Subgroups — #16807069

Order 3328.

598 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131203	C₂	1	2	yes	maximal, complemented
2	#13131201	C₂	1	2	yes	maximal, complemented
3	#13131196	C₂	1	2	yes	maximal, complemented
4	#13131194	C₂	1	2	yes	maximal, complemented
5	#13131172	C₂	1	2	yes	maximal, complemented
6	#13131067	C₂	1	2	yes	maximal
7	#13131043	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₁₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	#13123258	C₂	1	1	yes	maximal, characteristic
10	C₂×#88161	C₂²	1	1	yes	characteristic
11	C₂×#88155	C₄	1	1	yes	characteristic
12	#11719227	C₄	1	1	yes	characteristic
13	C₂×#88098	C₂²	1	2	yes	—
14	C₂×D₁₃⋊C₁₆	C₂²	1	2	yes	—
15	C₁₃×#405	C₄	1	1	yes	characteristic
16	M₁₆×Dic₁₃	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
18	#11718658	C₂²	1	1	yes	characteristic
19	#11718369	C₂²	1	1	yes	characteristic
20	#11718369	C₂²	1	1	yes	characteristic
21	#11718368	C₄	1	1	yes	characteristic
22	#11718368	C₄	1	1	yes	characteristic
23	#11718350	C₄	1	1	yes	characteristic
24	#11718350	C₄	1	1	yes	characteristic
25	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
26	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
27	C₂²⋊#6092	C₂²	1	4	yes	—
28	#11718260	C₂²	1	4	yes	—
29	#11718259	C₂²	1	4	yes	—
30	#11718257	C₂²	1	2	yes	—
31	#11718257	C₂²	1	2	yes	—
32	#11718231	C₂²	1	2	yes	—
33	C₄⋊#6092	C₂²	1	4	yes	—
34	C₄×#6092	C₂²	1	2	yes	—
35	C₂×#6127	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×#6099	C₂×C₄	1	1	yes	characteristic
41	C₂²×#1055	C₂³	1	1	yes	characteristic
42	#88161	—	2	4	no	—
43	#88155	—	2	4	no	—
44	C₂×#6094	C₂×C₄	1	1	yes	characteristic
45	C₂×#6094	C₂×C₄	1	1	yes	characteristic
46	C₂×#6094	C₂×C₄	1	1	yes	characteristic
47	C₂×#6094	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
52	#88147	C₈	1	2	yes	—
53	#88141	C₈	1	2	yes	—
54	#88135	C₈	1	2	yes	—
55	#88135	C₈	1	2	yes	—
56	#88126	C₂×C₄	1	1	yes	characteristic
57	#88126	C₂×C₄	1	1	yes	characteristic
58	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
59	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
60	C₂×#6092	C₂³	1	4	yes	—
61	C₂×#6092	C₂³	1	4	yes	—
62	#88098	—	2	4	no	—
63	D₁₃⋊C₁₆	—	2	4	no	—
64	C₁₃×#97	C₂×C₄	1	1	yes	characteristic
65	C₁₃×#97	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
68	#88055	C₂×C₄	1	1	yes	characteristic
69	#88055	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
72	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
73	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
74	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
75	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
76	#88043	C₂×C₄	1	1	yes	characteristic
77	#88043	C₂×C₄	1	1	yes	characteristic
78	C₄×#1055	C₂³	1	1	yes	characteristic
79	C₄×#1055	C₂³	1	1	yes	characteristic
80	#33420	—	13	1	no	maximal, Sylow
81	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₂²⋊D₁₃	C₂×C₈	1	1	yes	characteristic
83	C₂×C₂²⋊D₁₃	C₂×C₈	1	1	yes	characteristic
84	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
85	#6127	—	2	2	no	—
86	#6127	—	2	2	no	—
87	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
88	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
89	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
90	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
93	C₁₃×M₁₆	—	2	2	no	—
94	C₂×C₈×C₁₃	—	2	2	no	—
95	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
100	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
101	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
103	#6103	C₂×C₈	1	2	yes	—
104	#6103	C₂×C₈	1	2	yes	—
105	#6101	C₂×C₈	1	2	yes	—
106	#6101	C₂×C₈	1	2	yes	—
107	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
108	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
109	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
110	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
111	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
112	#6099	—	2	2	no	—
113	C₂×#1055	—	2	2	no	—
114	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
115	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
116	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
117	C₂×#1055	C₂⁴	1	1	yes	characteristic
118	#6094	—	2	2	no	—
119	#6094	—	2	2	no	—
120	#6094	—	2	2	no	—
121	#6094	—	2	2	no	—
122	C₈×D₁₃	—	2	2	no	—
123	C₈×D₁₃	—	2	2	no	—
124	C₈×D₁₃	M₁₆	1	4	yes	—
125	C₈×D₁₃	#41	1	4	yes	—
126	#6092	—	2	4	no	—
127	#6092	—	2	4	no	—
128	C₄×#131	—	13	1	no	—
129	#2164	—	13	2	no	—
130	#2164	—	13	2	no	—
131	#2162	—	13	2	no	—
132	#2162	—	13	2	no	—
133	#2151	—	13	2	no	—
134	#2113	—	13	2	no	—
135	C₄×M₃₂	—	13	1	no	—
136	C₂×C₄×C₁₆	—	13	1	no	—
137	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
138	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
139	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₄×C₁₃	—	2	1	no	—
141	C₂×C₄×C₁₃	—	2	1	no	—
142	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
144	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
145	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
146	C₂²⋊D₁₃	—	2	2	no	—
147	C₂²⋊D₁₃	—	2	2	no	—
148	C₂×Dic₁₃	—	2	4	no	—
149	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
150	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
151	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
152	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
153	D₅₂	—	2	2	no	—
154	C₄×D₁₃	—	2	1	no	—
155	C₄×D₁₃	—	2	1	no	—
156	C₄×D₁₃	—	2	4	no	—
157	C₄×D₁₃	—	2	4	no	—
158	C₄×D₁₃	#117	1	2	yes	—
159	C₄×D₁₃	C₂×M₁₆	1	2	yes	—
160	Dic₂₆	—	2	2	no	—
161	C₈×C₁₃	—	2	1	no	—
162	C₈×C₁₃	—	2	1	no	—
163	C₈×C₁₃	#117	1	2	yes	—
164	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
165	#1055	—	2	1	no	—
166	#1055	—	2	1	no	—
167	#1055	C₂×M₁₆	1	2	yes	—
168	#1055	C₂×#41	1	2	yes	—
169	C₂×#131	—	13	1	no	—
170	C₂×#131	—	13	1	no	—
171	C₄×#41	—	13	1	no	—
172	C₂×M₃₂	—	13	2	no	—
173	C₂²×C₁₆	—	13	2	no	—
174	#405	—	13	1	no	—
175	#405	—	13	1	no	—
176	#405	—	13	1	no	—
177	#405	—	13	1	no	—
178	#405	—	13	1	no	—
179	#405	#261	1	1	yes	characteristic
180	C₄×M₁₆	—	13	1	no	—
181	C₄×M₁₆	—	13	1	no	—
182	C₄×M₁₆	—	13	1	no	—
183	C₂×C₄×C₈	—	13	1	no	—
184	C₂×C₄×C₈	—	13	1	no	—
185	C₂×C₄×C₈	—	13	1	no	—
186	C₄⋊C₁₆	—	13	4	no	—
187	C₄⋊C₁₆	—	13	4	no	—
188	C₂²⋊C₁₆	—	13	4	no	—
189	C₂²⋊C₁₆	—	13	4	no	—
190	#346	—	13	2	no	—
191	C₄×C₁₆	—	13	2	no	—
192	C₄×C₁₆	—	13	2	no	—
193	C₄×C₁₆	—	13	2	no	—
194	C₂²×C₁₃	—	2	2	no	—
195	C₂²×C₁₃	—	2	1	no	—
196	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
197	C₂×D₁₃	—	2	2	no	—
198	C₂×D₁₃	—	2	2	no	—
199	C₂×D₁₃	—	2	1	no	—
200	C₂×D₁₃	#431	1	2	yes	—
201	C₄×C₁₃	—	2	2	no	—
202	C₄×C₁₃	—	2	2	no	—
203	C₄×C₁₃	—	2	1	no	—
204	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
205	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
206	Dic₁₃	—	2	2	no	—
207	Dic₁₃	—	2	1	no	—
208	Dic₁₃	#431	1	2	yes	—
209	Dic₁₃	#504	1	4	yes	—
210	C₂×#41	—	13	1	no	—
211	#131	—	26	4	no	—
212	#131	—	26	4	no	—
213	C₂×M₁₆	—	13	1	no	—
214	C₂×M₁₆	—	13	1	no	—
215	C₂×M₁₆	—	13	1	no	—
216	C₂×M₁₆	—	13	1	no	—
217	C₂×M₁₆	—	13	1	no	—
218	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
219	C₂²×C₈	—	13	1	no	—
220	C₂²×C₈	—	13	1	no	—
221	C₂²×C₈	—	13	1	no	—
222	C₂²×C₈	—	13	1	no	—
223	C₂²×C₈	—	13	1	no	—
224	C₂²×C₈	C₂×#261	1	1	yes	characteristic
225	C₄×Q₈	—	13	2	no	—
226	C₄×D₄	—	13	2	no	—
227	C₄×D₄	—	13	2	no	—
228	C₄×D₄	—	13	2	no	—
229	#117	—	13	1	no	—
230	#117	—	13	1	no	—
231	#117	C₂×#261	1	1	yes	characteristic
232	C₂×C₄²	—	13	1	no	—
233	C₂×C₄²	—	13	1	no	—
234	C₂×C₄²	—	13	1	no	—
235	M₃₂	—	26	4	no	—
236	C₂×C₁₆	—	13	4	no	—
237	C₂×C₁₆	—	13	4	no	—
238	C₂×C₁₆	—	26	4	no	—
239	#97	—	13	1	no	—
240	#97	—	13	1	no	—
241	#97	—	13	1	no	—
242	#97	—	13	1	no	—
243	#97	C₂×#261	1	1	yes	characteristic
244	#97	C₂×#261	1	1	yes	characteristic
245	C₄×C₈	—	13	1	no	—
246	C₄×C₈	—	13	1	no	—
247	C₄×C₈	—	13	1	no	—
248	C₄×C₈	—	13	1	no	—
249	C₄×C₈	—	13	1	no	—
250	C₄×C₈	—	13	1	no	—
251	C₄×C₈	—	13	1	no	—
252	C₄×C₈	—	13	1	no	—
253	C₄×C₈	C₂×#261	1	1	yes	characteristic
254	C₄×C₈	C₂×#261	1	1	yes	characteristic
255	C₂×C₁₃	—	4	1	no	—
256	C₂×C₁₃	C₂×#431	1	1	yes	characteristic
257	C₂×C₁₃	C₂×#504	1	2	yes	—
258	D₁₃	—	2	2	no	—
259	D₁₃	—	4	1	no	—
260	#41	—	26	2	no	—
261	#41	—	26	2	no	—
262	C₂×Q₈	—	13	1	no	—
263	C₂×D₄	—	13	1	no	—
264	C₂×D₄	—	13	1	no	—
265	C₂×D₄	—	13	1	no	—
266	C₂²×C₄	—	13	1	no	—
267	C₂²×C₄	—	13	2	no	—
268	C₂²×C₄	—	13	2	no	—
269	C₂²×C₄	—	13	2	no	—
270	C₂²×C₄	—	13	1	no	—
271	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
272	M₁₆	—	2	2	no	—
273	M₁₆	—	26	2	no	—
274	M₁₆	—	26	2	no	—
275	M₁₆	—	26	2	no	—
276	M₁₆	—	26	2	no	—
277	M₁₆	—	26	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	13	1	no	—
280	C₂×C₈	—	13	1	no	—
281	C₂×C₈	—	13	4	no	—
282	C₂×C₈	—	13	4	no	—
283	C₂×C₈	—	13	1	no	—
284	C₂×C₈	—	13	1	no	—
285	C₂×C₈	—	26	2	no	—
286	C₂×C₈	—	26	2	no	—
287	C₂×C₈	—	26	2	no	—
288	C₂×C₈	C₂²×#261	1	1	yes	characteristic
289	C₂×C₈	C₂²×#261	1	1	yes	characteristic
290	C₂×C₈	C₂²×#261	1	1	yes	characteristic
291	C₂×C₈	C₂²×#261	1	1	yes	characteristic
292	C₄⋊C₄	—	13	2	no	—
293	C₄⋊C₄	—	13	2	no	—
294	C₄⋊C₄	D₁₃⋊C₈	1	2	yes	—
295	C₂²⋊C₄	—	13	2	no	—
296	C₂²⋊C₄	—	13	2	no	—
297	C₂²⋊C₄	D₁₃⋊C₈	1	2	yes	—
298	C₄²	—	13	2	no	—
299	C₄²	—	13	2	no	—
300	C₄²	—	13	2	no	—
301	C₄²	—	13	1	no	—
302	C₄²	—	13	1	no	—
303	C₄²	C₂²×#261	1	1	yes	characteristic
304	C₄²	C₂²×#261	1	1	yes	characteristic
305	C₁₆	—	26	4	no	—
306	C₁₆	—	26	4	no	—
307	C₁₃	#33420	1	1	yes	characteristic, complemented, Sylow
308	C₂³	—	13	1	no	—
309	C₂³	—	13	1	no	—
310	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
311	Q₈	—	26	2	no	—
312	D₄	—	26	2	no	—
313	D₄	—	26	2	no	—
314	D₄	—	26	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	13	2	no	—
318	C₂×C₄	—	13	1	no	—
319	C₂×C₄	—	13	2	no	—
320	C₂×C₄	—	13	2	no	—
321	C₂×C₄	—	13	2	no	—
322	C₂×C₄	—	13	1	no	—
323	C₂×C₄	—	26	4	no	—
324	C₂×C₄	—	26	4	no	—
325	C₂×C₄	—	26	4	no	—
326	C₂×C₄	—	26	1	no	—
327	C₂×C₄	—	26	1	no	—
328	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
329	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
330	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
331	C₂×C₄	C₂³×#261	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	13	2	no	—
335	C₈	—	13	2	no	—
336	C₈	—	26	1	no	—
337	C₈	—	26	1	no	—
338	C₈	D₁₃⋊M₁₆	1	2	yes	—
339	C₈	#88237	1	2	yes	—
340	C₂²	—	2	2	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	13	2	no	—
343	C₂²	—	26	2	no	—
344	C₂²	—	26	2	no	—
345	C₂²	—	26	1	no	—
346	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
347	C₄	—	2	2	no	—
348	C₄	—	2	2	no	—
349	C₄	—	2	1	no	—
350	C₄	—	13	4	no	—
351	C₄	—	13	2	no	—
352	C₄	—	26	2	no	—
353	C₄	—	26	1	no	—
354	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
355	C₄	C₂×#88237	1	1	yes	characteristic
356	C₂	—	4	1	no	—
357	C₂	—	26	2	no	—
358	C₂	—	52	1	no	—
359	C₂	D₁₃⋊#431	1	1	yes	characteristic
360	C₂	D₁₃⋊#504	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₁₃×#405	C₄
prime radical	C₂×#6127	C₂×C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	#405	#261
hypocenter	C₁₃	#33420
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
1. #13131203	C₂	semidirect
2. #13131201	C₂	semidirect
3. #13131196	C₂	semidirect
4. #13131194	C₂	semidirect
5. #13131172	C₂	semidirect
7. #13131043	C₂	semidirect
8. C₄×D₁₃⋊C₁₆	C₂	semidirect
307. C₁₃	#33420	semidirect

Maximal subgroups

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

subgroup	index
#13131203	2
#13131201	2
#13131196	2
#13131194	2
#13131172	2
#13131067	2
#13131043	2
C₄×D₁₃⋊C₁₆	2
#13123258	2
#33420	13