
64	C₂²×Dic₁₃	C₂⁴	1	16	yes	—
65	C₂²⋊Dic₁₃	—	2	16	no	—
66	C₂²⋊Dic₁₃	—	4	48	no	—
67	C₄⋊Dic₁₃	—	2	48	no	—
68	C₄×Dic₁₃	—	2	48	no	—
69	C₂³×#41	D₁₃	1	1	yes	characteristic
70	C₂×#519	—	13	2	no	—
71	#2341	—	13	16	no	—
72	#2309	—	13	6	no	—
73	#2280	—	13	6	no	—
74	C₂³×C₁₃	—	2	4	no	—
75	C₂³×C₁₃	—	2	24	no	—
76	C₂³×C₁₃	—	2	12	no	—
77	C₂³×C₁₃	—	2	24	no	—
78	C₂³×C₁₃	—	4	24	no	—
79	C₂³×C₁₃	—	4	48	no	—
80	C₂³×C₁₃	C₂³×C₄	1	12	yes	—
81	C₂³×C₁₃	C₂³×C₄	1	2	yes	—
82	C₂³×C₁₃	C₂²×D₄	1	4	yes	—
83	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
84	Q₈×C₁₃	C₂×C₂²⋊C₄	1	64	yes	—
85	D₄×C₁₃	—	2	96	no	—
86	C₂×C₄×C₁₃	—	2	8	no	—
87	C₂×C₄×C₁₃	—	2	4	no	—
88	C₂×C₄×C₁₃	—	2	96	no	—
89	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	48	yes	—
90	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
91	C₂×C₄×C₁₃	C₂³×C₄	1	12	yes	—
92	C₂×C₄×C₁₃	C₂²×D₄	1	24	yes	—
93	C₂×Dic₁₃	—	2	16	no	—
94	C₂×Dic₁₃	—	2	32	no	—
95	C₂²×#41	—	2	16	no	—
96	C₂²×#41	Dic₁₃	1	16	yes	complemented
97	C₂²×#41	C₂×D₁₃	1	8	yes	—
98	C₂³×Q₈	C₂×D₁₃	1	1	yes	characteristic
99	C₂³×D₄	C₂×D₁₃	1	3	yes	—
100	C₂⁴×C₄	C₂×D₁₃	1	3	yes	—
101	#519	—	26	16	no	—
102	C₂×C₄×Q₈	—	13	16	no	—
103	C₂×#117	—	13	12	no	—
104	#390	—	13	48	no	—
105	#381	—	13	4	no	—
106	#380	—	13	24	no	—
107	#379	—	13	12	no	—
108	C₄×C₂²⋊C₄	—	13	24	no	—
109	C₂²×C₁₃	—	2	4	no	—
110	C₂²×C₁₃	—	2	8	no	—
111	C₂²×C₁₃	—	2	12	no	—
112	C₂²×C₁₃	—	2	24	no	—
113	C₂²×C₁₃	—	4	48	no	—
114	C₂²×C₁₃	—	4	24	no	—
115	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	4	yes	—
116	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
117	C₂²×C₁₃	C₂³×D₄	1	2	yes	—
118	C₂²×C₁₃	#585	1	4	yes	—
119	C₄×C₁₃	—	2	4	no	—
120	C₄×C₁₃	C₂²×C₂²⋊C₄	1	24	yes	—
121	Dic₁₃	—	4	16	no	—
122	C₂⁵	C₂²×D₁₃	1	3	yes	—
123	C₂×#41	—	2	128	no	—
124	C₂×#41	—	2	64	no	—
125	C₂×#41	C₂×Dic₁₃	1	64	yes	—
126	C₂²×Q₈	C₂×Dic₁₃	1	8	yes	—
127	C₂²×Q₈	C₂²⋊D₁₃	1	16	yes	—
128	C₂²×Q₈	C₂²×D₁₃	1	4	yes	—
129	C₂²×D₄	—	2	24	no	—
130	C₂²×D₄	C₂×Dic₁₃	1	24	yes	—
131	C₂²×D₄	C₂²×D₁₃	1	12	yes	—
132	C₂³×C₄	—	2	24	no	—
133	C₂³×C₄	C₂×Dic₁₃	1	24	yes	—
134	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
135	C₂³×C₄	C₂²×D₁₃	1	12	yes	—
136	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
137	C₄×Q₈	—	26	64	no	—
138	#117	—	26	48	no	—
139	C₂×C₄⋊C₄	—	13	24	no	—
140	C₂×C₂²⋊C₄	—	13	8	no	—
141	C₂×C₂²⋊C₄	—	13	48	no	—
142	C₂×C₄²	—	13	24	no	—
143	#95	—	13	16	no	—
144	C₂×C₁₃	—	2	4	no	—
145	C₂×C₁₃	—	4	12	no	—
146	C₂×C₁₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₁₃	#3428	1	2	yes	—
148	C₂×C₁₃	#3487	1	4	yes	—
149	C₂⁴	—	2	12	no	—
150	C₂⁴	—	4	12	no	—
151	C₂⁴	C₂²×Dic₁₃	1	12	yes	—
152	C₂⁴	C₂³×D₁₃	1	6	yes	—
153	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
154	#41	—	2	256	no	—
155	C₂×Q₈	C₂²⋊Dic₁₃	1	64	yes	complemented
156	C₂×Q₈	C₂²×Dic₁₃	1	16	yes	—
157	C₂×Q₈	C₂×C₂²⋊D₁₃	1	32	yes	—
158	C₂×D₄	—	2	96	no	—
159	C₂×D₄	—	2	48	no	—
160	C₂×D₄	C₂²×Dic₁₃	1	48	yes	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	96	no	—
163	C₂²×C₄	—	2	48	no	—
164	C₂²×C₄	—	13	16	no	—
165	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
166	C₂²×C₄	C₂²×Dic₁₃	1	48	yes	—
167	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
168	C₂²×C₄	C₂×C₂²⋊D₁₃	1	24	yes	—
169	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
170	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
171	C₄⋊C₄	—	26	48	no	—
172	C₂²⋊C₄	—	26	16	no	—
173	C₂²⋊C₄	—	52	48	no	—
174	C₄²	—	26	48	no	—
175	C₁₃	#34174	1	1	yes	characteristic, complemented, Sylow
176	C₂³	—	2	4	no	—
177	C₂³	—	2	24	no	—
178	C₂³	—	2	24	no	—
179	C₂³	—	2	12	no	—
180	C₂³	—	4	24	no	—
181	C₂³	—	4	48	no	—
182	C₂³	C₂³×Dic₁₃	1	12	yes	—
183	C₂³	C₂³×Dic₁₃	1	2	yes	—
184	C₂³	C₂²×C₂²⋊D₁₃	1	4	yes	—
185	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
186	Q₈	C₂×C₂²⋊Dic₁₃	1	64	yes	—
187	D₄	—	2	96	no	—
188	C₂×C₄	—	2	8	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	96	no	—
191	C₂×C₄	—	26	32	no	—
192	C₂×C₄	—	26	16	no	—
193	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	48	yes	—
194	C₂×C₄	C₂³×Dic₁₃	1	4	yes	—
195	C₂×C₄	C₂³×Dic₁₃	1	12	yes	—
196	C₂×C₄	C₂²×C₂²⋊D₁₃	1	24	yes	—
197	C₂²	—	2	8	no	—
198	C₂²	—	2	4	no	—
199	C₂²	—	2	12	no	—
200	C₂²	—	2	24	no	—
201	C₂²	—	4	48	no	—
202	C₂²	—	4	24	no	—
203	C₂²	C₂²×C₂²⋊Dic₁₃	1	4	yes	—
204	C₂²	#11719648	1	4	yes	—
205	C₂²	C₂⁴×Dic₁₃	1	1	yes	characteristic
206	C₂²	C₂³×C₂²⋊D₁₃	1	2	yes	—
207	C₄	—	2	4	no	—
208	C₄	—	52	16	no	—
209	C₄	C₂²×C₂²⋊Dic₁₃	1	24	yes	—
210	C₂	—	2	4	no	—
211	C₂	—	4	12	no	—
212	C₂	C₂³×C₂²⋊Dic₁₃	1	1	yes	characteristic
213	C₂	#13128039	1	2	yes	—
214	C₂	#13128494	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₁₃
commutator	C₂²×C₁₃	C₂⁴×C₄
Fitting, prime radical	C₂³×C₁₃×#41	C₂
socle, cosocle	C₂³×C₁₃	C₂⁵
hypercenter	C₂³×#41	D₁₃
hypocenter	C₁₃	#34174
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#11719517	C₂	semidirect
2. #13117677	C₂	semidirect
3. #13117418	C₂	semidirect
4. #13117219	C₂	semidirect
7. C₂²×C₁₃×#41	C₄	semidirect
13. C₂×Q₈×Dic₁₃	C₂²	semidirect
16. #11718882	C₂²	semidirect
17. #11718873	C₂²	semidirect
19. C₄×C₂²⋊Dic₁₃	C₂²	semidirect
49. C₂×Q₈×C₁₃	C₂²⋊C₄	semidirect
96. C₂²×#41	Dic₁₃	semidirect
155. C₂×Q₈	C₂²⋊Dic₁₃	semidirect
175. C₁₃	#34174	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
#13117677	2
#13117418	2
#13117219	2
C₂³×C₁₃×#41	2
#34174	13