Subgroups — #16016003

Order 3328.

590 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×#11718762	C₂	1	1	yes	maximal, characteristic
2	#13116640	C₂	1	8	yes	maximal, complemented
3	#13116591	C₂	1	2	yes	maximal, complemented
4	#13116447	C₂	1	1	yes	maximal, characteristic
5	M₃₂×Dic₁₃	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2112	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88180	C₄	1	1	yes	characteristic
9	C₂²×#6099	C₄	1	1	yes	characteristic
10	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
11	#11718762	C₄	1	4	yes	—
12	#11718762	C₂²	1	4	yes	—
13	C₂×#88055	C₄	1	1	yes	characteristic
14	C₂×#88055	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
17	C₂×#88052	C₂²	1	2	yes	—
18	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
19	C₂²⋊#6090	C₂²	1	4	yes	—
20	#11718094	C₂²	1	4	yes	complemented
21	#11718093	C₂²	1	8	yes	complemented
22	C₁₆×Dic₁₃	C₂²	1	4	yes	complemented
23	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	—	2	4	no	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
28	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₂²⋊Dic₁₃	C₈	1	2	yes	—
30	#88180	C₈	1	4	yes	—
31	#88180	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊Dic₁₃	C₈	1	2	yes	—
33	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×#6099	—	2	4	no	—
36	C₂×#6099	C₂×C₄	1	4	yes	—
37	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
38	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
39	C₁₃×M₃₂	—	2	4	no	—
40	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
41	#88055	C₂×C₄	1	4	yes	—
42	#88055	C₂×C₄	1	4	yes	—
43	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
44	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
45	C₈×Dic₁₃	C₂³	1	2	yes	—
46	C₈×Dic₁₃	C₂³	1	2	yes	—
47	#88052	—	2	4	no	—
48	C₂×#6090	C₂³	1	4	yes	—
49	#33419	—	13	1	no	maximal, Sylow
50	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	—	2	2	no	—
53	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
58	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
59	C₂×C₈×C₁₃	—	2	2	no	—
60	C₂×C₈×C₁₃	—	2	4	no	—
61	C₂×C₈×C₁₃	—	2	4	no	—
62	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₃	#41	1	4	yes	—
65	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
66	C₂²⋊Dic₁₃	C₂×C₈	1	8	yes	—
67	C₄⋊Dic₁₃	C₂×C₈	1	8	yes	—
68	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
69	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
70	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
71	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
72	#6099	—	2	8	no	—
73	C₂×#1055	—	2	2	no	—
74	C₂×#1055	M₁₆	1	4	yes	—
75	C₂×#1055	C₂²×C₄	1	2	yes	—
76	C₂×#1055	C₂²×C₄	1	2	yes	—
77	C₁₃×C₁₆	—	2	4	no	—
78	#6090	—	2	4	no	—
79	C₂×#405	—	13	1	no	—
80	#2164	—	13	8	no	—
81	#2150	—	13	2	no	—
82	#2112	—	13	1	no	—
83	#2112	D₁₃	1	1	yes	characteristic
84	C₄×M₃₂	—	13	2	no	—
85	C₂³×C₁₃	—	2	2	no	—
86	C₂³×C₁₃	—	4	2	no	—
87	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
88	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	4	no	—
93	C₂×C₄×C₁₃	#117	1	2	yes	—
94	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×Dic₁₃	—	4	2	no	—
99	C₂×Dic₁₃	—	4	2	no	—
100	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
101	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
102	C₈×C₁₃	—	2	2	no	—
103	C₈×C₁₃	#117	1	2	yes	—
104	C₈×C₁₃	C₂×#41	1	2	yes	—
105	#1055	—	2	2	no	—
106	#1055	C₂×M₁₆	1	4	yes	—
107	C₂²×M₁₆	—	13	1	no	—
108	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
109	C₂×#117	—	13	1	no	—
110	C₂×M₃₂	—	13	2	no	—
111	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
112	#405	—	13	4	no	—
113	#405	—	13	4	no	—
114	C₂×#97	—	13	1	no	—
115	C₂×#97	—	13	1	no	—
116	C₂×C₄×C₈	—	13	1	no	—
117	C₂×C₄×C₈	—	13	1	no	—
118	C₄⋊C₁₆	—	13	8	no	—
119	C₂²⋊C₁₆	—	13	4	no	—
120	C₂²⋊C₁₆	C₂×D₁₃	1	4	yes	—
121	#346	—	13	4	no	—
122	C₄×C₁₆	—	13	4	no	—
123	C₂²×C₁₃	—	2	2	no	—
124	C₂²×C₁₃	—	2	4	no	—
125	C₂²×C₁₃	—	2	2	no	—
126	C₂²×C₁₃	—	4	4	no	—
127	C₂²×C₁₃	#431	1	2	yes	—
128	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
129	C₄×C₁₃	—	2	2	no	—
130	C₄×C₁₃	#431	1	2	yes	—
131	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
132	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
133	Dic₁₃	—	2	4	no	—
134	Dic₁₃	—	2	4	no	—
135	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
136	C₂×M₁₆	—	13	4	no	—
137	C₂×M₁₆	—	26	4	no	—
138	C₂²×C₈	—	2	4	no	—
139	C₂²×C₈	—	13	1	no	—
140	C₂²×C₈	—	13	1	no	—
141	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
142	C₂²×C₈	C₄×D₁₃	1	2	yes	—
143	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
144	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
145	#117	—	13	4	no	—
146	#117	—	13	4	no	—
147	C₂×C₄⋊C₄	—	13	2	no	—
148	C₂×C₂²⋊C₄	—	13	2	no	—
149	C₂×C₄²	—	13	1	no	—
150	C₂×C₄²	—	13	1	no	—
151	M₃₂	—	2	4	no	—
152	M₃₂	—	26	4	no	—
153	C₂×C₁₆	—	13	4	no	—
154	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
155	#97	—	13	4	no	—
156	#97	—	13	4	no	—
157	C₄×C₈	—	13	2	no	—
158	C₄×C₈	—	13	2	no	—
159	C₄×C₈	—	13	2	no	—
160	C₄×C₈	—	13	2	no	—
161	C₂×C₁₃	—	2	2	no	—
162	C₂×C₁₃	—	4	2	no	—
163	C₂×C₁₃	C₂×#431	1	1	yes	characteristic
164	C₂×C₁₃	#3407	1	2	yes	—
165	C₂⁴	C₈×D₁₃	1	1	yes	characteristic
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	13	2	no	—
169	C₂²×C₄	—	13	2	no	—
170	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
171	C₂²×C₄	C₈×D₁₃	1	2	yes	—
172	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
173	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
174	M₁₆	—	26	8	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	13	4	no	—
179	C₂×C₈	—	13	2	no	—
180	C₂×C₈	—	13	2	no	—
181	C₂×C₈	—	26	2	no	—
182	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
183	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
184	C₂×C₈	#6129	1	4	yes	—
185	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
186	C₄⋊C₄	—	13	8	no	—
187	C₂²⋊C₄	—	13	8	no	—
188	C₄²	—	13	2	no	—
189	C₄²	—	13	2	no	—
190	C₄²	—	13	2	no	—
191	C₄²	—	13	2	no	—
192	C₁₆	—	2	4	no	—
193	C₁₆	—	26	4	no	—
194	C₁₃	#33419	1	1	yes	characteristic, complemented, Sylow
195	C₂³	—	2	2	no	—
196	C₂³	—	4	2	no	—
197	C₂³	C₂×C₈×D₁₃	1	2	yes	—
198	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	13	4	no	—
204	C₂×C₄	—	13	4	no	—
205	C₂×C₄	—	52	2	no	—
206	C₂×C₄	—	52	2	no	—
207	C₂×C₄	#88131	1	2	yes	—
208	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
209	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
210	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
211	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	—	13	4	no	—
214	C₈	—	26	2	no	—
215	C₈	#88131	1	2	yes	—
216	C₈	C₂×#6129	1	2	yes	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	4	4	no	—
221	C₂²	#11718379	1	2	yes	—
222	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	26	4	no	—
225	C₄	—	26	4	no	—
226	C₄	#11718379	1	2	yes	—
227	C₄	C₂×#88131	1	1	yes	characteristic
228	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	—	4	2	no	—
231	C₂	C₂×#11718379	1	1	yes	characteristic
232	C₂	#13127938	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#2112	C₂
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2112	D₁₃
hypocenter	C₁₃	#33419
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ #2112

Complements

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

N	H ≅ G/N	type
2. #13116640	C₂	semidirect
3. #13116591	C₂	semidirect
5. M₃₂×Dic₁₃	C₂	semidirect
20. #11718094	C₂²	semidirect
21. #11718093	C₂²	semidirect
22. C₁₆×Dic₁₃	C₂²	semidirect
194. C₁₃	#33419	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718762	2
#13116640	2
#13116591	2
#13116447	2
M₃₂×Dic₁₃	2
C₁₃×#2112	2
#33419	13