Subgroups — #16015922

Order 3328.

684 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127938	C₂	1	2	yes	maximal
2	#13123258	C₂	1	1	yes	maximal, characteristic
3	#13116602	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116600	C₂	1	4	yes	maximal, complemented
5	C₂×#11718103	C₂	1	2	yes	maximal, complemented
6	D₁₃×#364	C₂	1	4	yes	maximal, complemented
7	C₁₃×#2153	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#88161	C₄	1	1	yes	characteristic
9	C₂×#88155	C₂²	1	1	yes	characteristic
10	#11719227	C₄	1	1	yes	characteristic
11	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
12	C₁₃×#405	C₂²	1	1	yes	characteristic
13	M₁₆×Dic₁₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88052	C₂²	1	2	yes	—
16	#11718658	C₄	1	1	yes	characteristic
17	#11718568	—	2	4	no	—
18	D₁₃×M₃₂	C₂²	1	8	yes	—
19	#11718562	—	2	4	no	—
20	#11718369	C₄	1	2	yes	—
21	#11718368	C₄	1	2	yes	—
22	#11718350	C₂²	1	2	yes	—
23	C₄×C₈×D₁₃	C₂²	1	2	yes	—
24	C₁₃×#364	C₂²	1	4	yes	—
25	#11718103	C₂²	1	4	yes	—
26	#11718103	C₂²	1	4	yes	—
27	#11718055	C₂²	1	4	yes	—
28	C₂×#6127	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×#6099	C₂×C₄	1	1	yes	characteristic
34	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
35	#88161	C₈	1	8	yes	—
36	#88155	D₄	1	4	yes	—
37	#88155	Q₈	1	4	yes	—
38	C₂×#6094	C₂×C₄	1	2	yes	—
39	C₂×#6094	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₁₃	C₂³	1	2	yes	—
42	#88147	—	2	1	no	—
43	#88141	—	2	1	no	—
44	#88135	—	2	2	no	—
45	#88126	C₂×C₄	1	2	yes	—
46	C₄²×D₁₃	C₂×C₄	1	2	yes	—
47	C₁₃×M₃₂	—	2	2	no	—
48	C₁₃×M₃₂	C₂³	1	4	yes	—
49	C₂×C₁₃×C₁₆	—	2	2	no	—
50	C₁₃×#97	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₁₃	C₂³	1	2	yes	—
52	#88055	C₂×C₄	1	2	yes	—
53	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
54	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
55	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
56	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
57	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
58	#88052	—	2	2	no	—
59	#88052	C₂³	1	4	yes	—
60	C₂×#6090	—	2	2	no	—
61	#88043	C₂×C₄	1	2	yes	—
62	C₄×#1055	C₂×C₄	1	2	yes	—
63	#88038	—	2	8	no	—
64	C₁₆×D₁₃	—	2	8	no	—
65	#33401	—	13	1	no	maximal, Sylow
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₂²⋊D₁₃	M₁₆	1	2	yes	—
68	C₂²×Dic₁₃	—	2	1	no	—
69	#6127	C₄⋊C₄	1	4	yes	—
70	#6127	C₂×C₈	1	4	yes	—
71	C₂×D₅₂	M₁₆	1	1	yes	characteristic
72	C₂×C₄×D₁₃	—	2	2	no	—
73	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
74	C₂×Dic₂₆	M₁₆	1	1	yes	characteristic
75	C₁₃×M₁₆	C₂×C₈	1	4	yes	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
80	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₁₃×C₄⋊C₄	—	2	1	no	—
82	C₁₃×C₂²⋊C₄	—	2	1	no	—
83	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
84	#6103	—	2	2	no	—
85	#6101	—	2	2	no	—
86	C₄×Dic₁₃	—	2	1	no	—
87	C₄×Dic₁₃	—	2	2	no	—
88	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
90	#6099	C₄⋊C₄	1	4	yes	—
91	C₂×#1055	C₂×C₈	1	4	yes	—
92	C₂×#1055	C₂²×C₄	1	2	yes	—
93	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
94	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
95	#6094	C₄⋊C₄	1	8	yes	—
96	#6094	C₂×C₈	1	8	yes	—
97	C₈×D₁₃	C₂×C₈	1	8	yes	—
98	C₈×D₁₃	C₂×D₄	1	4	yes	—
99	C₈×D₁₃	C₂×Q₈	1	4	yes	—
100	C₁₃×C₁₆	—	2	4	no	—
101	#6090	—	2	4	no	—
102	#3407	—	13	2	no	—
103	C₄×#131	—	13	1	no	—
104	#2153	—	13	1	no	—
105	#2153	—	13	4	no	—
106	#2153	D₁₃	1	1	yes	characteristic, complemented
107	C₂×#364	—	13	2	no	—
108	C₂×#364	—	13	4	no	—
109	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂²×D₁₃	C₂×M₁₆	1	2	yes	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
116	C₂²⋊D₁₃	C₄⋊C₈	1	8	yes	—
117	C₂×Dic₁₃	—	2	1	no	—
118	C₂×Dic₁₃	—	2	1	no	—
119	C₂×Dic₁₃	—	4	2	no	—
120	C₂×Dic₁₃	C₂×M₁₆	1	2	yes	—
121	D₅₂	C₄⋊C₈	1	4	yes	—
122	C₄×D₁₃	—	4	4	no	—
123	C₄×D₁₃	C₂×C₄⋊C₄	1	4	yes	—
124	C₄×D₁₃	C₂²×C₈	1	4	yes	—
125	Dic₂₆	C₄⋊C₈	1	4	yes	—
126	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₈×C₁₃	C₂²×C₈	1	4	yes	—
128	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
129	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
130	#1055	C₂×C₄⋊C₄	1	4	yes	—
131	#1055	C₂²×C₈	1	2	yes	—
132	#1055	C₂²×C₈	1	2	yes	—
133	C₂×#131	—	13	1	no	—
134	C₂×#131	—	13	1	no	—
135	C₄×#41	—	13	1	no	—
136	#504	—	26	4	no	—
137	#504	—	26	4	no	—
138	C₂×M₃₂	—	13	2	no	—
139	C₂×M₃₂	—	13	8	no	—
140	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
141	#405	—	13	1	no	—
142	#405	—	13	2	no	—
143	#405	—	13	2	no	—
144	#405	C₂×D₁₃	1	1	yes	characteristic
145	C₄×M₁₆	—	13	1	no	—
146	C₄×M₁₆	—	13	2	no	—
147	C₂×C₄×C₈	—	13	1	no	—
148	C₂×C₄×C₈	—	13	2	no	—
149	#364	—	13	4	no	—
150	#364	—	13	4	no	—
151	#364	—	13	4	no	—
152	#364	C₂×D₁₃	1	4	yes	—
153	C₂²×C₁₃	—	4	1	no	—
154	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
155	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
156	C₂×D₁₃	—	4	2	no	—
157	C₂×D₁₃	C₂×C₄⋊C₈	1	4	yes	—
158	C₄×C₁₃	—	4	2	no	—
159	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
160	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
161	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
162	Dic₁₃	—	2	2	no	—
163	Dic₁₃	—	4	1	no	—
164	Dic₁₃	C₂×C₄⋊C₈	1	4	yes	—
165	C₂×#41	—	13	1	no	—
166	#131	—	13	8	no	—
167	#131	—	13	4	no	—
168	#131	—	13	4	no	—
169	C₂×M₁₆	—	13	1	no	—
170	C₂×M₁₆	—	13	2	no	—
171	C₂×M₁₆	—	13	2	no	—
172	C₂×M₁₆	C₄×D₁₃	1	1	yes	characteristic
173	C₂²×C₈	—	13	1	no	—
174	C₂²×C₈	—	13	2	no	—
175	C₂²×C₈	—	13	2	no	—
176	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
177	C₄×Q₈	—	26	1	no	—
178	C₄×D₄	—	26	2	no	—
179	C₄×D₄	—	26	1	no	—
180	#117	—	13	2	no	—
181	#117	C₄×D₁₃	1	1	yes	characteristic
182	C₂×C₄²	—	13	1	no	—
183	C₂×C₄²	—	13	2	no	—
184	M₃₂	—	2	2	no	—
185	M₃₂	—	13	4	no	—
186	M₃₂	—	26	8	no	—
187	M₃₂	—	26	2	no	—
188	M₃₂	C₂²×D₁₃	1	4	yes	—
189	C₂×C₁₆	—	2	2	no	—
190	C₂×C₁₆	—	26	8	no	—
191	C₂×C₁₆	—	26	2	no	—
192	#97	—	13	2	no	—
193	#97	—	13	2	no	—
194	#97	C₄×D₁₃	1	2	yes	—
195	C₄×C₈	—	13	1	no	—
196	C₄×C₈	—	13	2	no	—
197	C₄×C₈	—	13	1	no	—
198	C₄×C₈	—	13	1	no	—
199	C₄×C₈	—	13	1	no	—
200	C₄×C₈	—	13	2	no	—
201	C₄×C₈	C₂²×D₁₃	1	2	yes	—
202	C₂×C₁₃	—	2	1	no	—
203	C₂×C₁₃	—	2	2	no	—
204	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
205	D₁₃	—	2	4	no	—
206	#41	—	13	4	no	—
207	#41	—	13	4	no	—
208	C₂×Q₈	—	13	1	no	—
209	C₂×D₄	—	13	2	no	—
210	C₂×D₄	—	13	1	no	—
211	C₂²×C₄	—	13	2	no	—
212	C₂²×C₄	—	26	2	no	—
213	C₂²×C₄	—	26	1	no	—
214	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
215	M₁₆	—	13	4	no	—
216	M₁₆	—	13	8	no	—
217	M₁₆	—	13	8	no	—
218	M₁₆	C₈×D₁₃	1	4	yes	—
219	C₂×C₈	—	13	4	no	—
220	C₂×C₈	—	13	1	no	—
221	C₂×C₈	—	13	1	no	—
222	C₂×C₈	—	13	2	no	—
223	C₂×C₈	—	13	8	no	—
224	C₂×C₈	—	13	4	no	—
225	C₂×C₈	—	13	4	no	—
226	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
228	C₂×C₈	D₄×D₁₃	1	2	yes	—
229	C₂×C₈	Q₈×D₁₃	1	2	yes	—
230	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	26	2	no	—
233	C₂²⋊C₄	—	2	1	no	—
234	C₂²⋊C₄	—	26	2	no	—
235	C₄²	—	13	1	no	—
236	C₄²	—	13	1	no	—
237	C₄²	—	26	2	no	—
238	C₄²	—	26	1	no	—
239	C₄²	C₂×C₄×D₁₃	1	2	yes	—
240	C₁₆	—	2	4	no	—
241	C₁₆	—	26	4	no	—
242	C₁₃	#33401	1	1	yes	characteristic, complemented, Sylow
243	C₂³	—	13	2	no	—
244	C₂³	M₁₆×D₁₃	1	1	yes	characteristic
245	Q₈	—	13	4	no	—
246	D₄	—	13	8	no	—
247	D₄	—	13	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	13	2	no	—
250	C₂×C₄	—	13	4	no	—
251	C₂×C₄	—	13	4	no	—
252	C₂×C₄	—	26	1	no	—
253	C₂×C₄	—	26	1	no	—
254	C₂×C₄	—	52	4	no	—
255	C₂×C₄	—	52	2	no	—
256	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
257	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
258	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
260	C₈	—	13	2	no	—
261	C₈	—	13	2	no	—
262	C₈	—	13	4	no	—
263	C₈	C₄⋊C₄×D₁₃	1	2	yes	—
264	C₈	C₂×C₈×D₁₃	1	4	yes	—
265	C₈	C₂×D₄×D₁₃	1	1	yes	characteristic
266	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
267	C₂²	—	4	1	no	—
268	C₂²	—	13	4	no	—
269	C₂²	—	52	2	no	—
270	C₂²	D₁₃×C₄⋊C₈	1	2	yes	—
271	C₂²	C₂×M₁₆×D₁₃	1	1	yes	characteristic
272	C₄	—	4	2	no	—
273	C₄	—	13	4	no	—
274	C₄	—	26	2	no	—
275	C₄	—	52	1	no	—
276	C₄	D₁₃×C₄⋊C₈	1	2	yes	—
277	C₄	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
278	C₄	C₂²×C₈×D₁₃	1	1	yes	characteristic
279	C₂	—	2	1	no	—
280	C₂	—	2	2	no	—
281	C₂	—	26	4	no	—
282	C₂	C₂×D₁₃×C₄⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₁₃
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2153	C₂
socle	C₂×C₁₃	C₂²×C₄⋊C₈
prime radical	C₂×#6127	C₂×C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2153	D₁₃
hypocenter	C₁₃	#33401
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

Complements

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

N	H ≅ G/N	type
3. #13116602	C₂	semidirect
4. #13116600	C₂	semidirect
5. C₂×#11718103	C₂	semidirect
6. D₁₃×#364	C₂	semidirect
7. C₁₃×#2153	C₂	semidirect
106. #2153	D₁₃	semidirect
242. C₁₃	#33401	semidirect

Maximal subgroups

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

subgroup	index
#13127938	2
#13123258	2
#13116602	2
#13116600	2
C₂×#11718103	2
D₁₃×#364	2
C₁₃×#2153	2
#33401	13