Subgroups — #15866273

Order 3328.

998 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718767	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88055	C₂	1	1	yes	maximal, characteristic
3	#13117320	C₂	1	2	yes	maximal
4	#13115580	C₂	1	4	yes	maximal, complemented
5	#13115575	C₂	1	4	yes	maximal, complemented
6	#13114552	C₂	1	2	yes	maximal
7	C₂×C₁₃×#427	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#117	C₂²	1	2	yes	—
10	C₂×#88180	C₂²	1	2	yes	—
11	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂³×#1055	C₂²	1	1	yes	characteristic
13	C₁₃×#427	C₄	1	8	yes	complemented
14	C₂×C₁₃×#107	C₂²	1	2	yes	—
15	C₂×C₁₃×#106	C₂²	1	2	yes	—
16	#11718767	—	2	4	no	—
17	C₂×#88058	C₂²	1	2	yes	—
18	C₂×#88057	C₂²	1	2	yes	—
19	C₂×#88055	—	2	4	no	—
20	C₂×#88055	C₂²	1	2	yes	—
21	C₂×#88055	C₂²	1	2	yes	—
22	#11718646	C₂²	1	8	yes	—
23	#11718611	—	2	4	no	—
24	C₄×C₄⋊Dic₁₃	—	2	4	no	—
25	#11718121	C₂²	1	8	yes	—
26	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
27	C₂³×Dic₁₃	D₄	1	2	yes	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₁₃	D₄	1	4	yes	—
30	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₁₃×#117	—	2	4	no	—
34	C₁₃×#117	C₂×C₄	1	8	yes	—
35	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
36	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
37	C₂×C₄²×C₁₃	—	2	2	no	—
38	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
39	#88180	—	2	4	no	—
40	#88180	—	2	4	no	—
41	C₂×C₄⋊Dic₁₃	—	2	8	no	—
42	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
43	C₂×C₄×Dic₁₃	—	2	2	no	—
44	C₂×C₄×Dic₁₃	—	2	2	no	—
45	C₂×C₄×Dic₁₃	—	2	2	no	—
46	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
47	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
48	C₂²×#1055	—	2	2	no	—
49	C₂²×#1055	—	2	2	no	—
50	C₂²×#1055	Q₈	1	4	yes	—
51	C₂²×#1055	C₂³	1	2	yes	—
52	C₁₃×#107	C₂×C₄	1	8	yes	complemented
53	C₁₃×#106	C₂×C₄	1	8	yes	complemented
54	#88071	—	2	8	no	—
55	#88058	—	2	4	no	—
56	#88057	—	2	4	no	—
57	#88055	—	2	4	no	—
58	#88055	—	2	4	no	—
59	#21154	—	13	1	no	maximal, Sylow
60	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	—	2	4	no	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₁₃	#41	1	2	yes	—
68	C₂²×C₄×C₁₃	#41	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₁₃	—	2	4	no	—
71	C₂²×Dic₁₃	—	2	4	no	—
72	C₂²×Dic₁₃	—	4	4	no	—
73	C₂²×Dic₁₃	C₂×D₄	1	4	yes	—
74	C₂×C₈×C₁₃	C₄⋊C₄	1	16	yes	complemented
75	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
80	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
81	C₁₃×C₄⋊C₄	—	2	4	no	—
82	C₁₃×C₄⋊C₄	C₂²×C₄	1	8	yes	—
83	C₁₃×C₂²⋊C₄	—	2	8	no	—
84	C₄²×C₁₃	—	2	8	no	—
85	C₂²⋊Dic₁₃	—	4	4	no	—
86	C₄⋊Dic₁₃	—	2	4	no	—
87	C₄⋊Dic₁₃	—	2	4	no	—
88	C₄⋊Dic₁₃	—	4	16	no	—
89	C₄×Dic₁₃	—	2	4	no	—
90	C₄×Dic₁₃	—	2	2	no	—
91	C₄×Dic₁₃	—	2	2	no	—
92	C₄×Dic₁₃	—	4	4	no	—
93	C₂×#1055	—	2	8	no	—
94	C₂×#1055	—	2	2	no	—
95	C₂×#1055	—	2	2	no	—
96	C₂×#1055	C₂×Q₈	1	4	yes	—
97	C₂×#427	—	13	1	no	—
98	C₂×#427	D₁₃	1	1	yes	characteristic
99	C₂²×#97	—	13	1	no	—
100	#2296	—	13	2	no	—
101	#1944	—	13	4	no	—
102	#1943	—	13	4	no	—
103	#1735	—	13	2	no	—
104	C₂³×C₁₃	—	2	4	no	—
105	C₂³×C₁₃	C₄×D₄	1	2	yes	—
106	C₂³×C₁₃	#121	1	2	yes	—
107	C₂³×C₁₃	C₄⋊Q₈	1	2	yes	—
108	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	—	2	8	no	—
110	C₂×C₄×C₁₃	—	4	8	no	—
111	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
112	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
114	C₂×C₄×C₁₃	#121	1	2	yes	—
115	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
116	C₂×C₄×C₁₃	#125	1	4	yes	—
117	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
118	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
124	C₂×Dic₁₃	—	2	4	no	—
125	C₂×Dic₁₃	—	2	8	no	—
126	C₂×Dic₁₃	—	4	8	no	—
127	C₂×Dic₁₃	—	4	4	no	—
128	C₂×Dic₁₃	—	4	8	no	—
129	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
130	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
131	#1055	—	2	4	no	—
132	C₂³×C₈	—	13	1	no	—
133	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
134	C₂×#117	—	13	2	no	—
135	C₂×#117	C₂×D₁₃	1	2	yes	—
136	C₂²×C₄²	—	13	1	no	—
137	#427	—	26	4	no	—
138	#427	Dic₁₃	1	8	yes	complemented
139	C₂×#107	—	13	2	no	—
140	C₂×#107	C₂×D₁₃	1	2	yes	—
141	C₂×#106	—	13	2	no	—
142	C₂×#106	C₂×D₁₃	1	2	yes	—
143	C₂×#97	—	13	2	no	—
144	C₂×#97	—	13	2	no	—
145	C₂×#97	—	26	4	no	—
146	#389	—	13	8	no	—
147	#385	—	26	4	no	—
148	C₄×C₄⋊C₄	—	26	4	no	—
149	#340	—	13	8	no	—
150	C₂²×C₁₃	—	2	4	no	—
151	C₂²×C₁₃	—	2	4	no	—
152	C₂²×C₁₃	—	2	4	no	—
153	C₂²×C₁₃	#389	1	4	yes	—
154	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
155	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
156	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
157	C₂²×C₁₃	#575	1	4	yes	—
158	C₄×C₁₃	—	4	8	no	—
159	C₄×C₁₃	#389	1	4	yes	—
160	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
161	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
162	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
163	C₄×C₁₃	C₂×#125	1	1	yes	characteristic
164	Dic₁₃	—	4	4	no	—
165	Dic₁₃	—	8	4	no	—
166	C₂³×C₄	—	13	2	no	—
167	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
168	C₂²×C₈	—	13	4	no	—
169	C₂²×C₈	—	13	2	no	—
170	C₂²×C₈	—	26	2	no	—
171	C₂²×C₈	—	26	2	no	—
172	C₂²×C₈	Dic₂₆	1	4	yes	—
173	C₂²×C₈	D₅₂	1	4	yes	—
174	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
175	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
177	#117	—	2	4	no	—
178	#117	—	26	4	no	—
179	#117	—	26	4	no	—
180	#117	C₂×Dic₁₃	1	8	yes	—
181	C₂×C₄⋊C₄	—	13	4	no	—
182	C₂×C₄⋊C₄	—	26	8	no	—
183	C₂×C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
184	C₂×C₂²⋊C₄	—	2	2	no	—
185	C₂×C₂²⋊C₄	—	26	2	no	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	13	2	no	—
188	C₂×C₄²	—	13	2	no	—
189	C₂×C₄²	—	26	2	no	—
190	C₂×C₄²	—	26	2	no	—
191	C₂×C₄²	—	26	2	no	—
192	#107	—	26	4	no	—
193	#107	C₂×Dic₁₃	1	8	yes	complemented
194	#106	—	26	4	no	—
195	#106	C₂×Dic₁₃	1	8	yes	complemented
196	#97	—	26	4	no	—
197	#97	—	26	4	no	—
198	#95	—	26	8	no	—
199	C₂×C₁₃	—	2	4	no	—
200	C₂×C₁₃	C₂×#389	1	1	yes	characteristic
201	C₂×C₁₃	#2943	1	2	yes	—
202	C₂×C₁₃	#3054	1	2	yes	—
203	C₂×C₁₃	#3162	1	2	yes	—
204	C₂⁴	D₄×D₁₃	1	1	yes	characteristic
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	13	4	no	—
207	C₂²×C₄	—	26	4	no	—
208	C₂²×C₄	—	26	4	no	—
209	C₂²×C₄	—	52	4	no	—
210	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
211	C₂²×C₄	C₂×D₅₂	1	2	yes	—
212	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
213	C₂²×C₄	#6129	1	2	yes	—
214	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
215	C₂²×C₄	#6131	1	2	yes	—
216	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
217	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
218	C₂×C₈	—	13	4	no	—
219	C₂×C₈	—	26	8	no	—
220	C₂×C₈	—	26	2	no	—
221	C₂×C₈	—	26	2	no	—
222	C₂×C₈	C₄⋊Dic₁₃	1	16	yes	complemented
223	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
224	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
225	C₂×C₈	C₂×D₅₂	1	2	yes	—
226	C₂×C₈	C₂×D₅₂	1	2	yes	—
227	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
228	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	26	4	no	—
231	C₄⋊C₄	—	26	4	no	—
232	C₄⋊C₄	—	52	16	no	—
233	C₄⋊C₄	C₂²×Dic₁₃	1	8	yes	—
234	C₂²⋊C₄	—	2	8	no	—
235	C₂²⋊C₄	—	52	4	no	—
236	C₄²	—	2	8	no	—
237	C₄²	—	26	4	no	—
238	C₄²	—	26	2	no	—
239	C₄²	—	26	2	no	—
240	C₄²	—	52	4	no	—
241	C₁₃	#21154	1	1	yes	characteristic, complemented, Sylow
242	C₂³	—	2	4	no	—
243	C₂³	#88142	1	2	yes	—
244	C₂³	#88149	1	2	yes	—
245	C₂³	D₄×Dic₁₃	1	2	yes	—
246	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	4	8	no	—
249	C₂×C₄	—	26	4	no	—
250	C₂×C₄	—	26	8	no	—
251	C₂×C₄	—	52	8	no	—
252	C₂×C₄	—	52	4	no	—
253	C₂×C₄	—	52	8	no	—
254	C₂×C₄	#88142	1	2	yes	—
255	C₂×C₄	#88144	1	4	yes	—
256	C₂×C₄	#88149	1	2	yes	—
257	C₂×C₄	#88151	1	4	yes	—
258	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
259	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
260	C₂×C₄	Q₈×Dic₁₃	1	4	yes	—
261	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
262	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
263	C₂×C₄	C₂×#6129	1	1	yes	characteristic
264	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
265	C₂×C₄	C₂×#6131	1	1	yes	characteristic
266	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
267	C₈	—	26	4	no	—
268	C₈	C₂×C₄⋊Dic₁₃	1	4	yes	—
269	C₈	C₂×C₄⋊Dic₁₃	1	4	yes	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	2	4	no	—
273	C₂²	#11718646	1	4	yes	—
274	C₂²	C₂×#88142	1	1	yes	characteristic
275	C₂²	C₂×#88149	1	1	yes	characteristic
276	C₂²	#11719473	1	4	yes	—
277	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
278	C₄	—	4	8	no	—
279	C₄	—	52	4	no	—
280	C₄	—	104	4	no	—
281	C₄	#11718646	1	4	yes	—
282	C₄	C₂×#88144	1	1	yes	characteristic
283	C₄	C₂×#88151	1	1	yes	characteristic
284	C₄	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
285	C₄	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
286	C₂	—	2	4	no	—
287	C₂	C₂×#11718646	1	1	yes	characteristic
288	C₂	#13123637	1	2	yes	—
289	C₂	#13124446	1	2	yes	—
290	C₂	#13125359	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#427	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#427	D₁₃
hypocenter	C₁₃	#21154
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
4. #13115580	C₂	semidirect
5. #13115575	C₂	semidirect
13. C₁₃×#427	C₄	semidirect
52. C₁₃×#107	C₂×C₄	semidirect
53. C₁₃×#106	C₂×C₄	semidirect
74. C₂×C₈×C₁₃	C₄⋊C₄	semidirect
138. #427	Dic₁₃	semidirect
193. #107	C₂×Dic₁₃	semidirect
195. #106	C₂×Dic₁₃	semidirect
222. C₂×C₈	C₄⋊Dic₁₃	semidirect
241. C₁₃	#21154	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718767	2
C₂²×#88055	2
#13117320	2
#13115580	2
#13115575	2
#13114552	2
C₂×C₁₃×#427	2
#21154	13