Subgroups — #15846213

Order 3328.

658 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125781	C₂	1	1	yes	maximal, characteristic, complemented
2	#13125774	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718355	C₂	1	1	yes	maximal, characteristic
4	#13114404	C₂	1	4	yes	maximal, complemented
5	#13114280	C₂	1	4	yes	maximal, complemented
6	#13113584	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113522	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113505	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#2960	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#533	C₂²	1	1	yes	characteristic
11	C₂×#88128	C₂²	1	1	yes	characteristic
12	C₂×#88123	C₂²	1	1	yes	characteristic
13	C₁₃×#446	—	2	1	no	—
14	C₁₃×#446	C₂²	1	2	yes	complemented
15	C₈×Q₈×C₁₃	—	2	1	no	—
16	C₈×Q₈×C₁₃	C₂²	1	2	yes	complemented
17	C₁₃×#424	—	2	1	no	—
18	C₁₃×#423	C₂²	1	1	yes	characteristic
19	C₁₃×#405	—	2	1	no	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	C₂×#88061	—	2	1	no	—
22	C₂×#88061	—	2	1	no	—
23	#11718754	—	2	1	no	—
24	#11718753	—	2	1	no	—
25	#11718749	—	2	1	no	—
26	#11718746	—	2	1	no	—
27	#11718711	—	2	2	no	—
28	#11718707	—	2	2	no	—
29	#11718692	C₂²	1	4	yes	complemented
30	#11718688	C₂²	1	4	yes	—
31	#11718586	C₂²	1	1	yes	characteristic
32	#11718586	C₂²	1	1	yes	characteristic
33	#11718355	—	2	1	no	—
34	#11718355	—	2	1	no	—
35	#11718355	C₂²	1	2	yes	—
36	#11718355	C₂²	1	2	yes	—
37	#11718078	C₂²	1	4	yes	complemented
38	#11718077	C₂²	1	4	yes	—
39	#11718050	C₂²	1	2	yes	complemented
40	#11718049	C₂²	1	2	yes	complemented
41	C₂²×D₅₂	—	2	1	no	—
42	C₂²×D₅₂	—	2	1	no	—
43	C₂²×D₅₂	—	2	1	no	—
44	C₂²×Dic₂₆	—	2	1	no	—
45	C₂×C₁₃×M₁₆	—	2	1	no	—
46	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
48	C₁₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
49	C₁₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
50	C₁₃×#125	—	2	1	no	—
51	C₁₃×C₂²⋊Q₈	—	2	1	no	—
52	C₁₃×C₂²⋊Q₈	D₄	1	2	yes	—
53	C₄×Q₈×C₁₃	—	2	1	no	—
54	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
55	C₁₃×#117	—	2	1	no	—
56	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
57	C₂×#6107	—	2	4	no	—
58	C₂×#6106	—	2	4	no	—
59	C₂×C₄⋊Dic₁₃	—	2	1	no	—
60	C₂×C₄⋊Dic₁₃	—	2	1	no	—
61	#88128	—	2	1	no	—
62	#88128	—	2	1	no	—
63	#88128	—	2	1	no	—
64	#88128	C₂³	1	2	yes	—
65	#88123	—	2	1	no	—
66	#88123	—	2	1	no	—
67	#88123	—	2	1	no	—
68	#88123	C₂³	1	2	yes	—
69	C₁₃×C₄⋊C₈	—	2	2	no	—
70	C₁₃×C₄⋊C₈	—	2	2	no	—
71	C₁₃×C₄⋊C₈	—	2	1	no	—
72	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
73	C₁₃×#97	—	2	2	no	—
74	C₄×C₈×C₁₃	—	2	2	no	—
75	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
76	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
79	C₂²⋊#1055	—	2	1	no	—
80	C₂²⋊#1055	—	2	1	no	—
81	#88061	—	2	2	no	—
82	#88061	—	2	2	no	—
83	#88061	—	2	2	no	—
84	#88061	—	2	2	no	—
85	#88050	—	2	4	no	—
86	#88049	—	2	4	no	—
87	C₄⋊#1055	C₂³	1	2	yes	—
88	C₄⋊#1055	C₂³	1	2	yes	—
89	C₄×#1055	—	2	1	no	—
90	C₄×#1055	—	2	1	no	—
91	#19214	—	13	1	no	maximal, Sylow
92	C₂³×D₁₃	—	4	1	no	—
93	C₂×Q₈×C₁₃	—	2	1	no	—
94	C₂×Q₈×C₁₃	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
98	C₂²×Dic₁₃	—	4	1	no	—
99	C₂×D₅₂	—	2	2	no	—
100	C₂×D₅₂	—	2	2	no	—
101	C₂×D₅₂	—	2	2	no	—
102	C₂×D₅₂	—	4	2	no	—
103	C₂×D₅₂	—	4	2	no	—
104	C₂×D₅₂	—	4	1	no	—
105	C₂×D₅₂	—	4	1	no	—
106	C₂×D₅₂	—	4	1	no	—
107	C₂×D₅₂	—	8	2	no	—
108	C₂×D₅₂	—	8	2	no	—
109	C₂×Dic₂₆	—	2	2	no	—
110	C₂×Dic₂₆	—	4	1	no	—
111	C₂×Dic₂₆	—	8	2	no	—
112	C₁₃×M₁₆	—	4	2	no	—
113	C₂×C₈×C₁₃	—	2	2	no	—
114	C₂×C₈×C₁₃	—	2	2	no	—
115	C₂×C₈×C₁₃	—	2	2	no	—
116	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
120	C₁₃×C₄⋊C₄	—	2	1	no	—
121	C₁₃×C₄⋊C₄	—	2	1	no	—
122	C₁₃×C₄⋊C₄	—	2	1	no	—
123	C₁₃×C₄⋊C₄	—	2	2	no	—
124	C₁₃×C₄⋊C₄	—	2	2	no	—
125	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
126	C₁₃×C₂²⋊C₄	—	2	2	no	—
127	C₄²×C₁₃	—	2	2	no	—
128	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₁₃	#41	1	1	yes	characteristic
131	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
132	#6107	—	8	4	no	—
133	#6106	—	8	4	no	—
134	C₄⋊Dic₁₃	—	2	2	no	—
135	C₄⋊Dic₁₃	—	2	2	no	—
136	C₄⋊Dic₁₃	—	4	1	no	—
137	C₄⋊Dic₁₃	—	4	1	no	—
138	C₂×#1055	—	2	2	no	—
139	C₂×#1055	—	2	2	no	—
140	#3209	—	13	1	no	—
141	#3208	—	13	1	no	—
142	C₂×#486	—	13	1	no	—
143	#2960	D₁₃	1	1	yes	characteristic, complemented
144	#1702	—	13	4	no	—
145	#1682	—	13	4	no	—
146	#1564	—	13	1	no	—
147	#1549	—	13	1	no	—
148	#1545	—	13	1	no	—
149	C₂³×C₁₃	#121	1	1	yes	characteristic
150	C₂²×D₁₃	—	4	2	no	—
151	C₂²×D₁₃	—	8	2	no	—
152	C₂²×D₁₃	—	8	2	no	—
153	C₂²×D₁₃	—	8	2	no	—
154	Q₈×C₁₃	—	2	4	no	—
155	Q₈×C₁₃	—	4	2	no	—
156	C₂×C₄×C₁₃	—	2	2	no	—
157	C₂×C₄×C₁₃	—	2	2	no	—
158	C₂×C₄×C₁₃	—	2	2	no	—
159	C₂×C₄×C₁₃	—	2	2	no	—
160	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
161	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
162	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
163	C₂×C₄×C₁₃	#135	1	2	yes	—
164	C₂×C₄×C₁₃	#137	1	2	yes	—
165	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
168	C₂×Dic₁₃	—	4	2	no	—
169	C₂×Dic₁₃	—	8	2	no	—
170	D₅₂	—	4	2	no	—
171	D₅₂	—	4	2	no	—
172	D₅₂	—	4	4	no	—
173	D₅₂	—	8	2	no	—
174	D₅₂	—	8	2	no	—
175	D₅₂	—	8	2	no	—
176	D₅₂	—	8	2	no	—
177	Dic₂₆	—	4	4	no	—
178	Dic₂₆	—	8	2	no	—
179	C₈×C₁₃	—	2	1	no	—
180	C₈×C₁₃	—	2	1	no	—
181	C₈×C₁₃	—	2	1	no	—
182	C₈×C₁₃	—	2	1	no	—
183	C₈×C₁₃	—	4	2	no	—
184	#1055	—	4	2	no	—
185	#1055	—	4	2	no	—
186	#533	C₂×D₁₃	1	1	yes	characteristic
187	C₂×C₄⋊Q₈	—	13	1	no	—
188	C₂×#127	—	13	1	no	—
189	#494	—	26	1	no	—
190	#492	—	26	1	no	—
191	#487	—	26	1	no	—
192	#486	—	13	2	no	—
193	#486	—	13	2	no	—
194	#486	—	26	1	no	—
195	#486	—	26	1	no	—
196	#486	—	26	1	no	—
197	#462	—	13	4	no	—
198	#460	—	13	4	no	—
199	#451	—	26	2	no	—
200	#450	—	26	2	no	—
201	#446	—	2	1	no	—
202	#446	C₂×D₁₃	1	2	yes	complemented
203	C₈×Q₈	—	2	1	no	—
204	C₈×Q₈	C₂×D₁₃	1	2	yes	complemented
205	#431	—	13	1	no	—
206	#431	—	13	1	no	—
207	#424	—	2	1	no	—
208	#423	C₂×D₁₃	1	1	yes	characteristic
209	C₂×#102	—	26	1	no	—
210	C₂×#102	—	26	1	no	—
211	#405	—	2	1	no	—
212	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
213	#335	—	13	2	no	—
214	#334	—	13	2	no	—
215	#332	—	13	4	no	—
216	#331	—	13	4	no	—
217	C₂²×C₁₃	—	2	1	no	—
218	C₂²×C₁₃	—	2	1	no	—
219	C₂²×C₁₃	#464	1	2	yes	—
220	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
221	C₂×D₁₃	—	8	2	no	—
222	C₂×D₁₃	—	8	2	no	—
223	C₂×D₁₃	—	8	2	no	—
224	C₂×D₁₃	—	16	4	no	—
225	C₄×C₁₃	—	2	1	no	—
226	C₄×C₁₃	—	2	1	no	—
227	C₄×C₁₃	—	4	2	no	—
228	C₄×C₁₃	—	4	2	no	—
229	C₄×C₁₃	#464	1	2	yes	—
230	C₄×C₁₃	#471	1	2	yes	—
231	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
232	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
233	C₄×C₁₃	#576	1	2	yes	—
234	Dic₁₃	—	8	2	no	—
235	C₂²×Q₈	—	26	1	no	—
236	C₂²×D₄	—	26	1	no	—
237	C₂²×D₄	—	26	1	no	—
238	C₂²×D₄	—	26	1	no	—
239	C₂×Q₁₆	—	26	4	no	—
240	C₂×SD₁₆	—	26	4	no	—
241	C₂×M₁₆	—	2	1	no	—
242	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
243	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
244	C₄⋊Q₈	—	13	2	no	—
245	C₄⋊Q₈	—	26	1	no	—
246	C₄⋊Q₈	—	26	1	no	—
247	C₄⋊Q₈	—	26	1	no	—
248	C₄⋊Q₈	D₅₂	1	1	yes	characteristic
249	C₄⋊Q₈	D₅₂	1	1	yes	characteristic
250	#127	—	13	2	no	—
251	#127	—	26	1	no	—
252	#127	—	26	1	no	—
253	#127	—	26	1	no	—
254	#125	—	2	1	no	—
255	C₂²⋊Q₈	—	2	1	no	—
256	C₂²⋊Q₈	D₅₂	1	2	yes	—
257	C₄×Q₈	—	2	1	no	—
258	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
259	#117	—	2	1	no	—
260	C₂×C₄⋊C₄	—	26	1	no	—
261	C₂×C₄⋊C₄	—	26	1	no	—
262	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
263	C₄⋊C₈	—	2	2	no	—
264	C₄⋊C₈	—	2	2	no	—
265	C₄⋊C₈	—	2	1	no	—
266	C₄⋊C₈	—	13	2	no	—
267	C₄⋊C₈	—	13	2	no	—
268	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
269	#103	—	26	4	no	—
270	#102	—	26	2	no	—
271	#102	—	26	2	no	—
272	#102	—	26	2	no	—
273	#102	—	26	2	no	—
274	#102	—	26	4	no	—
275	C₂²⋊C₈	—	26	1	no	—
276	C₂²⋊C₈	—	26	1	no	—
277	#97	—	2	2	no	—
278	C₄×C₈	—	2	2	no	—
279	C₄×C₈	—	26	1	no	—
280	C₄×C₈	—	26	1	no	—
281	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
282	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
283	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
285	C₂×C₁₃	—	2	2	no	—
286	C₂×C₁₃	C₂×#464	1	1	yes	characteristic
287	C₂×C₁₃	#3058	1	1	yes	characteristic
288	C₂×C₁₃	C₂×#471	1	1	yes	characteristic
289	D₁₃	—	16	2	no	—
290	C₂⁴	—	52	1	no	—
291	C₂×Q₈	—	2	1	no	—
292	C₂×Q₈	—	26	2	no	—
293	C₂×Q₈	—	52	1	no	—
294	C₂×Q₈	—	104	2	no	—
295	C₂×Q₈	C₂×D₅₂	1	2	yes	—
296	C₂×D₄	—	26	2	no	—
297	C₂×D₄	—	26	2	no	—
298	C₂×D₄	—	26	2	no	—
299	C₂×D₄	—	52	1	no	—
300	C₂×D₄	—	52	1	no	—
301	C₂×D₄	—	52	1	no	—
302	C₂×D₄	—	52	2	no	—
303	C₂×D₄	—	52	2	no	—
304	C₂×D₄	—	104	2	no	—
305	C₂×D₄	—	104	2	no	—
306	C₂²×C₄	—	52	1	no	—
307	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
308	C₂²×C₄	#6127	1	1	yes	characteristic
309	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
310	Q₁₆	—	104	4	no	—
311	SD₁₆	—	104	4	no	—
312	M₁₆	—	4	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	26	2	no	—
317	C₂×C₈	—	26	2	no	—
318	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
319	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
320	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
321	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	—	2	1	no	—
324	C₄⋊C₄	—	2	1	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	26	2	no	—
328	C₄⋊C₄	—	26	2	no	—
329	C₄⋊C₄	—	52	1	no	—
330	C₄⋊C₄	—	52	1	no	—
331	C₄⋊C₄	C₂×D₅₂	1	2	yes	—
332	C₂²⋊C₄	—	2	2	no	—
333	C₄²	—	2	2	no	—
334	C₄²	C₂×D₅₂	1	1	yes	characteristic
335	C₄²	#6127	1	1	yes	characteristic
336	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
337	C₄²	C₂³×D₁₃	1	1	yes	characteristic
338	C₁₃	#19214	1	1	yes	characteristic, complemented, Sylow
339	C₂³	—	52	2	no	—
340	C₂³	—	104	2	no	—
341	C₂³	—	104	2	no	—
342	C₂³	—	104	2	no	—
343	C₂³	#88184	1	1	yes	characteristic
344	Q₈	—	2	4	no	—
345	Q₈	—	4	2	no	—
346	Q₈	—	52	4	no	—
347	Q₈	—	104	2	no	—
348	D₄	—	52	4	no	—
349	D₄	—	52	2	no	—
350	D₄	—	52	2	no	—
351	D₄	—	104	2	no	—
352	D₄	—	104	2	no	—
353	D₄	—	104	2	no	—
354	D₄	—	104	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	52	2	no	—
360	C₂×C₄	—	104	2	no	—
361	C₂×C₄	#88184	1	1	yes	characteristic
362	C₂×C₄	#88184	1	1	yes	characteristic
363	C₂×C₄	#88184	1	1	yes	characteristic
364	C₂×C₄	#88196	1	2	yes	—
365	C₂×C₄	#88204	1	2	yes	—
366	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
367	C₂×C₄	C₂×#6127	1	1	yes	characteristic
368	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	—	4	2	no	—
374	C₈	—	52	2	no	—
375	C₈	—	52	2	no	—
376	C₂²	—	2	1	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	104	2	no	—
379	C₂²	—	104	2	no	—
380	C₂²	—	104	2	no	—
381	C₂²	—	208	4	no	—
382	C₂²	#11718689	1	2	yes	—
383	C₂²	C₂×#88184	1	1	yes	characteristic
384	C₄	—	2	1	no	—
385	C₄	—	2	1	no	—
386	C₄	—	4	2	no	—
387	C₄	—	4	2	no	—
388	C₄	—	104	2	no	—
389	C₄	#11718689	1	2	yes	—
390	C₄	#11718802	1	2	yes	—
391	C₄	#11719457	1	2	yes	—
392	C₄	C₂×#88196	1	1	yes	characteristic
393	C₄	C₂×#88204	1	1	yes	characteristic
394	C₂	—	2	2	no	—
395	C₂	—	208	2	no	—
396	C₂	C₂×#11718689	1	1	yes	characteristic
397	C₂	#13124484	1	1	yes	characteristic
398	C₂	C₂×#11718802	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88184
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#2960	C₂
socle	C₂²×C₁₃	C₂×#121
prime radical	C₂×#88128	C₂²
hypercenter	#2960	D₁₃
hypocenter	C₁₃	#19214
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
1. #13125781	C₂	semidirect
4. #13114404	C₂	semidirect
5. #13114280	C₂	semidirect
6. #13113584	C₂	semidirect
7. #13113522	C₂	semidirect
9. C₁₃×#2960	C₂	semidirect
14. C₁₃×#446	C₂²	semidirect
16. C₈×Q₈×C₁₃	C₂²	semidirect
29. #11718692	C₂²	semidirect
37. #11718078	C₂²	semidirect
39. #11718050	C₂²	semidirect
40. #11718049	C₂²	semidirect
143. #2960	D₁₃	semidirect
202. #446	C₂×D₁₃	semidirect
204. C₈×Q₈	C₂×D₁₃	semidirect
338. C₁₃	#19214	semidirect

Maximal subgroups

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

subgroup	index
#13125781	2
#13125774	2
C₂×#11718355	2
#13114404	2
#13114280	2
#13113584	2
#13113522	2
#13113505	2
C₁₃×#2960	2
#19214	13