Subgroups — #15863458

Order 3328.

982 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	#13117243	C₂	1	2	yes	maximal
2	#13114996	C₂	1	2	yes	maximal, complemented
3	#13114964	C₂	1	2	yes	maximal, complemented
4	#13114865	C₂	1	4	yes	maximal
5	C₄×#88058	C₂	1	4	yes	maximal, complemented
6	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88180	—	2	2	no	—
10	C₂×C₄×C₈×C₁₃	C₄	1	8	yes	complemented
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	#11718873	—	2	2	no	—
15	#11718873	C₂²	1	4	yes	—
16	C₂×#88058	C₂²	1	8	yes	—
17	#11718597	—	2	2	no	—
18	#11718596	C₂²	1	4	yes	—
19	C₄×C₄⋊Dic₁₃	—	2	2	no	—
20	C₄×C₄⋊Dic₁₃	C₂²	1	4	yes	—
21	#11718143	C₂²	1	8	yes	—
22	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	—	2	8	no	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
27	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
28	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
33	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
35	C₂×C₂²⋊Dic₁₃	—	2	4	no	—
36	#88180	—	4	8	no	—
37	C₂×C₄⋊Dic₁₃	—	2	4	no	—
38	C₂×C₄⋊Dic₁₃	—	2	4	no	—
39	C₂×C₄⋊Dic₁₃	C₂³	1	8	yes	—
40	C₂×C₄×Dic₁₃	—	2	4	no	—
41	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	—
43	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
44	#88071	—	2	4	no	—
45	#88071	—	2	4	no	—
46	#88058	—	2	16	no	—
47	#20817	—	13	1	no	maximal, Sylow
48	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	—	2	4	no	—
50	C₂²×C₄×C₁₃	—	2	4	no	—
51	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₁₃	D₈	1	4	yes	—
54	C₂²×C₄×C₁₃	Q₁₆	1	4	yes	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
63	C₂²×Dic₁₃	—	2	8	no	—
64	C₂×C₈×C₁₃	—	2	4	no	—
65	C₂×C₈×C₁₃	—	2	16	no	—
66	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
67	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
68	C₂×C₈×C₁₃	#41	1	8	yes	—
69	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
70	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
71	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
72	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
73	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
74	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
75	C₂²⋊Dic₁₃	—	4	8	no	—
76	C₄⋊Dic₁₃	—	2	16	no	—
77	C₄⋊Dic₁₃	—	4	8	no	—
78	C₄⋊Dic₁₃	—	4	8	no	—
79	C₄×Dic₁₃	—	4	8	no	—
80	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
81	#2285	—	13	2	no	—
82	#1845	—	13	2	no	—
83	#1834	—	13	2	no	—
84	#1808	—	13	4	no	—
85	C₄×#107	—	13	4	no	—
86	C₂³×C₁₃	—	2	2	no	—
87	C₂³×C₁₃	—	2	2	no	—
88	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
89	C₂³×C₁₃	C₂×D₈	1	2	yes	—
90	C₂³×C₁₃	C₂×Q₁₆	1	2	yes	—
91	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	4	no	—
95	C₂×C₄×C₁₃	—	2	4	no	—
96	C₂×C₄×C₁₃	—	2	8	no	—
97	C₂×C₄×C₁₃	#107	1	16	yes	complemented
98	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
105	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
106	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
107	C₂×C₄×C₁₃	#135	1	4	yes	—
108	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
111	C₂×Dic₁₃	—	4	8	no	—
112	C₂×Dic₁₃	—	4	8	no	—
113	C₂×Dic₁₃	—	4	8	no	—
114	C₈×C₁₃	—	2	4	no	—
115	C₈×C₁₃	#117	1	8	yes	complemented
116	C₂³×C₈	C₂×D₁₃	1	2	yes	—
117	C₂×#117	—	26	2	no	—
118	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
119	C₂×#107	—	13	8	no	—
120	C₂×C₄×C₈	Dic₁₃	1	8	yes	complemented
121	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
123	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
124	#384	—	13	4	no	—
125	#382	—	26	2	no	—
126	#380	—	13	4	no	—
127	#380	—	26	2	no	—
128	C₄×C₄⋊C₄	—	13	4	no	—
129	C₄×C₄⋊C₄	—	26	2	no	—
130	#340	—	13	8	no	—
131	C₂²×C₁₃	—	2	2	no	—
132	C₂²×C₁₃	—	2	4	no	—
133	C₂²×C₁₃	—	2	4	no	—
134	C₂²×C₁₃	—	2	2	no	—
135	C₂²×C₁₃	C₂×#107	1	4	yes	—
136	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₁₃	#532	1	2	yes	—
138	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
139	C₂²×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
140	C₂²×C₁₃	C₂×#135	1	2	yes	—
141	C₄×C₁₃	—	2	2	no	—
142	C₄×C₁₃	—	2	2	no	—
143	C₄×C₁₃	C₂×#107	1	4	yes	—
144	C₄×C₁₃	C₂×#107	1	4	yes	—
145	C₄×C₁₃	#427	1	4	yes	—
146	C₄×C₁₃	C₂×#117	1	2	yes	—
147	C₄×C₁₃	#533	1	2	yes	—
148	Dic₁₃	—	8	8	no	—
149	C₂³×C₄	D₅₂	1	1	yes	characteristic
150	C₂³×C₄	D₅₂	1	1	yes	characteristic
151	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
152	C₂²×C₈	—	2	8	no	—
153	C₂²×C₈	C₂×Dic₁₃	1	8	yes	—
154	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
155	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
156	#117	—	52	8	no	—
157	C₂×C₄⋊C₄	—	13	8	no	—
158	C₂×C₄⋊C₄	—	26	4	no	—
159	C₂×C₄⋊C₄	—	26	4	no	—
160	C₂×C₂²⋊C₄	—	26	4	no	—
161	C₂×C₄²	—	26	4	no	—
162	C₂×C₄²	Dic₂₆	1	4	yes	—
163	C₂×C₄²	D₅₂	1	1	yes	characteristic
164	C₂×C₄²	D₅₂	1	1	yes	characteristic
165	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
166	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
167	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
168	#107	—	26	16	no	—
169	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
170	C₄×C₈	C₂×Dic₁₃	1	8	yes	—
171	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
172	#95	—	26	4	no	—
173	#95	—	26	4	no	—
174	C₂×C₁₃	—	2	4	no	—
175	C₂×C₁₃	#2285	1	1	yes	characteristic
176	C₂×C₁₃	C₂²×#107	1	1	yes	characteristic
177	C₂×C₁₃	C₂×#427	1	1	yes	characteristic
178	C₂×C₁₃	#3169	1	2	yes	—
179	C₂×C₁₃	#3170	1	2	yes	—
180	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	26	8	no	—
184	C₂²×C₄	D₁₀₄	1	4	yes	—
185	C₂²×C₄	Dic₅₂	1	4	yes	—
186	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
187	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
188	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
189	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
190	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
194	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
195	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	2	16	no	—
198	C₂×C₈	#6127	1	8	yes	—
199	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
200	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
201	C₄⋊C₄	—	26	16	no	—
202	C₄⋊C₄	—	52	8	no	—
203	C₄⋊C₄	—	52	8	no	—
204	C₂²⋊C₄	—	52	8	no	—
205	C₄²	—	52	8	no	—
206	C₄²	C₄⋊Dic₁₃	1	4	yes	—
207	C₄²	C₄⋊Dic₁₃	1	4	yes	—
208	C₄²	C₂×Dic₂₆	1	2	yes	—
209	C₄²	C₂×Dic₂₆	1	2	yes	—
210	C₄²	C₂²×Dic₁₃	1	2	yes	—
211	C₄²	C₂²×Dic₁₃	1	2	yes	—
212	C₁₃	#20817	1	1	yes	characteristic, complemented, Sylow
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	C₂×D₁₀₄	1	2	yes	—
216	C₂³	C₂×Dic₅₂	1	2	yes	—
217	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
218	C₂³	C₂²×D₅₂	1	1	yes	characteristic
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	52	8	no	—
225	C₂×C₄	—	52	8	no	—
226	C₂×C₄	—	52	8	no	—
227	C₂×C₄	#88058	1	16	yes	complemented
228	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
229	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
230	C₂×C₄	#88158	1	4	yes	—
231	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
232	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
233	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
234	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
235	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
236	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
237	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
238	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
239	C₂×C₄	C₂×#6127	1	4	yes	—
240	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
241	C₈	—	2	4	no	—
242	C₈	#88180	1	8	yes	complemented
243	C₂²	—	2	4	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	C₂×#88058	1	4	yes	—
248	C₂²	#11719182	1	2	yes	—
249	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
250	C₂²	C₂×#88158	1	2	yes	—
251	C₂²	C₂²×Dic₅₂	1	1	yes	characteristic
252	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	104	8	no	—
256	C₄	C₂×#88058	1	4	yes	—
257	C₄	C₂×#88058	1	4	yes	—
258	C₄	#11718767	1	4	yes	—
259	C₄	#11719175	1	2	yes	—
260	C₄	C₂×#88180	1	2	yes	—
261	C₂	—	2	4	no	—
262	C₂	#13117243	1	1	yes	characteristic
263	C₂	C₂²×#88058	1	1	yes	characteristic
264	C₂	C₂×#11718767	1	1	yes	characteristic
265	C₂	#13125415	1	2	yes	—
266	C₂	#13125422	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	#20817
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13114996	C₂	semidirect
3. #13114964	C₂	semidirect
5. C₄×#88058	C₂	semidirect
10. C₂×C₄×C₈×C₁₃	C₄	semidirect
97. C₂×C₄×C₁₃	#107	semidirect
115. C₈×C₁₃	#117	semidirect
120. C₂×C₄×C₈	Dic₁₃	semidirect
212. C₁₃	#20817	semidirect
227. C₂×C₄	#88058	semidirect
242. C₈	#88180	semidirect

Maximal subgroups

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

subgroup	index
#13117243	2
#13114996	2
#13114964	2
#13114865	2
C₄×#88058	2
C₂²×C₄×C₈×C₁₃	2
#20817	13