Subgroups — #15833781

Order 3328.

544 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125422	C₂	1	1	yes	maximal, characteristic
2	#13123930	C₂	1	2	yes	maximal
3	#13113709	C₂	1	4	yes	maximal, complemented
4	#13113688	C₂	1	4	yes	maximal, complemented
5	#13113599	C₂	1	1	yes	maximal, characteristic
6	#13113310	C₂	1	2	yes	maximal
7	C₁₃×#1452	C₂	1	1	yes	maximal, characteristic
8	#11719175	C₂²	1	2	yes	—
9	C₁₃×#431	C₂²	1	1	yes	characteristic
10	C₁₃×#431	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	#11718777	C₄	1	2	yes	—
13	#11718777	C₄	1	2	yes	—
14	#11718587	C₂²	1	2	yes	—
15	#11718489	—	2	4	no	—
16	#11718381	—	2	4	no	—
17	#11718360	C₄	1	1	yes	characteristic
18	C₄×Dic₅₂	C₂²	1	2	yes	—
19	C₄×Dic₅₂	C₂²	1	2	yes	—
20	#11718343	C₄	1	1	yes	characteristic
21	#11718342	C₄	1	2	yes	—
22	#11718340	C₂²	1	4	yes	—
23	C₈×Dic₂₆	C₂²	1	4	yes	—
24	C₁₃×#322	C₂²	1	2	yes	—
25	C₁₃×#322	C₂²	1	2	yes	—
26	#11718050	C₂²	1	2	yes	—
27	#11718049	C₂²	1	2	yes	—
28	#11718048	C₂²	1	8	yes	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
32	#88180	—	2	2	no	—
33	#88178	—	2	2	no	—
34	#88178	C₂×C₄	1	4	yes	—
35	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
36	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
37	#88124	—	2	2	no	—
38	#88123	C₂×C₄	1	2	yes	—
39	#88123	C₂×C₄	1	2	yes	—
40	C₄×Dic₂₆	—	2	2	no	—
41	C₄×Dic₂₆	C₂³	1	4	yes	—
42	C₁₃×C₄⋊C₈	—	2	1	no	—
43	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
44	C₁₃×C₂²⋊C₈	—	2	1	no	—
45	C₁₃×C₂²⋊C₈	D₄	1	2	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	2	yes	—
49	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₁₃	C₂³	1	2	yes	—
52	C₂²⋊#1055	—	2	2	no	—
53	#88058	C₂×C₄	1	2	yes	—
54	#88058	C₂×C₄	1	2	yes	—
55	#88056	C₂×C₄	1	4	yes	—
56	#88056	C₂×C₄	1	4	yes	—
57	#88055	—	2	4	no	—
58	#88054	—	2	4	no	—
59	#88054	—	2	4	no	—
60	C₈×Dic₁₃	—	2	4	no	—
61	C₄⋊#1055	C₂³	1	4	yes	—
62	#88043	—	2	2	no	—
63	#17991	—	13	1	no	maximal, Sylow
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
67	C₂×Dic₂₆	—	2	2	no	—
68	C₂×Dic₂₆	C₂²×C₄	1	4	yes	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	—	4	1	no	—
71	C₂×C₈×C₁₃	—	4	1	no	—
72	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
77	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₃	#41	1	1	yes	characteristic
80	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₂²⋊Dic₁₃	—	2	4	no	—
82	C₄⋊Dic₁₃	—	2	2	no	—
83	C₄⋊Dic₁₃	—	2	2	no	—
84	C₄⋊Dic₁₃	—	2	2	no	—
85	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
86	#6101	—	2	4	no	—
87	#6101	—	2	4	no	—
88	C₄×Dic₁₃	—	2	4	no	—
89	C₂×#1055	—	2	4	no	—
90	Dic₅₂	—	4	2	no	—
91	Dic₅₂	—	4	2	no	—
92	#3170	—	13	1	no	—
93	#2985	—	13	2	no	—
94	#1587	—	13	4	no	—
95	#1584	—	13	4	no	—
96	#1568	—	13	1	no	—
97	#1497	—	13	2	no	—
98	#1452	D₁₃	1	1	yes	characteristic
99	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	4	1	no	—
104	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
110	C₂×Dic₁₃	—	2	4	no	—
111	C₂×Dic₁₃	—	2	4	no	—
112	Dic₂₆	—	2	4	no	—
113	Dic₂₆	—	2	4	no	—
114	Dic₂₆	—	8	2	no	—
115	C₈×C₁₃	—	2	1	no	—
116	C₈×C₁₃	—	2	1	no	—
117	C₈×C₁₃	—	2	1	no	—
118	C₈×C₁₃	—	2	1	no	—
119	C₈×C₁₃	—	2	4	no	—
120	C₈×C₁₃	—	4	2	no	—
121	#1055	—	4	4	no	—
122	#533	—	13	2	no	—
123	#500	—	13	1	no	—
124	#494	—	13	1	no	—
125	#487	—	13	2	no	—
126	#467	—	13	2	no	—
127	#467	—	13	2	no	—
128	#446	—	13	4	no	—
129	#446	—	26	4	no	—
130	C₈×Q₈	—	13	4	no	—
131	C₄×Q₁₆	—	13	2	no	—
132	C₄×Q₁₆	—	13	2	no	—
133	#433	—	26	4	no	—
134	#432	—	13	2	no	—
135	#431	C₂×D₁₃	1	1	yes	characteristic
136	#431	C₂×D₁₃	1	1	yes	characteristic
137	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
138	#335	—	13	2	no	—
139	#334	—	13	2	no	—
140	Q₈⋊C₈	—	13	8	no	—
141	#322	C₂×D₁₃	1	2	yes	—
142	#322	C₂×D₁₃	1	2	yes	—
143	C₂²×C₁₃	—	2	1	no	—
144	C₂²×C₁₃	—	2	1	no	—
145	C₂²×C₁₃	—	2	1	no	—
146	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
147	C₄×C₁₃	—	2	1	no	—
148	C₄×C₁₃	—	2	1	no	—
149	C₄×C₁₃	—	2	1	no	—
150	C₄×C₁₃	—	2	1	no	—
151	C₄×C₁₃	#568	1	1	yes	characteristic
152	C₄×C₁₃	#568	1	1	yes	characteristic
153	C₄×C₁₃	#578	1	2	yes	—
154	Dic₁₃	—	4	4	no	—
155	Dic₁₃	—	4	4	no	—
156	C₂×Q₁₆	—	13	2	no	—
157	C₂×Q₁₆	—	13	2	no	—
158	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
159	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
160	C₄⋊Q₈	—	13	2	no	—
161	C₄⋊Q₈	—	13	2	no	—
162	#125	—	26	2	no	—
163	C₂²⋊Q₈	—	13	4	no	—
164	C₂²⋊Q₈	—	26	2	no	—
165	C₄×Q₈	—	13	4	no	—
166	C₄×Q₈	—	26	2	no	—
167	#117	—	26	2	no	—
168	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
169	#107	—	13	2	no	—
170	#107	—	13	2	no	—
171	C₄⋊C₈	—	2	1	no	—
172	C₄⋊C₈	—	13	4	no	—
173	C₄⋊C₈	—	26	4	no	—
174	C₄⋊C₈	—	26	4	no	—
175	C₄⋊C₈	D₅₂	1	2	yes	—
176	#103	—	13	4	no	—
177	#103	—	13	4	no	—
178	C₂²⋊C₈	—	2	1	no	—
179	C₂²⋊C₈	—	26	2	no	—
180	C₂²⋊C₈	D₅₂	1	2	yes	—
181	#97	—	26	4	no	—
182	#97	—	26	2	no	—
183	C₄×C₈	—	26	4	no	—
184	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
185	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
186	C₄×C₈	C₂²×D₁₃	1	2	yes	—
187	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
189	C₄×C₈	C₂²×D₁₃	1	2	yes	—
190	C₂×C₁₃	—	4	1	no	—
191	C₂×C₁₃	#2932	1	1	yes	characteristic
192	C₂×C₁₃	#2950	1	1	yes	characteristic
193	C₂×C₁₃	#2957	1	1	yes	characteristic
194	C₂×Q₈	—	13	4	no	—
195	C₂×Q₈	—	26	2	no	—
196	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
198	C₂²×C₄	#6127	1	1	yes	characteristic
199	Q₁₆	—	52	2	no	—
200	Q₁₆	—	52	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	26	4	no	—
205	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
206	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
207	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
208	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
209	C₂×C₈	C₂×D₅₂	1	4	yes	—
210	C₄⋊C₄	—	13	4	no	—
211	C₄⋊C₄	—	26	2	no	—
212	C₄⋊C₄	—	26	2	no	—
213	C₄⋊C₄	—	26	4	no	—
214	C₄⋊C₄	—	26	2	no	—
215	C₄⋊C₄	—	26	4	no	—
216	C₂²⋊C₄	—	26	4	no	—
217	C₄²	—	26	4	no	—
218	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
219	C₄²	C₂×D₅₂	1	1	yes	characteristic
220	C₄²	#6127	1	1	yes	characteristic
221	C₄²	C₂³×D₁₃	1	1	yes	characteristic
222	C₁₃	#17991	1	1	yes	characteristic, complemented, Sylow
223	C₂³	C₄×D₅₂	1	1	yes	characteristic
224	Q₈	—	26	4	no	—
225	Q₈	—	26	4	no	—
226	Q₈	—	104	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	26	4	no	—
232	C₂×C₄	—	26	4	no	—
233	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
234	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
235	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
236	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
237	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
238	C₂×C₄	C₂×#6127	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	2	4	no	—
244	C₈	—	4	2	no	—
245	C₈	—	52	4	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	52	4	no	—
255	C₄	—	52	4	no	—
256	C₄	#11719450	1	1	yes	characteristic
257	C₄	#11719450	1	1	yes	characteristic
258	C₄	#11719458	1	2	yes	—
259	C₂	—	4	1	no	—
260	C₂	#13123555	1	1	yes	characteristic
261	C₂	#13123705	1	1	yes	characteristic
262	C₂	#13123754	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1452	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1452	D₁₃
hypocenter	C₁₃	#17991
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1452

Complements

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

N	H ≅ G/N	type
3. #13113709	C₂	semidirect
4. #13113688	C₂	semidirect
222. C₁₃	#17991	semidirect

Maximal subgroups

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

subgroup	index
#13125422	2
#13123930	2
#13113709	2
#13113688	2
#13113599	2
#13113310	2
C₁₃×#1452	2
#17991	13