Subgroups — #16789068

Order 3328.

586 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃⋊M₃₂	C₂	1	2	yes	maximal
2	#13129881	C₂	1	2	yes	maximal, complemented
3	#13129880	C₂	1	2	yes	maximal, complemented
4	C₂×#11718369	C₂	1	1	yes	maximal, characteristic
5	C₂²×#6094	C₄	1	1	yes	characteristic
6	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
7	C₂×#88126	C₄	1	1	yes	characteristic
8	C₂×#88106	C₂²	1	4	yes	—
9	D₁₃⋊M₃₂	—	2	4	no	—
10	D₁₃⋊M₃₂	C₄	1	8	yes	complemented
11	C₂×#88098	—	2	2	no	—
12	C₂×D₁₃⋊C₁₆	—	2	2	no	—
13	C₂×C₁₃×#97	C₄	1	1	yes	characteristic
14	C₂×#88055	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
17	#11718369	—	2	4	no	—
18	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×#6103	—	2	1	no	—
23	C₂×#6101	—	2	1	no	—
24	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
26	C₂²×#1055	C₂³	1	1	yes	characteristic
27	C₂×#6094	—	2	2	no	—
28	C₂×#6094	—	2	2	no	—
29	C₂×#6094	C₈	1	4	yes	—
30	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
31	C₂×C₈×D₁₃	D₄	1	2	yes	—
32	C₂×C₈×D₁₃	D₄	1	4	yes	—
33	C₂×C₈×D₁₃	Q₈	1	2	yes	—
34	#88126	—	2	2	no	—
35	#88126	C₈	1	4	yes	—
36	#88106	—	2	4	no	—
37	#88106	C₂×C₄	1	8	yes	—
38	C₂×#6092	—	2	4	no	—
39	#88098	—	2	8	no	—
40	D₁₃⋊C₁₆	—	2	8	no	—
41	C₁₃×#97	—	2	2	no	—
42	#88055	—	2	2	no	—
43	C₈×Dic₁₃	—	2	2	no	—
44	C₄×#1055	—	2	2	no	—
45	#12549	—	13	1	no	maximal, Sylow
46	C₂³×D₁₃	M₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	—	2	1	no	—
48	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×Dic₁₃	—	2	1	no	—
50	C₂²×Dic₁₃	M₁₆	1	1	yes	characteristic
51	C₂×C₄×D₁₃	C₄²	1	2	yes	—
52	C₂×C₄×D₁₃	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
54	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
55	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
56	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
57	C₂×C₈×C₁₃	—	2	1	no	—
58	C₂×C₈×C₁₃	—	2	1	no	—
59	C₂×C₈×C₁₃	C₄²	1	2	yes	—
60	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
62	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
63	C₄²×C₁₃	—	2	1	no	—
64	C₄²×C₁₃	C₂×C₈	1	2	yes	—
65	#6103	—	2	4	no	—
66	#6101	—	2	4	no	—
67	C₄×Dic₁₃	—	2	1	no	—
68	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
69	C₂×#1055	—	2	1	no	—
70	C₂×#1055	—	2	1	no	—
71	C₂×#1055	C₂×C₈	1	2	yes	—
72	C₂×#1055	C₂²×C₄	1	2	yes	—
73	C₂×#1055	C₂×D₄	1	1	yes	characteristic
74	C₂×#1055	C₂×D₄	1	2	yes	—
75	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
76	#6094	—	2	8	no	—
77	C₈×D₁₃	C₂²⋊C₄	1	4	yes	—
78	C₈×D₁₃	C₂²⋊C₄	1	4	yes	—
79	C₈×D₁₃	C₄⋊C₄	1	8	yes	—
80	#6092	—	2	8	no	—
81	C₂²×M₃₂	—	13	2	no	—
82	C₂×#405	—	13	1	no	—
83	#1376	—	13	2	no	—
84	#1376	—	13	2	no	—
85	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
86	C₂²×D₁₃	—	4	2	no	—
87	C₂²×D₁₃	#97	1	2	yes	—
88	C₂²×D₁₃	C₂²⋊C₈	1	2	yes	—
89	C₂²×D₁₃	C₄⋊C₈	1	2	yes	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	4	2	no	—
92	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
98	C₂×Dic₁₃	—	2	2	no	—
99	C₂×Dic₁₃	—	4	2	no	—
100	C₂×Dic₁₃	#97	1	2	yes	—
101	C₂×Dic₁₃	C₂²⋊C₈	1	2	yes	—
102	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
103	C₄×D₁₃	#95	1	4	yes	—
104	C₄×D₁₃	C₄×C₈	1	4	yes	—
105	C₄×D₁₃	C₂²⋊C₈	1	4	yes	—
106	C₄×D₁₃	C₄⋊C₈	1	4	yes	—
107	C₈×C₁₃	—	2	2	no	—
108	C₈×C₁₃	#95	1	4	yes	—
109	#1055	—	2	2	no	—
110	#1055	C₂×C₂²⋊C₄	1	2	yes	—
111	#1055	C₂×C₄⋊C₄	1	2	yes	—
112	C₂²×M₁₆	—	13	1	no	—
113	C₂³×C₈	—	13	1	no	—
114	C₂×#117	—	13	1	no	—
115	C₂×M₃₂	—	13	4	no	—
116	C₂×M₃₂	—	13	8	no	—
117	C₂×M₃₂	—	26	2	no	—
118	C₂×M₃₂	—	26	4	no	—
119	C₂²×C₁₆	—	26	2	no	—
120	#405	—	26	4	no	—
121	C₂×#97	—	13	1	no	—
122	C₂×#97	#261	1	1	yes	characteristic
123	C₂×C₄×C₈	—	13	1	no	—
124	C₂×C₄×C₈	—	13	1	no	—
125	C₂²×C₁₃	—	2	2	no	—
126	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
127	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
128	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
129	C₂×D₁₃	—	2	4	no	—
130	C₂×D₁₃	—	4	2	no	—
131	C₂×D₁₃	—	4	2	no	—
132	C₂×D₁₃	#336	1	4	yes	—
133	C₄×C₁₃	—	4	2	no	—
134	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
135	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
136	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
137	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
138	Dic₁₃	—	4	2	no	—
139	Dic₁₃	#336	1	4	yes	—
140	C₂³×C₄	—	13	1	no	—
141	C₂×M₁₆	—	13	4	no	—
142	C₂×M₁₆	—	26	2	no	—
143	C₂×M₁₆	—	26	2	no	—
144	C₂²×C₈	—	13	1	no	—
145	C₂²×C₈	—	13	4	no	—
146	C₂²×C₈	—	13	2	no	—
147	C₂²×C₈	—	13	2	no	—
148	C₂²×C₈	—	13	4	no	—
149	C₂²×C₈	—	13	1	no	—
150	C₂²×C₈	C₂×#261	1	1	yes	characteristic
151	C₂²×C₈	C₂×#261	1	1	yes	characteristic
152	#117	—	13	4	no	—
153	#117	—	26	2	no	—
154	C₂×C₄⋊C₄	—	26	1	no	—
155	C₂×C₂²⋊C₄	—	26	1	no	—
156	C₂×C₄²	—	13	1	no	—
157	C₂×C₄²	C₂×#261	1	1	yes	characteristic
158	M₃₂	—	13	8	no	—
159	M₃₂	—	26	8	no	—
160	M₃₂	—	26	4	no	—
161	C₂×C₁₆	—	26	8	no	—
162	C₂×C₁₆	—	26	4	no	—
163	#97	—	2	2	no	—
164	#97	—	26	2	no	—
165	C₄×C₈	—	26	2	no	—
166	C₄×C₈	—	26	2	no	—
167	C₂×C₁₃	—	2	1	no	—
168	C₂×C₁₃	—	2	1	no	—
169	C₂×C₁₃	C₂×#336	1	1	yes	characteristic
170	C₂×C₁₃	#2153	1	2	yes	—
171	D₁₃	—	2	4	no	—
172	C₂⁴	—	13	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	13	1	no	—
175	C₂²×C₄	—	13	2	no	—
176	C₂²×C₄	—	13	2	no	—
177	C₂²×C₄	—	13	2	no	—
178	C₂²×C₄	—	13	2	no	—
179	C₂²×C₄	—	13	2	no	—
180	C₂²×C₄	—	13	2	no	—
181	C₂²×C₄	—	26	1	no	—
182	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
183	M₁₆	—	26	8	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	13	2	no	—
187	C₂×C₈	—	13	1	no	—
188	C₂×C₈	—	13	1	no	—
189	C₂×C₈	—	13	2	no	—
190	C₂×C₈	—	13	4	no	—
191	C₂×C₈	—	13	4	no	—
192	C₂×C₈	—	13	8	no	—
193	C₂×C₈	—	13	2	no	—
194	C₂×C₈	—	26	1	no	—
195	C₂×C₈	—	26	1	no	—
196	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
197	C₂×C₈	C₄×#261	1	2	yes	—
198	C₂×C₈	C₄⋊#261	1	2	yes	—
199	C₂×C₈	C₂²⋊#261	1	2	yes	—
200	C₄⋊C₄	—	26	4	no	—
201	C₂²⋊C₄	—	26	4	no	—
202	C₄²	—	2	1	no	—
203	C₄²	—	13	2	no	—
204	C₄²	—	26	1	no	—
205	C₄²	D₁₃⋊C₈	1	2	yes	—
206	C₁₆	—	26	8	no	—
207	C₁₃	#12549	1	1	yes	characteristic, complemented, Sylow
208	C₂³	—	13	2	no	—
209	C₂³	—	13	2	no	—
210	C₂³	—	13	2	no	—
211	C₂³	—	52	2	no	—
212	C₂³	D₁₃⋊M₁₆	1	1	yes	characteristic
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	13	2	no	—
216	C₂×C₄	—	13	2	no	—
217	C₂×C₄	—	13	2	no	—
218	C₂×C₄	—	13	4	no	—
219	C₂×C₄	—	13	4	no	—
220	C₂×C₄	—	13	4	no	—
221	C₂×C₄	—	13	4	no	—
222	C₂×C₄	—	26	2	no	—
223	C₂×C₄	—	52	2	no	—
224	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
225	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
226	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
227	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
228	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
229	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	—	13	2	no	—
232	C₈	—	13	2	no	—
233	C₈	—	26	2	no	—
234	C₈	(C₂×#261)⋊C₄	1	4	yes	—
235	C₂²	—	2	2	no	—
236	C₂²	—	13	4	no	—
237	C₂²	—	26	4	no	—
238	C₂²	—	52	2	no	—
239	C₂²	—	52	2	no	—
240	C₂²	#11719047	1	1	yes	characteristic
241	C₂²	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
242	C₂²	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
243	C₄	—	4	2	no	—
244	C₄	—	13	4	no	—
245	C₄	—	52	2	no	—
246	C₄	C₄×D₁₃⋊C₈	1	1	yes	characteristic
247	C₄	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
248	C₄	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
249	C₄	C₂×(C₂×#261)⋊C₄	1	1	yes	characteristic
250	C₂	—	2	1	no	—
251	C₂	—	2	1	no	—
252	C₂	—	26	4	no	—
253	C₂	#13130497	1	1	yes	characteristic
254	C₂	#13131178	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#261
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₁₃×#97	C₄
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×D₁₃	M₁₆
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×#97	#261
hypocenter	C₁₃	#12549
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
2. #13129881	C₂	semidirect
3. #13129880	C₂	semidirect
10. D₁₃⋊M₃₂	C₄	semidirect
207. C₁₃	#12549	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₃⋊M₃₂	2
#13129881	2
#13129880	2
C₂×#11718369	2
#12549	13