Subgroups — #15863526

Order 3328.

810 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	1	yes	maximal, characteristic
2	C₂²×C₄×#1055	C₂	1	1	yes	maximal, characteristic
3	#13115007	C₂	1	2	yes	maximal, complemented
4	#13114973	C₂	1	2	yes	maximal, complemented
5	#13114872	C₂	1	4	yes	maximal
6	C₄×#88062	C₂	1	4	yes	maximal, complemented
7	C₁₃×#2920	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂³×#1055	C₂²	1	2	yes	—
10	C₁₃×#432	—	2	2	no	—
11	C₁₃×#431	—	2	2	no	—
12	C₁₃×#423	C₂²	1	2	yes	—
13	C₁₃×#407	C₂²	1	2	yes	—
14	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
15	#11718868	C₂²	1	2	yes	—
16	C₂×#88062	C₂²	1	8	yes	—
17	#11718587	—	2	2	no	—
18	#11718586	—	2	2	no	—
19	#11718585	C₂²	1	2	yes	—
20	C₄×#6099	C₂²	1	2	yes	—
21	C₂×C₄×#1055	C₄	1	4	yes	—
22	C₂×C₄×#1055	C₄	1	4	yes	—
23	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
24	C₂×C₄×#1055	C₂²	1	2	yes	—
25	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
26	#11718150	C₂²	1	8	yes	—
27	C₂³×C₄×C₁₃	Q₈	1	2	yes	—
28	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₃	D₄	1	2	yes	—
33	C₂×C₄²×C₁₃	D₄	1	2	yes	—
34	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
35	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×#6099	C₂³	1	4	yes	—
38	C₂²×#1055	—	2	8	no	—
39	C₂²×#1055	C₂×C₄	1	8	yes	—
40	C₂²×#1055	C₂³	1	2	yes	—
41	C₂²×#1055	C₂³	1	2	yes	—
42	C₁₃×C₄⋊C₈	—	2	4	no	—
43	C₁₃×C₂²⋊C₈	—	2	4	no	—
44	C₁₃×#97	—	2	2	no	—
45	C₄×C₈×C₁₃	—	2	2	no	—
46	C₂²⋊#1055	—	2	4	no	—
47	#88062	—	2	16	no	—
48	C₄⋊#1055	—	2	4	no	—
49	#88043	—	2	2	no	—
50	C₄×#1055	—	2	2	no	—
51	C₄×#1055	C₂×C₄	1	2	yes	—
52	C₄×#1055	C₂×C₄	1	2	yes	—
53	C₄×#1055	C₂×C₄	1	8	yes	—
54	C₄×#1055	C₂×C₄	1	2	yes	—
55	C₄×#1055	C₂×C₄	1	2	yes	—
56	#20826	—	13	1	no	maximal, Sylow
57	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	—	2	8	no	—
59	C₂²×C₄×C₁₃	—	2	4	no	—
60	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
70	C₁₃×M₁₆	—	2	8	no	—
71	C₂×C₈×C₁₃	—	2	4	no	—
72	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
73	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
74	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
75	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
76	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
78	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₃	C₂×D₄	1	2	yes	—
81	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
82	#6099	—	2	8	no	—
83	C₂×#1055	—	2	4	no	—
84	C₂×#1055	—	2	16	no	—
85	C₂×#1055	—	2	4	no	—
86	C₂×#1055	C₂²×C₄	1	4	yes	—
87	C₂×#1055	C₂²×C₄	1	4	yes	—
88	C₂×#1055	#41	1	8	yes	—
89	#2920	—	13	1	no	—
90	#2920	D₁₃	1	1	yes	characteristic
91	C₂²×C₄×C₈	—	13	1	no	—
92	#1848	—	13	2	no	—
93	#1837	—	13	2	no	—
94	#1810	—	13	4	no	—
95	C₄×#108	—	13	4	no	—
96	C₂³×C₁₃	—	2	2	no	—
97	C₂³×C₁₃	—	2	4	no	—
98	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
99	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	4	no	—
102	C₂×C₄×C₁₃	—	2	4	no	—
103	C₂×C₄×C₁₃	—	2	4	no	—
104	C₂×C₄×C₁₃	—	2	4	no	—
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	—	2	4	no	—
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	#108	1	8	yes	complemented
109	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₁₃	C₂×C₄⋊C₄	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	4	yes	—
117	C₈×C₁₃	—	4	4	no	—
118	#1055	—	2	4	no	—
119	#1055	—	4	4	no	—
120	#1055	#117	1	8	yes	—
121	C₂³×C₈	—	13	2	no	—
122	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
123	#432	—	2	2	no	—
124	#432	—	26	2	no	—
125	#431	—	2	2	no	—
126	#431	—	26	2	no	—
127	C₂×#108	—	13	8	no	—
128	#423	—	13	2	no	—
129	#423	C₂×D₁₃	1	2	yes	—
130	#407	—	13	2	no	—
131	#407	C₂×D₁₃	1	2	yes	—
132	C₄×M₁₆	—	13	2	no	—
133	C₄×M₁₆	C₂×D₁₃	1	2	yes	—
134	C₂×C₄×C₈	—	13	4	no	—
135	C₂×C₄×C₈	—	13	4	no	—
136	C₂×C₄×C₈	—	13	1	no	—
137	C₂×C₄×C₈	—	13	1	no	—
138	C₂×C₄×C₈	—	13	2	no	—
139	#341	—	13	8	no	—
140	C₂²×C₁₃	—	2	8	no	—
141	C₂²×C₁₃	—	2	2	no	—
142	C₂²×C₁₃	—	2	2	no	—
143	C₂²×C₁₃	—	2	2	no	—
144	C₂²×C₁₃	C₂×#108	1	4	yes	—
145	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₂²×C₁₃	#533	1	2	yes	—
147	C₄×C₁₃	—	2	2	no	—
148	C₄×C₁₃	—	2	2	no	—
149	C₄×C₁₃	—	2	2	no	—
150	C₄×C₁₃	C₂×#108	1	4	yes	—
151	C₄×C₁₃	C₂×#108	1	4	yes	—
152	C₄×C₁₃	C₂×#117	1	2	yes	—
153	C₄×C₁₃	#532	1	2	yes	—
154	C₂³×C₄	Dic₂₆	1	2	yes	—
155	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
156	C₂×M₁₆	—	13	4	no	—
157	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
158	C₂²×C₈	—	13	2	no	—
159	C₂²×C₈	—	13	8	no	—
160	C₂²×C₈	—	13	2	no	—
161	C₂²×C₈	—	26	8	no	—
162	C₂×C₄²	Dic₂₆	1	2	yes	—
163	C₂×C₄²	C₄×D₁₃	1	2	yes	—
164	C₂×C₄²	C₄×D₁₃	1	2	yes	—
165	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
166	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
167	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
168	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
169	#108	—	26	16	no	—
170	C₄⋊C₈	—	2	4	no	—
171	C₄⋊C₈	—	26	4	no	—
172	C₂²⋊C₈	—	2	4	no	—
173	C₂²⋊C₈	—	26	4	no	—
174	#97	—	2	2	no	—
175	#97	—	26	2	no	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	—	13	2	no	—
178	C₄×C₈	—	13	2	no	—
179	C₄×C₈	—	13	8	no	—
180	C₄×C₈	—	13	2	no	—
181	C₄×C₈	—	13	2	no	—
182	C₄×C₈	—	26	2	no	—
183	C₂×C₁₃	—	2	2	no	—
184	C₂×C₁₃	—	2	4	no	—
185	C₂×C₁₃	#2285	1	1	yes	characteristic
186	C₂×C₁₃	C₂²×#108	1	2	yes	—
187	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	8	no	—
190	C₂²×C₄	#6101	1	2	yes	—
191	C₂²×C₄	#6101	1	2	yes	—
192	C₂²×C₄	#6101	1	2	yes	—
193	C₂²×C₄	#6101	1	2	yes	—
194	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
195	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
196	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
197	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
198	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
199	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
200	M₁₆	—	2	8	no	—
201	M₁₆	—	26	8	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	13	4	no	—
204	C₂×C₈	—	13	8	no	—
205	C₂×C₈	—	13	4	no	—
206	C₂×C₈	—	26	16	no	—
207	C₂×C₈	—	26	4	no	—
208	C₂×C₈	—	26	4	no	—
209	C₄²	#6101	1	2	yes	—
210	C₄²	#6101	1	2	yes	—
211	C₄²	#6101	1	2	yes	—
212	C₄²	#6101	1	2	yes	—
213	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
214	C₄²	C₂×C₄×D₁₃	1	2	yes	—
215	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
216	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
217	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
218	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
219	C₁₃	#20826	1	1	yes	characteristic, complemented, Sylow
220	C₂³	—	2	2	no	—
221	C₂³	—	2	4	no	—
222	C₂³	C₂×#6101	1	2	yes	—
223	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	#88062	1	8	yes	complemented
233	C₂×C₄	C₂×#6101	1	2	yes	—
234	C₂×C₄	C₂×#6101	1	2	yes	—
235	C₂×C₄	C₂×#6101	1	2	yes	—
236	C₂×C₄	C₂×#6101	1	2	yes	—
237	C₂×C₄	C₂×#6101	1	2	yes	—
238	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
239	C₂×C₄	#41×D₁₃	1	4	yes	—
240	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
241	C₈	—	4	4	no	—
242	C₈	—	13	8	no	—
243	C₈	—	26	4	no	—
244	C₈	—	52	4	no	—
245	C₂²	—	2	8	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	C₂×#88062	1	4	yes	—
250	C₂²	#11719222	1	2	yes	—
251	C₂²	C₂²×#6101	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	C₂×#88062	1	4	yes	—
256	C₄	C₂×#88062	1	4	yes	—
257	C₄	D₁₃×#117	1	2	yes	—
258	C₄	#11719537	1	2	yes	—
259	C₂	—	2	2	no	—
260	C₂	—	2	4	no	—
261	C₂	#13117239	1	1	yes	characteristic
262	C₂	C₂²×#88062	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#2920	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#2920	D₁₃
hypocenter	C₁₃	#20826
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
3. #13115007	C₂	semidirect
4. #13114973	C₂	semidirect
6. C₄×#88062	C₂	semidirect
108. C₂×C₄×C₁₃	#108	semidirect
219. C₁₃	#20826	semidirect
232. C₂×C₄	#88062	semidirect

Maximal subgroups

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

subgroup	index
#13123469	2
C₂²×C₄×#1055	2
#13115007	2
#13114973	2
#13114872	2
C₄×#88062	2
C₁₃×#2920	2
#20826	13