Subgroups — #15762910

Order 3328.

476 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
3	#13112677	C₂	1	2	yes	maximal
4	#13112677	C₂	1	2	yes	maximal
5	C₄×C₁₃×#125	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#125	C₄	1	1	yes	characteristic
7	C₁₃×#389	C₄	1	1	yes	characteristic
8	C₁₃×#389	C₄	1	1	yes	characteristic
9	C₁₃×#387	—	2	1	no	—
10	C₁₃×#387	—	2	1	no	—
11	C₁₃×#382	C₄	1	1	yes	characteristic
12	C₄×C₁₃×C₄⋊C₄	—	2	1	no	—
13	C₄×C₁₃×C₄⋊C₄	—	2	1	no	—
14	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₄³×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊#1055	—	2	1	no	—
18	C₂×C₄⋊#1055	C₂²	1	2	yes	—
19	C₂×C₄×#1055	—	2	1	no	—
20	C₂×C₄×#1055	C₂²	1	2	yes	—
21	#11718116	—	2	1	no	—
22	#11718116	—	2	1	no	—
23	C₁₃×#125	—	2	2	no	—
24	C₁₃×#125	C₈	1	4	yes	complemented
25	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
26	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
27	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
28	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
29	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	—	2	1	no	—
34	C₂×C₄²×C₁₃	—	2	1	no	—
35	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×#1055	—	2	2	no	—
43	C₂²×#1055	—	2	2	no	—
44	C₁₃×#95	—	2	1	no	—
45	C₁₃×#95	—	2	1	no	—
46	C₁₃×#95	—	2	1	no	—
47	C₁₃×#95	—	2	1	no	—
48	C₁₃×#95	—	2	1	no	—
49	C₁₃×#95	—	2	1	no	—
50	C₄⋊#1055	—	2	4	no	—
51	C₄⋊#1055	—	2	2	no	—
52	C₄⋊#1055	Q₈	1	4	yes	—
53	C₄×#1055	—	2	4	no	—
54	C₄×#1055	—	4	2	no	—
55	#9866	—	13	1	no	maximal, Sylow
56	C₂²×C₄×C₁₃	—	2	1	no	—
57	C₂²×C₄×C₁₃	—	2	1	no	—
58	C₂²×C₄×C₁₃	—	2	1	no	—
59	C₂²×C₄×C₁₃	—	2	1	no	—
60	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₁₃×C₄⋊C₄	—	2	4	no	—
68	C₁₃×C₄⋊C₄	—	2	4	no	—
69	C₁₃×C₄⋊C₄	—	2	1	no	—
70	C₁₃×C₄⋊C₄	—	2	1	no	—
71	C₁₃×C₄⋊C₄	—	2	2	no	—
72	C₁₃×C₄⋊C₄	—	2	2	no	—
73	C₁₃×C₄⋊C₄	—	4	2	no	—
74	C₁₃×C₄⋊C₄	—	4	2	no	—
75	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
76	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
77	C₄²×C₁₃	—	2	2	no	—
78	C₄²×C₁₃	—	2	2	no	—
79	C₄²×C₁₃	—	2	2	no	—
80	C₄²×C₁₃	—	2	1	no	—
81	C₄²×C₁₃	—	2	1	no	—
82	C₄²×C₁₃	—	4	1	no	—
83	C₄²×C₁₃	—	4	2	no	—
84	C₄²×C₁₃	—	4	1	no	—
85	C₄²×C₁₃	—	4	1	no	—
86	C₄²×C₁₃	C₂×C₈	1	2	yes	—
87	C₄²×C₁₃	M₁₆	1	2	yes	—
88	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
89	C₄²×C₁₃	#41	1	2	yes	—
90	C₂×#1055	—	2	4	no	—
91	C₂×#1055	—	2	4	no	—
92	C₂×#1055	—	4	2	no	—
93	C₂×#1055	—	4	2	no	—
94	C₂×#1055	—	4	2	no	—
95	C₂×#1055	—	4	2	no	—
96	C₄×#125	D₁₃	1	1	yes	characteristic
97	C₄×C₄⋊C₈	—	13	1	no	—
98	C₄×C₄⋊C₈	—	13	1	no	—
99	#1279	—	13	2	no	—
100	#1279	—	13	2	no	—
101	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	2	1	no	—
108	C₂×C₄×C₁₃	—	2	1	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	4	1	no	—
116	C₂×C₄×C₁₃	—	4	1	no	—
117	C₂×C₄×C₁₃	—	4	1	no	—
118	C₂×C₄×C₁₃	—	4	1	no	—
119	C₂×C₄×C₁₃	—	4	1	no	—
120	C₂×C₄×C₁₃	—	4	1	no	—
121	C₂×C₄×C₁₃	—	4	1	no	—
122	C₂×C₄×C₁₃	—	4	1	no	—
123	C₂×C₄×C₁₃	—	4	1	no	—
124	C₂×C₄×C₁₃	—	4	1	no	—
125	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
126	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
127	C₂×C₄×C₁₃	#104	1	2	yes	—
128	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
129	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
130	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
134	C₂×C₄×C₁₃	#131	1	2	yes	—
135	C₂×C₄×C₁₃	#135	1	2	yes	—
136	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
138	#1055	—	4	4	no	—
139	#1055	—	4	4	no	—
140	C₂×#125	Dic₁₃	1	1	yes	characteristic
141	C₂×C₄⋊C₈	—	13	2	no	—
142	C₂×C₄⋊C₈	—	26	1	no	—
143	C₂×C₄×C₈	—	13	2	no	—
144	C₂×C₄×C₈	—	26	1	no	—
145	#389	Dic₁₃	1	1	yes	characteristic
146	#389	Dic₁₃	1	1	yes	characteristic
147	#387	—	2	1	no	—
148	#387	—	2	1	no	—
149	#382	Dic₁₃	1	1	yes	characteristic
150	C₄×C₄⋊C₄	—	2	1	no	—
151	C₄×C₄⋊C₄	—	2	1	no	—
152	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
153	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
154	C₄³	C₂×D₁₃	1	1	yes	characteristic
155	#336	—	26	1	no	—
156	#336	—	26	1	no	—
157	C₂²×C₁₃	#391	1	1	yes	characteristic
158	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
159	C₂²×C₁₃	#408	1	1	yes	characteristic
160	C₂²×C₁₃	#416	1	1	yes	characteristic
161	C₂²×C₁₃	#417	1	1	yes	characteristic
162	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
163	C₂²×C₁₃	#421	1	1	yes	characteristic
164	C₄×C₁₃	—	2	1	no	—
165	C₄×C₁₃	—	2	1	no	—
166	C₄×C₁₃	—	2	2	no	—
167	C₄×C₁₃	—	2	2	no	—
168	C₄×C₁₃	—	2	2	no	—
169	C₄×C₁₃	—	4	2	no	—
170	C₄×C₁₃	—	4	2	no	—
171	C₄×C₁₃	—	4	1	no	—
172	C₄×C₁₃	—	8	1	no	—
173	C₄×C₁₃	—	8	1	no	—
174	C₄×C₁₃	C₈×Q₈	1	2	yes	—
175	C₄×C₁₃	#446	1	2	yes	—
176	C₄×C₁₃	#478	1	2	yes	—
177	C₄×C₁₃	#479	1	2	yes	—
178	C₂²×C₈	—	26	2	no	—
179	C₂²×C₈	—	26	2	no	—
180	#125	—	2	2	no	—
181	#125	#1055	1	4	yes	complemented
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
187	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
188	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
189	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	—	2	1	no	—
192	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
193	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
194	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
195	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
196	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
197	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
198	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
199	C₄⋊C₈	—	13	4	no	—
200	C₄⋊C₈	—	26	4	no	—
201	C₄⋊C₈	—	26	2	no	—
202	C₄×C₈	—	26	4	no	—
203	C₄×C₈	—	52	2	no	—
204	#95	—	2	1	no	—
205	#95	—	2	1	no	—
206	#95	—	2	1	no	—
207	#95	—	2	1	no	—
208	#95	—	2	1	no	—
209	#95	—	2	1	no	—
210	C₂×C₁₃	#1476	1	1	yes	characteristic
211	C₂×C₁₃	#1477	1	1	yes	characteristic
212	C₂×C₁₃	#1482	1	1	yes	characteristic
213	C₂×C₁₃	#1483	1	1	yes	characteristic
214	C₂×C₁₃	#1984	1	1	yes	characteristic
215	C₂×C₁₃	#1986	1	1	yes	characteristic
216	C₂×C₁₃	#1989	1	1	yes	characteristic
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
222	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
223	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
224	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
225	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
228	C₂×C₈	—	26	4	no	—
229	C₂×C₈	—	26	4	no	—
230	C₂×C₈	—	52	2	no	—
231	C₂×C₈	—	52	2	no	—
232	C₂×C₈	—	52	2	no	—
233	C₂×C₈	—	52	2	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	C₂×#1055	1	2	yes	—
243	C₄⋊C₄	C₂×#1055	1	2	yes	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	4	1	no	—
250	C₄²	—	4	1	no	—
251	C₄²	—	4	2	no	—
252	C₄²	—	4	1	no	—
253	C₄²	C₂×#1055	1	2	yes	—
254	C₄²	#6099	1	2	yes	—
255	C₄²	Q₈×D₁₃	1	2	yes	—
256	C₄²	#6131	1	2	yes	—
257	C₁₃	#9866	1	1	yes	characteristic, complemented, Sylow
258	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	C₂²⋊#1055	1	2	yes	—
283	C₂×C₄	C₂²⋊#1055	1	2	yes	—
284	C₂×C₄	#88077	1	2	yes	—
285	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
286	C₂×C₄	C₂×#6099	1	1	yes	characteristic
287	C₂×C₄	#88198	1	1	yes	characteristic
288	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
289	C₂×C₄	#88200	1	2	yes	—
290	C₂×C₄	#88201	1	1	yes	characteristic
291	C₂×C₄	#88202	1	2	yes	—
292	C₂×C₄	#88203	1	1	yes	characteristic
293	C₂×C₄	#88204	1	2	yes	—
294	C₂×C₄	#88205	1	1	yes	characteristic
295	C₈	—	52	4	no	—
296	C₈	—	52	4	no	—
297	C₂²	C₂×C₂²⋊#1055	1	1	yes	characteristic
298	C₂²	#11718891	1	1	yes	characteristic
299	C₂²	#11718893	1	1	yes	characteristic
300	C₂²	#11718894	1	1	yes	characteristic
301	C₂²	#11718895	1	1	yes	characteristic
302	C₂²	C₂×#88077	1	1	yes	characteristic
303	C₂²	#11718897	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	4	2	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	1	no	—
312	C₄	—	8	1	no	—
313	C₄	—	8	1	no	—
314	C₄	Q₈×#1055	1	2	yes	—
315	C₄	#11718685	1	2	yes	—
316	C₄	#11718724	1	2	yes	—
317	C₄	#11718727	1	2	yes	—
318	C₂	#13113218	1	1	yes	characteristic
319	C₂	#13113223	1	1	yes	characteristic
320	C₂	#13113244	1	1	yes	characteristic
321	C₂	#13113249	1	1	yes	characteristic
322	C₂	#13115773	1	1	yes	characteristic
323	C₂	#13115781	1	1	yes	characteristic
324	C₂	#13115794	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×#125	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×#125	D₁₃
hypocenter	C₁₃	#9866
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
24. C₁₃×#125	C₈	semidirect
181. #125	#1055	semidirect
257. C₁₃	#9866	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#1055	2
#13112677	2
C₄×C₁₃×#125	2
#9866	13