Subgroups — #15832393

Order 3328.

678 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123917	C₂	1	2	yes	maximal
2	#13113655	C₂	1	4	yes	maximal, complemented
3	Q₁₆×#1055	C₂	1	4	yes	maximal, complemented
4	#13113584	C₂	1	1	yes	maximal, characteristic
5	#13113231	C₂	1	2	yes	maximal
6	#13113119	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
10	C₄×C₁₃×Q₁₆	C₄	1	2	yes	—
11	C₄×C₁₃×Q₁₆	C₄	1	2	yes	—
12	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
13	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
14	C₂×C₁₃×#107	C₄	1	1	yes	characteristic
15	C₂×C₁₃×#103	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	Q₈×#1055	—	2	4	no	—
18	Q₈×#1055	C₂²	1	8	yes	—
19	#11718586	—	2	4	no	—
20	#11718586	C₂²	1	2	yes	—
21	#11718586	C₂²	1	1	yes	characteristic
22	#11718586	C₂²	1	1	yes	characteristic
23	Q₈⋊#1055	C₂²	1	8	yes	—
24	#11718050	C₂²	1	2	yes	—
25	#11718049	C₂²	1	2	yes	—
26	#11718047	C₂²	1	2	yes	—
27	C₈×#1055	C₂²	1	2	yes	—
28	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂×C₁₃×Q₁₆	C₈	1	8	yes	complemented
30	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
31	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₄×Q₈×C₁₃	—	2	2	no	—
35	C₄×Q₈×C₁₃	—	2	2	no	—
36	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
37	C₄×Q₈×C₁₃	C₂³	1	4	yes	—
38	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
39	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₁₃	—	2	2	no	—
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×#107	C₂×C₄	1	2	yes	—
43	C₁₃×#107	C₂×C₄	1	2	yes	—
44	C₁₃×#103	C₂×C₄	1	4	yes	—
45	C₁₃×#103	C₂×C₄	1	4	yes	—
46	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₂²⋊#1055	—	2	2	no	—
51	C₂²⋊#1055	—	2	1	no	—
52	C₂²⋊#1055	D₄	1	2	yes	—
53	C₄⋊#1055	—	2	1	no	—
54	C₄⋊#1055	—	2	8	no	—
55	C₄⋊#1055	D₄	1	2	yes	—
56	C₄⋊#1055	C₂³	1	4	yes	—
57	C₄×#1055	—	2	2	no	—
58	C₄×#1055	—	2	8	no	—
59	C₄×#1055	C₂³	1	2	yes	—
60	C₄×#1055	C₂³	1	2	yes	—
61	#17856	—	13	1	no	maximal, Sylow
62	C₂×Q₈×C₁₃	—	2	2	no	—
63	C₂×Q₈×C₁₃	—	2	4	no	—
64	C₂×Q₈×C₁₃	C₂×C₈	1	8	yes	—
65	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	—	2	2	no	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
71	C₁₃×Q₁₆	C₂×C₈	1	8	yes	—
72	C₁₃×Q₁₆	C₂×C₈	1	8	yes	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₁₃×C₄⋊C₄	—	2	4	no	—
80	C₁₃×C₄⋊C₄	—	2	2	no	—
81	C₁₃×C₄⋊C₄	—	2	4	no	—
82	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
83	C₄²×C₁₃	—	2	4	no	—
84	C₄²×C₁₃	—	2	4	no	—
85	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₃	#41	1	1	yes	characteristic
88	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₂×#1055	—	2	4	no	—
90	C₂×#1055	—	2	2	no	—
91	C₂×#1055	C₂×D₄	1	4	yes	—
92	#2982	—	13	2	no	—
93	C₂×C₄×Q₁₆	D₁₃	1	1	yes	characteristic
94	#1579	—	13	4	no	—
95	C₈×Q₁₆	—	13	4	no	—
96	#1564	—	13	1	no	—
97	#1479	—	13	2	no	—
98	#1450	—	13	1	no	—
99	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
100	Q₈×C₁₃	—	2	4	no	—
101	Q₈×C₁₃	—	2	8	no	—
102	Q₈×C₁₃	C₂²×C₈	1	8	yes	—
103	C₂×C₄×C₁₃	—	2	4	no	—
104	C₂×C₄×C₁₃	—	2	8	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	4	no	—
107	C₂×C₄×C₁₃	—	4	1	no	—
108	C₂×C₄×C₁₃	—	8	2	no	—
109	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
113	C₂×C₄×C₁₃	#131	1	2	yes	—
114	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
117	C₈×C₁₃	—	2	1	no	—
118	C₈×C₁₃	—	2	1	no	—
119	C₈×C₁₃	C₂²×C₈	1	2	yes	—
120	C₈×C₁₃	C₂²×C₈	1	2	yes	—
121	#1055	—	2	4	no	—
122	#1055	—	4	2	no	—
123	#1055	—	4	4	no	—
124	C₂²×Q₁₆	Dic₁₃	1	1	yes	characteristic
125	C₂×C₄×Q₈	C₂×D₁₃	1	2	yes	—
126	C₈×Q₈	—	13	8	no	—
127	C₈×Q₈	—	26	4	no	—
128	C₄×Q₁₆	Dic₁₃	1	2	yes	—
129	C₄×Q₁₆	Dic₁₃	1	2	yes	—
130	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
131	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
132	#431	—	13	1	no	—
133	#431	—	13	1	no	—
134	#431	—	13	2	no	—
135	#431	—	26	4	no	—
136	C₂×#107	Dic₁₃	1	1	yes	characteristic
137	C₂×#103	Dic₁₃	1	2	yes	—
138	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
139	#335	—	13	2	no	—
140	#334	—	13	2	no	—
141	Q₈⋊C₈	—	13	8	no	—
142	#322	—	13	2	no	—
143	C₈²	—	13	2	no	—
144	C₂²×C₁₃	—	2	1	no	—
145	C₂²×C₁₃	—	2	1	no	—
146	C₂²×C₁₃	C₈×D₄	1	2	yes	—
147	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
148	C₄×C₁₃	—	2	8	no	—
149	C₄×C₁₃	—	2	1	no	—
150	C₄×C₁₃	—	2	1	no	—
151	C₄×C₁₃	—	2	1	no	—
152	C₄×C₁₃	—	4	4	no	—
153	C₄×C₁₃	C₈×D₄	1	2	yes	—
154	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
155	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
156	C₄×C₁₃	#578	1	2	yes	—
157	C₂²×Q₈	C₂×Dic₁₃	1	2	yes	—
158	C₂×Q₁₆	#1055	1	8	yes	complemented
159	C₂×Q₁₆	C₂×Dic₁₃	1	2	yes	—
160	C₂×Q₁₆	C₂×Dic₁₃	1	2	yes	—
161	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
162	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
163	C₄×Q₈	—	2	2	no	—
164	C₄×Q₈	—	2	2	no	—
165	C₄×Q₈	C₂×Dic₁₃	1	4	yes	—
166	C₄×Q₈	C₂²×D₁₃	1	4	yes	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
169	C₂×C₄²	—	2	2	no	—
170	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
171	#107	C₂×Dic₁₃	1	2	yes	—
172	#107	C₂×Dic₁₃	1	2	yes	—
173	C₄⋊C₈	—	13	2	no	—
174	C₄⋊C₈	—	13	4	no	—
175	C₄⋊C₈	—	26	8	no	—
176	C₄⋊C₈	—	26	1	no	—
177	#103	C₂×Dic₁₃	1	4	yes	—
178	#103	C₂×Dic₁₃	1	4	yes	—
179	C₂²⋊C₈	—	13	2	no	—
180	C₂²⋊C₈	—	26	2	no	—
181	C₂²⋊C₈	—	26	1	no	—
182	C₄×C₈	—	13	2	no	—
183	C₄×C₈	—	13	2	no	—
184	C₄×C₈	—	26	8	no	—
185	C₄×C₈	—	26	2	no	—
186	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
187	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
189	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
190	C₂×C₁₃	—	2	2	no	—
191	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
192	C₂×C₁₃	#2950	1	1	yes	characteristic
193	C₂×C₁₃	#2957	1	1	yes	characteristic
194	C₂×Q₈	—	2	2	no	—
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	C₂×#1055	1	8	yes	—
197	C₂×Q₈	C₂²×Dic₁₃	1	4	yes	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
201	C₂²×C₄	#6129	1	1	yes	characteristic
202	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
203	Q₁₆	C₂×#1055	1	8	yes	—
204	Q₁₆	C₂×#1055	1	8	yes	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	13	4	no	—
207	C₂×C₈	—	26	4	no	—
208	C₂×C₈	—	26	2	no	—
209	C₂×C₈	C₂×#1055	1	4	yes	—
210	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
211	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
212	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
213	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	C₂²×Dic₁₃	1	4	yes	—
218	C₄²	—	2	4	no	—
219	C₄²	—	2	4	no	—
220	C₄²	D₄×D₁₃	1	1	yes	characteristic
221	C₄²	#6129	1	1	yes	characteristic
222	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
223	C₄²	C₂³×D₁₃	1	1	yes	characteristic
224	C₁₃	#17856	1	1	yes	characteristic, complemented, Sylow
225	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
226	Q₈	—	2	8	no	—
227	Q₈	—	2	4	no	—
228	Q₈	C₂²×#1055	1	8	yes	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	C₂²×#1055	1	2	yes	—
236	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
237	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
238	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
239	C₂×C₄	#88202	1	2	yes	—
240	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
241	C₂×C₄	C₂×#6129	1	1	yes	characteristic
242	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	26	4	no	—
246	C₈	—	52	2	no	—
247	C₈	—	52	4	no	—
248	C₈	C₂²×#1055	1	2	yes	—
249	C₈	C₂²×#1055	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	D₄×#1055	1	2	yes	—
253	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	8	no	—
258	C₄	—	4	4	no	—
259	C₄	D₄×#1055	1	2	yes	—
260	C₄	#11719471	1	2	yes	—
261	C₄	C₂³×#1055	1	1	yes	characteristic
262	C₄	C₂×#88202	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	C₂×D₄×#1055	1	1	yes	characteristic
265	C₂	#13123704	1	1	yes	characteristic
266	C₂	#13123753	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×Q₁₆	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₁₃
hypocenter	C₁₃	#17856
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #13113655	C₂	semidirect
3. Q₁₆×#1055	C₂	semidirect
29. C₂×C₁₃×Q₁₆	C₈	semidirect
158. C₂×Q₁₆	#1055	semidirect
224. C₁₃	#17856	semidirect

Maximal subgroups

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

subgroup	index
#13123917	2
#13113655	2
Q₁₆×#1055	2
#13113584	2
#13113231	2
#13113119	2
C₂×C₄×C₁₃×Q₁₆	2
#17856	13