Subgroups — #15779390

Order 3328.

420 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718055	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11718103	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116446	C₂	1	1	yes	maximal, characteristic
5	C₈×#6103	C₂	1	1	yes	maximal, characteristic
6	#13112917	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×#88060	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂×#88052	C₂²	1	1	yes	characteristic
15	C₂×#88052	C₂²	1	1	yes	characteristic
16	C₄×#6103	C₄	1	1	yes	characteristic
17	D₁₃×M₃₂	—	2	2	no	—
18	D₁₃×M₃₂	—	2	2	no	—
19	C₂×#88038	—	2	1	no	—
20	C₂×C₁₆×D₁₃	—	2	1	no	—
21	#11718142	C₄	1	1	yes	characteristic
22	#11718116	C₄	1	1	yes	characteristic
23	#11718103	—	2	2	no	—
24	#11718099	—	2	1	no	—
25	#11718099	—	2	1	no	—
26	#11718055	—	2	2	no	—
27	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×#6103	—	2	1	no	—
32	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
34	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₁₃	—	2	1	no	—
36	C₂×C₈×D₁₃	—	2	1	no	—
37	C₂×C₈×D₁₃	—	2	1	no	—
38	C₂×C₈×D₁₃	—	2	1	no	—
39	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
40	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
41	C₂×C₈×D₁₃	Q₈	1	1	yes	characteristic
42	C₂×C₈×D₁₃	Q₈	1	1	yes	characteristic
43	C₁₃×M₃₂	—	2	1	no	—
44	C₁₃×M₃₂	—	2	1	no	—
45	C₁₃×M₃₂	—	2	1	no	—
46	C₁₃×M₃₂	D₄	1	2	yes	—
47	C₂×C₁₃×C₁₆	—	2	1	no	—
48	C₂×C₁₃×C₁₆	—	2	1	no	—
49	C₄×C₈×C₁₃	—	2	1	no	—
50	C₄×C₈×C₁₃	—	2	1	no	—
51	#88071	—	2	1	no	—
52	#88060	C₈	1	2	yes	—
53	#88060	C₈	1	2	yes	—
54	C₈×Dic₁₃	—	2	1	no	—
55	C₈×Dic₁₃	—	2	1	no	—
56	#88052	—	2	1	no	—
57	#88052	—	2	1	no	—
58	#88052	—	2	1	no	—
59	#88052	D₄	1	2	yes	—
60	C₂×#6090	—	2	1	no	—
61	C₂×#6090	—	2	1	no	—
62	#88038	—	2	4	no	—
63	C₁₆×D₁₃	—	2	4	no	—
64	#12914	—	13	1	no	maximal, Sylow
65	C₂³×D₁₃	M₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	—	2	1	no	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×Dic₁₃	—	2	1	no	—
69	C₂²×Dic₁₃	M₁₆	1	1	yes	characteristic
70	C₂×C₄×D₁₃	—	2	1	no	—
71	C₂×C₄×D₁₃	—	2	1	no	—
72	C₂×C₄×D₁₃	—	2	1	no	—
73	C₂×C₄×D₁₃	—	2	1	no	—
74	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
77	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
78	C₂×C₈×C₁₃	—	2	1	no	—
79	C₂×C₈×C₁₃	—	2	1	no	—
80	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
81	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
86	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
87	C₄²×C₁₃	—	2	1	no	—
88	C₄²×C₁₃	—	2	1	no	—
89	#6103	—	2	4	no	—
90	C₄×Dic₁₃	—	2	1	no	—
91	C₄×Dic₁₃	—	2	1	no	—
92	C₂×#1055	—	2	1	no	—
93	C₂×#1055	—	2	1	no	—
94	C₂×#1055	—	2	1	no	—
95	C₂×#1055	—	2	1	no	—
96	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
97	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
98	C₂×#1055	C₂×C₈	1	2	yes	—
99	C₈×D₁₃	—	2	1	no	—
100	C₈×D₁₃	—	2	1	no	—
101	C₈×D₁₃	—	2	1	no	—
102	C₈×D₁₃	—	2	1	no	—
103	C₈×D₁₃	—	2	1	no	—
104	C₈×D₁₃	—	2	1	no	—
105	C₈×D₁₃	—	2	1	no	—
106	C₈×D₁₃	—	2	1	no	—
107	C₁₃×C₁₆	—	2	2	no	—
108	C₁₃×C₁₆	—	4	1	no	—
109	#6090	—	2	2	no	—
110	#6090	—	4	1	no	—
111	C₂²×M₃₂	—	13	1	no	—
112	C₂×#364	—	13	1	no	—
113	C₂×#364	—	13	1	no	—
114	#2112	—	13	1	no	—
115	C₈×C₂²⋊C₄	—	13	1	no	—
116	#1376	—	13	1	no	—
117	#1376	D₁₃	1	1	yes	characteristic, complemented
118	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
119	C₂²×D₁₃	—	2	1	no	—
120	C₂²×D₁₃	—	2	1	no	—
121	C₂²×D₁₃	—	2	2	no	—
122	C₂²×D₁₃	—	4	1	no	—
123	C₂²×D₁₃	C₄⋊C₈	1	1	yes	characteristic
124	C₂²×D₁₃	C₄⋊C₈	1	1	yes	characteristic
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	4	1	no	—
127	C₂×C₄×C₁₃	—	4	1	no	—
128	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
133	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
134	C₂×Dic₁₃	—	2	1	no	—
135	C₂×Dic₁₃	—	2	1	no	—
136	C₂×Dic₁₃	—	2	2	no	—
137	C₂×Dic₁₃	—	4	1	no	—
138	C₂×Dic₁₃	—	4	1	no	—
139	C₂×Dic₁₃	C₄⋊C₈	1	1	yes	characteristic
140	C₂×Dic₁₃	C₄⋊C₈	1	1	yes	characteristic
141	C₄×D₁₃	—	2	1	no	—
142	C₄×D₁₃	—	2	1	no	—
143	C₄×D₁₃	—	2	1	no	—
144	C₄×D₁₃	—	2	1	no	—
145	C₄×D₁₃	—	2	1	no	—
146	C₄×D₁₃	—	2	1	no	—
147	C₄×D₁₃	—	2	1	no	—
148	C₄×D₁₃	—	2	1	no	—
149	C₈×C₁₃	—	2	2	no	—
150	C₈×C₁₃	#120	1	1	yes	characteristic
151	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
152	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
153	C₈×C₁₃	#123	1	1	yes	characteristic
154	#1055	—	2	1	no	—
155	#1055	—	2	1	no	—
156	#1055	—	2	2	no	—
157	C₂³×C₈	—	13	1	no	—
158	C₂×M₃₂	—	13	1	no	—
159	C₂×M₃₂	—	13	1	no	—
160	C₂×M₃₂	—	26	1	no	—
161	C₂×M₃₂	—	26	2	no	—
162	C₂×M₃₂	—	26	2	no	—
163	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
164	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
165	C₂²×C₁₆	—	26	1	no	—
166	C₂×C₂²⋊C₈	—	13	1	no	—
167	C₂×C₄×C₈	—	13	1	no	—
168	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
169	C₄×C₂²⋊C₄	—	13	1	no	—
170	#364	—	26	2	no	—
171	#364	—	26	2	no	—
172	C₂²⋊C₁₆	—	26	1	no	—
173	C₂²⋊C₁₆	—	26	1	no	—
174	#336	—	13	1	no	—
175	#336	—	13	1	no	—
176	C₂²×C₁₃	—	2	1	no	—
177	C₂²×C₁₃	—	2	1	no	—
178	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
179	C₂²×C₁₃	#435	1	1	yes	characteristic
180	C₂²×C₁₃	#435	1	1	yes	characteristic
181	C₂×D₁₃	—	2	1	no	—
182	C₂×D₁₃	—	2	1	no	—
183	C₂×D₁₃	—	2	1	no	—
184	C₂×D₁₃	—	2	1	no	—
185	C₂×D₁₃	—	2	1	no	—
186	C₂×D₁₃	—	4	2	no	—
187	C₂×D₁₃	—	4	2	no	—
188	C₂×D₁₃	#364	1	2	yes	complemented
189	C₄×C₁₃	—	4	2	no	—
190	C₄×C₁₃	#385	1	1	yes	characteristic
191	C₄×C₁₃	#424	1	1	yes	characteristic
192	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
193	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
194	Dic₁₃	—	2	1	no	—
195	Dic₁₃	—	2	1	no	—
196	Dic₁₃	—	4	2	no	—
197	C₂³×C₄	—	13	1	no	—
198	C₂²×C₈	—	13	1	no	—
199	C₂²×C₈	—	13	1	no	—
200	C₂²×C₈	—	13	1	no	—
201	C₂²×C₈	—	13	1	no	—
202	C₂²×C₈	—	13	1	no	—
203	C₂²×C₈	—	13	1	no	—
204	C₂²×C₈	—	26	1	no	—
205	C₂²×C₈	—	26	1	no	—
206	C₂²×C₈	—	26	1	no	—
207	C₂²×C₈	—	26	1	no	—
208	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
209	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
210	C₂×C₂²⋊C₄	—	26	1	no	—
211	C₂×C₄²	—	13	1	no	—
212	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
213	M₃₂	—	2	1	no	—
214	M₃₂	—	2	1	no	—
215	M₃₂	—	2	1	no	—
216	M₃₂	—	13	2	no	—
217	M₃₂	—	26	4	no	—
218	M₃₂	—	26	1	no	—
219	M₃₂	—	26	1	no	—
220	M₃₂	—	26	1	no	—
221	M₃₂	D₅₂	1	2	yes	—
222	C₂×C₁₆	—	2	1	no	—
223	C₂×C₁₆	—	2	1	no	—
224	C₂×C₁₆	—	26	4	no	—
225	C₂×C₁₆	—	26	1	no	—
226	C₂×C₁₆	—	26	1	no	—
227	C₂²⋊C₈	—	13	2	no	—
228	C₂²⋊C₈	—	13	2	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	26	1	no	—
232	C₄×C₈	—	26	1	no	—
233	#95	—	26	1	no	—
234	C₂×C₁₃	—	2	1	no	—
235	C₂×C₁₃	—	2	1	no	—
236	C₂×C₁₃	#1850	1	1	yes	characteristic
237	C₂×C₁₃	C₂×#364	1	1	yes	characteristic
238	C₂×C₁₃	#2153	1	1	yes	characteristic
239	D₁₃	—	2	2	no	—
240	D₁₃	—	4	1	no	—
241	C₂⁴	—	13	1	no	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	13	1	no	—
244	C₂²×C₄	—	13	1	no	—
245	C₂²×C₄	—	13	1	no	—
246	C₂²×C₄	—	13	1	no	—
247	C₂²×C₄	—	13	1	no	—
248	C₂²×C₄	—	26	1	no	—
249	C₂²×C₄	—	26	1	no	—
250	C₂²×C₄	—	26	1	no	—
251	C₂²×C₄	—	26	1	no	—
252	C₂²×C₄	—	26	1	no	—
253	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	13	1	no	—
257	C₂×C₈	—	13	1	no	—
258	C₂×C₈	—	13	2	no	—
259	C₂×C₈	—	26	1	no	—
260	C₂×C₈	—	26	1	no	—
261	C₂×C₈	—	26	1	no	—
262	C₂×C₈	—	26	1	no	—
263	C₂×C₈	—	26	1	no	—
264	C₂×C₈	—	26	1	no	—
265	C₂×C₈	—	26	1	no	—
266	C₂×C₈	—	26	1	no	—
267	C₂×C₈	—	26	1	no	—
268	C₂×C₈	—	26	1	no	—
269	C₂×C₈	—	26	1	no	—
270	C₂×C₈	—	26	1	no	—
271	C₂×C₈	C₈×D₁₃	1	2	yes	—
272	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
273	C₂×C₈	#6127	1	1	yes	characteristic
274	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
275	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
276	C₂×C₈	#6129	1	1	yes	characteristic
277	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
278	C₂²⋊C₄	—	26	4	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	26	1	no	—
282	C₄²	—	26	1	no	—
283	C₁₆	—	2	2	no	—
284	C₁₆	—	4	1	no	—
285	C₁₆	—	26	2	no	—
286	C₁₆	—	52	1	no	—
287	C₁₃	#12914	1	1	yes	characteristic, complemented, Sylow
288	C₂³	—	13	1	no	—
289	C₂³	—	13	1	no	—
290	C₂³	—	26	2	no	—
291	C₂³	—	26	1	no	—
292	C₂³	—	26	1	no	—
293	C₂³	—	52	1	no	—
294	C₂³	M₁₆×D₁₃	1	1	yes	characteristic
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	13	1	no	—
299	C₂×C₄	—	13	1	no	—
300	C₂×C₄	—	26	1	no	—
301	C₂×C₄	—	26	1	no	—
302	C₂×C₄	—	26	1	no	—
303	C₂×C₄	—	26	1	no	—
304	C₂×C₄	—	26	1	no	—
305	C₂×C₄	—	26	1	no	—
306	C₂×C₄	—	26	1	no	—
307	C₂×C₄	—	26	1	no	—
308	C₂×C₄	—	26	1	no	—
309	C₂×C₄	—	26	1	no	—
310	C₂×C₄	—	26	2	no	—
311	C₂×C₄	—	52	1	no	—
312	C₂×C₄	—	52	1	no	—
313	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
314	C₂×C₄	#88135	1	1	yes	characteristic
315	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
316	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
317	C₂×C₄	#88155	1	1	yes	characteristic
318	C₂×C₄	#88161	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	26	1	no	—
321	C₈	—	26	1	no	—
322	C₈	—	26	2	no	—
323	C₈	#88136	1	1	yes	characteristic
324	C₈	#88137	1	1	yes	characteristic
325	C₈	#88150	1	1	yes	characteristic
326	C₈	#88151	1	1	yes	characteristic
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	13	2	no	—
330	C₂²	—	26	1	no	—
331	C₂²	—	26	1	no	—
332	C₂²	—	26	1	no	—
333	C₂²	—	26	1	no	—
334	C₂²	—	26	1	no	—
335	C₂²	—	52	2	no	—
336	C₂²	—	52	2	no	—
337	C₂²	#11718367	1	1	yes	characteristic
338	C₂²	#11718389	1	1	yes	characteristic
339	C₂²	D₁₃×C₄⋊C₈	1	1	yes	characteristic
340	C₄	—	4	2	no	—
341	C₄	—	26	1	no	—
342	C₄	—	26	1	no	—
343	C₄	—	52	2	no	—
344	C₄	#11718329	1	1	yes	characteristic
345	C₄	C₈×D₅₂	1	1	yes	characteristic
346	C₄	#11718386	1	1	yes	characteristic
347	C₄	#11718500	1	1	yes	characteristic
348	C₂	—	2	1	no	—
349	C₂	—	2	1	no	—
350	C₂	—	26	2	no	—
351	C₂	—	52	1	no	—
352	C₂	#13115015	1	1	yes	characteristic
353	C₂	D₁₃×#364	1	1	yes	characteristic
354	C₂	#13116600	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1376	C₂
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×D₁₃	M₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1376	D₁₃
hypocenter	C₁₃	#12914
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

Complements

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

N	H ≅ G/N	type
2. C₂×#11718055	C₂	semidirect
3. C₂×#11718103	C₂	semidirect
6. #13112917	C₂	semidirect
7. C₁₃×#1376	C₂	semidirect
117. #1376	D₁₃	semidirect
188. C₂×D₁₃	#364	semidirect
287. C₁₃	#12914	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₃×M₃₂	2
C₂×#11718055	2
C₂×#11718103	2
#13116446	2
C₈×#6103	2
#13112917	2
C₁₃×#1376	2
#12914	13