Subgroups — #15833599

Order 3328.

560 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123931	C₂	1	2	yes	maximal
2	#13113745	C₂	1	2	yes	maximal, complemented
3	#13113733	C₂	1	2	yes	maximal, complemented
4	#13113655	C₂	1	2	yes	maximal, complemented
5	Q₁₆×#1055	C₂	1	2	yes	maximal, complemented
6	#13113599	C₂	1	1	yes	maximal, characteristic
7	#13113309	C₂	1	2	yes	maximal
8	#13113119	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#3170	C₂	1	1	yes	maximal, characteristic
10	C₁₃×#533	C₂²	1	2	yes	—
11	C₁₃×#500	C₄	1	1	yes	characteristic
12	C₁₃×#494	C₄	1	1	yes	characteristic
13	C₁₃×#487	C₄	1	2	yes	—
14	C₁₃×#467	C₄	1	2	yes	—
15	C₁₃×#467	C₄	1	2	yes	—
16	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
17	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	#11718685	—	2	4	no	—
20	#11718685	C₂²	1	4	yes	—
21	Q₈×#1055	C₂²	1	4	yes	—
22	#11718661	—	2	4	no	—
23	#11718587	C₂²	1	2	yes	—
24	#11718586	C₂²	1	1	yes	characteristic
25	#11718586	C₂²	1	1	yes	characteristic
26	Q₈⋊#1055	C₂²	1	4	yes	—
27	Q₈⋊#1055	C₂²	1	4	yes	—
28	#11718050	C₂²	1	2	yes	—
29	#11718049	C₂²	1	2	yes	—
30	#11718047	C₂²	1	1	yes	characteristic
31	#11718047	C₂²	1	1	yes	characteristic
32	C₈×#1055	C₂²	1	1	yes	characteristic
33	C₈×#1055	C₂²	1	1	yes	characteristic
34	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
39	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
40	C₁₃×#125	—	2	2	no	—
41	C₁₃×C₂²⋊Q₈	—	2	2	no	—
42	C₁₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
43	C₄×Q₈×C₁₃	—	2	2	no	—
44	C₄×Q₈×C₁₃	C₂³	1	4	yes	—
45	C₁₃×#117	—	2	2	no	—
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×#107	C₂×C₄	1	2	yes	—
48	C₁₃×#107	C₂×C₄	1	2	yes	—
49	C₁₃×#103	C₂×C₄	1	4	yes	—
50	C₁₃×#103	C₂×C₄	1	4	yes	—
51	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
55	C₂²⋊#1055	—	2	2	no	—
56	C₂²⋊#1055	—	2	1	no	—
57	C₂²⋊#1055	D₄	1	2	yes	—
58	C₄⋊#1055	—	2	1	no	—
59	C₄⋊#1055	—	2	4	no	—
60	C₄⋊#1055	—	2	4	no	—
61	C₄⋊#1055	D₄	1	2	yes	—
62	C₄⋊#1055	C₂³	1	4	yes	—
63	#88043	—	2	2	no	—
64	#88043	—	2	4	no	—
65	C₄×#1055	—	2	4	no	—
66	C₄×#1055	C₂³	1	1	yes	characteristic
67	C₄×#1055	C₂³	1	1	yes	characteristic
68	C₄×#1055	C₂³	1	1	yes	characteristic
69	C₄×#1055	C₂³	1	1	yes	characteristic
70	#17972	—	13	1	no	maximal, Sylow
71	C₂×Q₈×C₁₃	—	2	2	no	—
72	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
76	C₁₃×Q₁₆	—	2	4	no	—
77	C₁₃×Q₁₆	—	2	4	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₁₃×C₄⋊C₄	—	2	2	no	—
85	C₁₃×C₄⋊C₄	—	2	2	no	—
86	C₁₃×C₄⋊C₄	—	2	2	no	—
87	C₁₃×C₄⋊C₄	—	2	4	no	—
88	C₁₃×C₄⋊C₄	—	2	4	no	—
89	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
90	C₁₃×C₂²⋊C₄	—	2	4	no	—
91	C₄²×C₁₃	—	2	4	no	—
92	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₁₃	#41	1	1	yes	characteristic
95	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
96	C₂×#1055	—	2	4	no	—
97	C₂×#1055	—	2	1	no	—
98	C₂×#1055	—	2	1	no	—
99	C₂×#1055	C₂×D₄	1	2	yes	—
100	C₂×#1055	C₂×D₄	1	2	yes	—
101	#3170	D₁₃	1	1	yes	characteristic
102	#2985	—	13	2	no	—
103	#1593	—	13	2	no	—
104	#1591	—	13	2	no	—
105	#1579	—	13	2	no	—
106	C₈×Q₁₆	—	13	2	no	—
107	#1568	—	13	1	no	—
108	#1497	—	13	2	no	—
109	#1450	—	13	1	no	—
110	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
111	Q₈×C₁₃	—	2	4	no	—
112	Q₈×C₁₃	—	2	4	no	—
113	Q₈×C₁₃	—	8	2	no	—
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	4	no	—
118	C₂×C₄×C₁₃	—	2	4	no	—
119	C₂×C₄×C₁₃	—	4	1	no	—
120	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
126	C₈×C₁₃	—	2	1	no	—
127	C₈×C₁₃	—	2	1	no	—
128	C₈×C₁₃	#131	1	2	yes	—
129	C₈×C₁₃	#131	1	2	yes	—
130	#1055	—	2	2	no	—
131	#1055	—	2	2	no	—
132	#1055	—	4	1	no	—
133	#1055	—	4	1	no	—
134	#1055	—	4	4	no	—
135	#533	C₂×D₁₃	1	2	yes	—
136	#500	Dic₁₃	1	1	yes	characteristic
137	#494	Dic₁₃	1	1	yes	characteristic
138	#487	Dic₁₃	1	2	yes	—
139	#467	Dic₁₃	1	2	yes	—
140	#467	Dic₁₃	1	2	yes	—
141	#446	—	13	4	no	—
142	#446	—	26	4	no	—
143	C₈×Q₈	—	13	4	no	—
144	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
145	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
146	#433	—	26	4	no	—
147	#432	—	13	2	no	—
148	#431	—	13	1	no	—
149	#431	—	13	1	no	—
150	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
151	#335	—	13	2	no	—
152	#334	—	13	2	no	—
153	Q₈⋊C₈	—	13	4	no	—
154	Q₈⋊C₈	—	13	4	no	—
155	#322	—	13	1	no	—
156	#322	—	13	1	no	—
157	C₈²	—	13	1	no	—
158	C₈²	—	13	1	no	—
159	C₂²×C₁₃	—	2	1	no	—
160	C₂²×C₁₃	—	2	1	no	—
161	C₂²×C₁₃	—	2	1	no	—
162	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
163	C₄×C₁₃	—	2	1	no	—
164	C₄×C₁₃	—	2	1	no	—
165	C₄×C₁₃	—	2	1	no	—
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	—	4	4	no	—
168	C₄×C₁₃	—	4	4	no	—
169	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
170	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
171	C₄×C₁₃	#578	1	2	yes	—
172	C₂×Q₁₆	C₂×Dic₁₃	1	2	yes	—
173	C₂×Q₁₆	C₂×Dic₁₃	1	2	yes	—
174	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
175	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
176	C₄⋊Q₈	C₂×Dic₁₃	1	2	yes	—
177	C₄⋊Q₈	C₂×Dic₁₃	1	2	yes	—
178	#125	—	2	2	no	—
179	C₂²⋊Q₈	—	2	2	no	—
180	C₂²⋊Q₈	C₂×Dic₁₃	1	4	yes	—
181	C₄×Q₈	—	2	2	no	—
182	C₄×Q₈	C₂²×D₁₃	1	4	yes	—
183	#117	—	2	2	no	—
184	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
185	#107	C₂×Dic₁₃	1	2	yes	—
186	#107	C₂×Dic₁₃	1	2	yes	—
187	C₄⋊C₈	—	13	2	no	—
188	C₄⋊C₈	—	13	4	no	—
189	C₄⋊C₈	—	26	4	no	—
190	C₄⋊C₈	—	26	4	no	—
191	C₄⋊C₈	—	26	1	no	—
192	#103	C₂×Dic₁₃	1	4	yes	—
193	#103	C₂×Dic₁₃	1	4	yes	—
194	C₂²⋊C₈	—	13	2	no	—
195	C₂²⋊C₈	—	26	2	no	—
196	C₂²⋊C₈	—	26	1	no	—
197	#97	—	26	4	no	—
198	#97	—	26	2	no	—
199	C₄×C₈	—	13	1	no	—
200	C₄×C₈	—	13	1	no	—
201	C₄×C₈	—	13	1	no	—
202	C₄×C₈	—	13	1	no	—
203	C₄×C₈	—	26	4	no	—
204	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
205	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
206	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
208	C₂×C₁₃	—	4	1	no	—
209	C₂×C₁₃	#2931	1	1	yes	characteristic
210	C₂×C₁₃	#2950	1	1	yes	characteristic
211	C₂×C₁₃	#2957	1	1	yes	characteristic
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	C₂²×Dic₁₃	1	4	yes	—
214	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
215	C₂²×C₄	#6129	1	1	yes	characteristic
216	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
217	Q₁₆	—	2	4	no	—
218	Q₁₆	—	2	4	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	13	2	no	—
222	C₂×C₈	—	13	2	no	—
223	C₂×C₈	—	26	4	no	—
224	C₂×C₈	—	26	1	no	—
225	C₂×C₈	—	26	1	no	—
226	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
227	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
228	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
229	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	C₂²×Dic₁₃	1	4	yes	—
236	C₂²⋊C₄	—	2	4	no	—
237	C₄²	—	2	4	no	—
238	C₄²	D₄×D₁₃	1	1	yes	characteristic
239	C₄²	#6129	1	1	yes	characteristic
240	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
241	C₄²	C₂³×D₁₃	1	1	yes	characteristic
242	C₁₃	#17972	1	1	yes	characteristic, complemented, Sylow
243	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
244	Q₈	—	2	4	no	—
245	Q₈	—	2	4	no	—
246	Q₈	—	8	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
254	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
255	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
257	C₂×C₄	C₂×#6129	1	1	yes	characteristic
258	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	26	2	no	—
262	C₈	—	26	2	no	—
263	C₈	—	52	1	no	—
264	C₈	—	52	1	no	—
265	C₈	—	52	4	no	—
266	C₈	#88202	1	2	yes	—
267	C₈	#88202	1	2	yes	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	4	no	—
277	C₄	—	4	4	no	—
278	C₄	#11719471	1	2	yes	—
279	C₄	C₂×#88202	1	1	yes	characteristic
280	C₄	C₂×#88202	1	1	yes	characteristic
281	C₂	—	4	1	no	—
282	C₂	#13123548	1	1	yes	characteristic
283	C₂	#13123704	1	1	yes	characteristic
284	C₂	#13123753	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#3170	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#3170	D₁₃
hypocenter	C₁₃	#17972
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #13113745	C₂	semidirect
3. #13113733	C₂	semidirect
4. #13113655	C₂	semidirect
5. Q₁₆×#1055	C₂	semidirect
242. C₁₃	#17972	semidirect

Maximal subgroups

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

subgroup	index
#13123931	2
#13113745	2
#13113733	2
#13113655	2
Q₁₆×#1055	2
#13113599	2
#13113309	2
#13113119	2
C₁₃×#3170	2
#17972	13