Subgroups — #15863538

Order 3328.

718 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	#13123474	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718658	C₂	1	1	yes	maximal, characteristic
3	#13115007	C₂	1	2	yes	maximal, complemented
4	#13114976	C₂	1	2	yes	maximal, complemented
5	#13114876	C₂	1	2	yes	maximal
6	#13114872	C₂	1	2	yes	maximal
7	#13114743	C₂	1	2	yes	maximal, complemented
8	C₄×#88062	C₂	1	2	yes	maximal, complemented
9	C₁₃×#2921	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
11	C₂²×#6099	C₂²	1	1	yes	characteristic
12	C₂³×#1055	C₂²	1	1	yes	characteristic
13	C₁₃×#433	—	2	4	no	—
14	C₁₃×#423	C₂²	1	2	yes	—
15	C₁₃×#407	C₂²	1	2	yes	—
16	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
17	#11718868	C₂²	1	2	yes	—
18	C₂×#88062	C₂²	1	4	yes	—
19	C₂×#88062	C₂²	1	4	yes	—
20	#11718661	—	2	4	no	—
21	#11718658	C₄	1	8	yes	—
22	#11718585	C₂²	1	2	yes	—
23	C₄×#6099	C₂²	1	2	yes	—
24	C₂×#88043	C₂²	1	1	yes	characteristic
25	C₂×#88043	C₂²	1	1	yes	characteristic
26	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
27	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
28	#11718150	C₂²	1	4	yes	—
29	#11718150	C₂²	1	4	yes	—
30	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
32	C₁₃×#117	C₂×C₄	1	4	yes	—
33	C₁₃×#117	D₄	1	4	yes	—
34	C₂×C₁₃×C₄⋊C₄	Q₈	1	2	yes	—
35	C₂×C₁₃×C₂²⋊C₄	Q₈	1	2	yes	—
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×#6099	—	2	4	no	—
39	C₂×#6099	C₂×C₄	1	4	yes	—
40	C₂×#6099	C₂³	1	4	yes	—
41	C₂²×#1055	—	2	4	no	—
42	C₂²×#1055	C₂×C₄	1	4	yes	—
43	C₂²×#1055	C₂³	1	1	yes	characteristic
44	C₂²×#1055	C₂³	1	1	yes	characteristic
45	C₂²×#1055	C₂³	1	1	yes	characteristic
46	C₂²×#1055	C₂³	1	1	yes	characteristic
47	C₁₃×C₄⋊C₈	—	2	4	no	—
48	C₁₃×C₂²⋊C₈	—	2	4	no	—
49	C₁₃×#97	—	2	2	no	—
50	C₄×C₈×C₁₃	—	2	2	no	—
51	C₂²⋊#1055	—	2	4	no	—
52	#88062	—	2	8	no	—
53	#88062	—	2	8	no	—
54	C₄⋊#1055	—	2	4	no	—
55	#88043	—	2	2	no	—
56	#88043	C₂×C₄	1	4	yes	—
57	#88043	C₂×C₄	1	4	yes	—
58	C₄×#1055	—	2	2	no	—
59	C₄×#1055	C₂×C₄	1	4	yes	—
60	C₄×#1055	C₂×C₄	1	4	yes	—
61	#20828	—	13	1	no	maximal, Sylow
62	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	—	2	2	no	—
64	C₂²×C₄×C₁₃	—	2	2	no	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
71	C₁₃×M₁₆	—	2	4	no	—
72	C₁₃×M₁₆	—	2	4	no	—
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₁₃×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
75	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
76	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
77	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
78	C₄²×C₁₃	C₂×D₄	1	2	yes	—
79	C₄²×C₁₃	C₂×D₄	1	2	yes	—
80	#6099	—	2	8	no	—
81	#6099	—	2	4	no	—
82	#6099	—	2	4	no	—
83	C₂×#1055	—	2	2	no	—
84	C₂×#1055	—	2	2	no	—
85	C₂×#1055	—	2	8	no	—
86	C₂×#1055	—	2	4	no	—
87	C₂×#1055	C₂²×C₄	1	2	yes	—
88	C₂×#1055	C₂²×C₄	1	2	yes	—
89	C₂×#1055	C₂²×C₄	1	2	yes	—
90	C₂×#1055	C₂²×C₄	1	2	yes	—
91	C₂×#1055	#41	1	4	yes	—
92	C₂×#1055	#41	1	4	yes	—
93	#2921	—	13	1	no	—
94	#2921	D₁₃	1	1	yes	characteristic
95	C₂×#405	—	13	1	no	—
96	#1848	—	13	2	no	—
97	#1838	—	13	2	no	—
98	#1811	—	13	2	no	—
99	#1810	—	13	2	no	—
100	#1778	—	13	2	no	—
101	C₄×#108	—	13	2	no	—
102	C₂³×C₁₃	—	2	2	no	—
103	C₂³×C₁₃	—	4	2	no	—
104	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
105	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	4	no	—
110	C₂×C₄×C₁₃	—	4	2	no	—
111	C₂×C₄×C₁₃	—	4	2	no	—
112	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
113	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
114	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
118	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
119	C₈×C₁₃	—	4	4	no	—
120	#1055	—	2	2	no	—
121	#1055	—	2	2	no	—
122	#1055	—	4	4	no	—
123	#1055	#117	1	4	yes	—
124	#1055	#117	1	4	yes	—
125	C₂²×M₁₆	—	13	1	no	—
126	C₂³×C₈	—	13	1	no	—
127	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
128	#433	—	2	4	no	—
129	#433	—	26	4	no	—
130	C₂×#108	—	13	4	no	—
131	C₂×#108	—	13	4	no	—
132	#423	—	13	2	no	—
133	#423	C₂×D₁₃	1	2	yes	—
134	#407	—	13	2	no	—
135	#407	C₂×D₁₃	1	2	yes	—
136	#405	—	13	8	no	—
137	C₄×M₁₆	—	13	2	no	—
138	C₄×M₁₆	C₂×D₁₃	1	2	yes	—
139	C₂×#97	—	13	1	no	—
140	C₂×#97	—	13	1	no	—
141	C₂×C₄×C₈	—	13	1	no	—
142	C₂×C₄×C₈	—	13	1	no	—
143	#341	—	13	4	no	—
144	#341	—	13	4	no	—
145	C₂²×C₁₃	—	2	2	no	—
146	C₂²×C₁₃	—	2	4	no	—
147	C₂²×C₁₃	—	2	2	no	—
148	C₂²×C₁₃	—	4	4	no	—
149	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
150	C₂²×C₁₃	#533	1	2	yes	—
151	C₄×C₁₃	—	2	4	no	—
152	C₄×C₁₃	—	2	4	no	—
153	C₄×C₁₃	—	2	2	no	—
154	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
155	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
156	C₄×C₁₃	#532	1	2	yes	—
157	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
158	C₂×M₁₆	—	13	4	no	—
159	C₂×M₁₆	—	13	4	no	—
160	C₂×M₁₆	—	26	4	no	—
161	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
162	C₂²×C₈	—	13	1	no	—
163	C₂²×C₈	—	13	1	no	—
164	C₂²×C₈	—	13	4	no	—
165	C₂²×C₈	—	13	1	no	—
166	C₂²×C₈	—	13	1	no	—
167	C₂²×C₈	—	26	4	no	—
168	#117	C₄×D₁₃	1	4	yes	—
169	#117	C₂²⋊D₁₃	1	4	yes	—
170	C₂×C₄⋊C₄	Dic₂₆	1	2	yes	—
171	C₂×C₂²⋊C₄	Dic₂₆	1	2	yes	—
172	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
173	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
174	#108	—	26	8	no	—
175	#108	—	26	8	no	—
176	C₄⋊C₈	—	2	4	no	—
177	C₄⋊C₈	—	26	4	no	—
178	C₂²⋊C₈	—	2	4	no	—
179	C₂²⋊C₈	—	26	4	no	—
180	#97	—	2	2	no	—
181	#97	—	13	4	no	—
182	#97	—	13	4	no	—
183	#97	—	26	2	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	—	13	4	no	—
186	C₄×C₈	—	13	4	no	—
187	C₄×C₈	—	26	2	no	—
188	C₂×C₁₃	—	2	2	no	—
189	C₂×C₁₃	—	4	2	no	—
190	C₂×C₁₃	#2285	1	1	yes	characteristic
191	C₂×C₁₃	#2916	1	2	yes	—
192	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
196	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
197	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
198	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
199	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
200	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
201	M₁₆	—	2	4	no	—
202	M₁₆	—	2	4	no	—
203	M₁₆	—	26	8	no	—
204	M₁₆	—	26	4	no	—
205	M₁₆	—	26	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	13	2	no	—
208	C₂×C₈	—	13	2	no	—
209	C₂×C₈	—	13	4	no	—
210	C₂×C₈	—	13	4	no	—
211	C₂×C₈	—	13	2	no	—
212	C₂×C₈	—	13	2	no	—
213	C₂×C₈	—	26	8	no	—
214	C₂×C₈	—	26	2	no	—
215	C₂×C₈	—	26	2	no	—
216	C₂×C₈	—	26	4	no	—
217	C₄⋊C₄	#6101	1	8	yes	—
218	C₂²⋊C₄	#6101	1	8	yes	—
219	C₄²	C₂×C₄×D₁₃	1	2	yes	—
220	C₄²	C₂×C₄×D₁₃	1	2	yes	—
221	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
222	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
223	C₁₃	#20828	1	1	yes	characteristic, complemented, Sylow
224	C₂³	—	2	2	no	—
225	C₂³	—	4	2	no	—
226	C₂³	C₂×#6101	1	2	yes	—
227	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	C₂×#6101	1	4	yes	—
235	C₂×C₄	C₂×#6101	1	4	yes	—
236	C₂×C₄	C₂×#6101	1	2	yes	—
237	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
238	C₂×C₄	#41×D₁₃	1	2	yes	—
239	C₂×C₄	#41×D₁₃	1	2	yes	—
240	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
241	C₈	—	4	4	no	—
242	C₈	—	13	4	no	—
243	C₈	—	13	4	no	—
244	C₈	—	26	2	no	—
245	C₈	—	26	2	no	—
246	C₈	—	52	4	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	4	no	—
250	C₂²	—	4	4	no	—
251	C₂²	#11719222	1	2	yes	—
252	C₂²	C₂²×#6101	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	4	no	—
255	C₄	—	2	4	no	—
256	C₄	D₁₃×#117	1	1	yes	characteristic
257	C₄	D₁₃×#117	1	1	yes	characteristic
258	C₄	#11719537	1	2	yes	—
259	C₂	—	2	2	no	—
260	C₂	—	4	2	no	—
261	C₂	#13117239	1	1	yes	characteristic
262	C₂	#13123445	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#2921	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#2921	D₁₃
hypocenter	C₁₃	#20828
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ #2921

Complements

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

N	H ≅ G/N	type
3. #13115007	C₂	semidirect
4. #13114976	C₂	semidirect
7. #13114743	C₂	semidirect
8. C₄×#88062	C₂	semidirect
223. C₁₃	#20828	semidirect

Maximal subgroups

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

subgroup	index
#13123474	2
C₂×#11718658	2
#13115007	2
#13114976	2
#13114876	2
#13114872	2
#13114743	2
C₄×#88062	2
C₁₃×#2921	2
#20828	13