Subgroups — #15832073

Order 3328.

638 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	#13123469	C₂	1	2	yes	maximal
2	#13113616	C₂	1	8	yes	maximal, complemented
3	#13113595	C₂	1	2	yes	maximal, complemented
4	#13113592	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#431	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
7	C₁₃×#431	C₄	1	4	yes	—
8	C₁₃×#431	C₂²	1	4	yes	—
9	C₂×C₁₃×C₄⋊C₈	C₄	1	2	yes	—
10	C₂×C₁₃×C₂²⋊C₈	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	#11718868	—	2	2	no	—
14	#11718587	—	2	2	no	—
15	#11718587	C₂²	1	4	yes	—
16	#11718586	—	2	2	no	—
17	#11718586	C₂²	1	4	yes	—
18	#11718585	C₂²	1	4	yes	—
19	C₄×#6099	—	2	2	no	—
20	#11718050	C₂²	1	8	yes	complemented
21	#11718049	C₂²	1	8	yes	complemented
22	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃	Q₈	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	D₄	1	2	yes	—
31	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
33	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
35	C₂×#6099	—	2	4	no	—
36	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
37	C₁₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
38	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₁₃	C₂³	1	2	yes	—
41	C₄×C₈×C₁₃	C₂³	1	2	yes	—
42	C₂²⋊#1055	—	2	4	no	—
43	C₂²⋊#1055	—	2	4	no	—
44	C₄⋊#1055	—	2	4	no	—
45	C₄⋊#1055	C₂³	1	8	yes	—
46	#88043	—	2	4	no	—
47	C₄×#1055	—	2	4	no	—
48	#17815	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	—	2	4	no	—
51	C₂²×C₄×C₁₃	—	4	2	no	—
52	C₂²×C₄×C₁₃	—	4	2	no	—
53	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
65	C₂×C₈×C₁₃	—	4	2	no	—
66	C₂×C₈×C₁₃	—	4	2	no	—
67	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
68	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
69	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
70	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
71	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
72	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
74	C₄²×C₁₃	C₂×D₄	1	2	yes	—
75	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
77	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
78	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
79	#6099	—	4	8	no	—
80	C₂×#1055	—	2	8	no	—
81	#2920	—	13	2	no	—
82	C₂×#431	D₁₃	1	1	yes	characteristic
83	#1572	—	13	8	no	—
84	#1567	—	13	2	no	—
85	#1566	—	13	2	no	—
86	C₂³×C₁₃	—	2	2	no	—
87	C₂³×C₁₃	—	4	1	no	—
88	C₂³×C₁₃	—	4	1	no	—
89	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
90	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	4	no	—
94	C₂×C₄×C₁₃	—	2	4	no	—
95	C₂×C₄×C₁₃	—	2	4	no	—
96	C₂×C₄×C₁₃	—	2	4	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	4	1	no	—
99	C₂×C₄×C₁₃	—	4	2	no	—
100	C₂×C₄×C₁₃	—	4	1	no	—
101	C₂×C₄×C₁₃	—	4	2	no	—
102	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#135	1	4	yes	—
109	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
112	C₈×C₁₃	—	2	4	no	—
113	C₈×C₁₃	—	2	4	no	—
114	#1055	—	4	8	no	—
115	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
116	#432	—	13	4	no	—
117	#432	—	26	2	no	—
118	#431	—	13	4	no	—
119	#431	—	26	2	no	—
120	#431	Dic₁₃	1	4	yes	—
121	#431	C₂×D₁₃	1	4	yes	—
122	#423	—	13	4	no	—
123	C₂×C₄⋊C₈	Dic₁₃	1	2	yes	—
124	#407	—	26	2	no	—
125	C₂×C₂²⋊C₈	Dic₁₃	1	2	yes	—
126	C₄×M₁₆	—	26	2	no	—
127	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
128	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
129	#335	—	13	8	no	—
130	#334	—	13	8	no	—
131	C₂²×C₁₃	—	2	1	no	—
132	C₂²×C₁₃	—	2	1	no	—
133	C₂²×C₁₃	—	2	2	no	—
134	C₂²×C₁₃	—	2	2	no	—
135	C₂²×C₁₃	—	2	2	no	—
136	C₂²×C₁₃	—	4	4	no	—
137	C₂²×C₁₃	#427	1	2	yes	—
138	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
139	C₄×C₁₃	—	2	1	no	—
140	C₄×C₁₃	—	2	2	no	—
141	C₄×C₁₃	—	2	2	no	—
142	C₄×C₁₃	—	2	2	no	—
143	C₄×C₁₃	—	2	1	no	—
144	C₄×C₁₃	#427	1	2	yes	—
145	C₄×C₁₃	#430	1	2	yes	—
146	C₄×C₁₃	#568	1	2	yes	—
147	C₄×C₁₃	C₂×#135	1	2	yes	—
148	C₂³×C₄	Dic₂₆	1	1	yes	characteristic
149	C₂³×C₄	D₅₂	1	1	yes	characteristic
150	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
151	C₂×M₁₆	—	26	4	no	—
152	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
153	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
154	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
155	C₂×C₄²	Dic₂₆	1	2	yes	—
156	C₂×C₄²	D₅₂	1	1	yes	characteristic
157	C₂×C₄²	D₅₂	1	2	yes	—
158	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
159	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
160	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
161	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
162	C₄⋊C₈	—	13	8	no	—
163	C₄⋊C₈	—	26	4	no	—
164	C₄⋊C₈	C₂×Dic₁₃	1	8	yes	—
165	C₂²⋊C₈	—	26	4	no	—
166	C₂²⋊C₈	—	26	4	no	—
167	C₂²⋊C₈	C₂×Dic₁₃	1	8	yes	—
168	#97	—	26	4	no	—
169	C₄×C₈	—	26	4	no	—
170	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
171	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
172	C₄×C₈	C₂²×D₁₃	1	2	yes	—
173	C₄×C₈	C₂²×D₁₃	1	2	yes	—
174	C₂×C₁₃	—	2	2	no	—
175	C₂×C₁₃	—	4	2	no	—
176	C₂×C₁₃	#2905	1	1	yes	characteristic
177	C₂×C₁₃	C₂×#427	1	1	yes	characteristic
178	C₂×C₁₃	C₂×#430	1	1	yes	characteristic
179	C₂⁴	C₄⋊Dic₁₃	1	1	yes	characteristic
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
184	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
185	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
186	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
187	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
188	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
189	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
190	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
191	C₂²×C₄	C₂×D₅₂	1	2	yes	—
192	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
193	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
194	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
195	M₁₆	—	52	8	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	26	8	no	—
199	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
200	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
201	C₄²	C₄⋊Dic₁₃	1	2	yes	—
202	C₄²	C₄⋊Dic₁₃	1	2	yes	—
203	C₄²	C₄⋊Dic₁₃	1	2	yes	—
204	C₄²	C₂×Dic₂₆	1	2	yes	—
205	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
206	C₄²	C₂×D₅₂	1	2	yes	—
207	C₄²	C₂×D₅₂	1	1	yes	characteristic
208	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
209	C₄²	C₂²×Dic₁₃	1	2	yes	—
210	C₄²	C₂³×D₁₃	1	1	yes	characteristic
211	C₁₃	#17815	1	1	yes	characteristic, complemented, Sylow
212	C₂³	—	2	2	no	—
213	C₂³	—	4	1	no	—
214	C₂³	—	4	1	no	—
215	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
216	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	#88158	1	4	yes	—
229	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
230	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
231	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
232	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
233	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
234	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
235	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
237	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
238	C₈	—	2	4	no	—
239	C₈	—	2	4	no	—
240	C₈	—	52	8	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	4	no	—
247	C₂²	#11718767	1	2	yes	—
248	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
249	C₄	—	2	1	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	#11718767	1	2	yes	—
255	C₄	#11718786	1	2	yes	—
256	C₄	C₂×#88158	1	2	yes	—
257	C₄	#11719524	1	2	yes	—
258	C₂	—	2	2	no	—
259	C₂	—	4	2	no	—
260	C₂	#13123400	1	1	yes	characteristic
261	C₂	C₂×#11718767	1	1	yes	characteristic
262	C₂	C₂×#11718786	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#431	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₄⋊C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂×#431	D₁₃
hypocenter	C₁₃	#17815
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
2. #13113616	C₂	semidirect
3. #13113595	C₂	semidirect
4. #13113592	C₂	semidirect
20. #11718050	C₂²	semidirect
21. #11718049	C₂²	semidirect
211. C₁₃	#17815	semidirect

Maximal subgroups

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

subgroup	index
#13123469	2
#13113616	2
#13113595	2
#13113592	2
C₂×C₁₃×#431	2
#17815	13