Subgroups — #15833771

Order 3328.

584 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125396	C₂	1	1	yes	maximal, characteristic
2	#13123911	C₂	1	1	yes	maximal, characteristic
3	#13113714	C₂	1	2	yes	maximal, complemented
4	#13113691	C₂	1	2	yes	maximal, complemented
5	#13113679	C₂	1	2	yes	maximal, complemented
6	#13113670	C₂	1	2	yes	maximal, complemented
7	#13113597	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113309	C₂	1	1	yes	maximal, characteristic, complemented
9	#13113304	C₂	1	1	yes	maximal, characteristic
10	#13113129	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2985	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#533	C₂²	1	1	yes	characteristic
13	#11719182	C₂²	1	1	yes	characteristic
14	C₁₃×#446	—	2	2	no	—
15	C₁₃×#446	C₂²	1	2	yes	—
16	C₈×Q₈×C₁₃	C₂²	1	2	yes	—
17	C₁₃×#433	—	2	2	no	—
18	C₁₃×#432	C₂²	1	1	yes	characteristic
19	C₁₃×#431	C₂²	1	1	yes	characteristic
20	#11718749	C₄	1	1	yes	characteristic
21	#11718746	C₄	1	1	yes	characteristic
22	#11718742	C₄	1	1	yes	characteristic
23	#11718740	C₄	1	1	yes	characteristic
24	#11718709	C₄	1	2	yes	—
25	#11718709	C₄	1	2	yes	—
26	C₄×#6106	C₂²	1	2	yes	—
27	C₄×#6106	C₂²	1	2	yes	—
28	#11718587	C₂²	1	1	yes	characteristic
29	#11718586	C₂²	1	1	yes	characteristic
30	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
31	#11718497	—	2	2	no	—
32	#11718388	—	2	2	no	—
33	#11718349	C₂²	1	2	yes	—
34	C₈×D₅₂	C₂²	1	2	yes	—
35	Q₈⋊#1055	C₂²	1	2	yes	—
36	Q₈⋊#1055	C₂²	1	2	yes	—
37	#11718075	C₂²	1	2	yes	—
38	#11718075	C₂²	1	2	yes	—
39	#11718072	C₂²	1	2	yes	—
40	#11718071	C₂²	1	2	yes	—
41	#11718047	C₂²	1	1	yes	characteristic
42	#11718047	C₂²	1	1	yes	characteristic
43	#11718046	C₂²	1	1	yes	characteristic
44	#11718046	C₂²	1	1	yes	characteristic
45	C₂×#6127	—	2	1	no	—
46	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
47	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
48	C₁₃×#125	—	2	1	no	—
49	C₁₃×C₂²⋊Q₈	—	2	1	no	—
50	C₁₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
51	C₄×Q₈×C₁₃	—	2	1	no	—
52	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
53	C₁₃×#117	—	2	1	no	—
54	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
55	C₂×#6106	C₂×C₄	1	2	yes	—
56	C₂×#6106	C₂×C₄	1	2	yes	—
57	#88184	—	2	1	no	—
58	#88184	C₂×C₄	1	2	yes	—
59	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
60	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
61	C₂×#6094	—	2	2	no	—
62	C₂×C₈×D₁₃	—	2	2	no	—
63	#88129	—	2	1	no	—
64	#88128	C₂×C₄	1	1	yes	characteristic
65	C₄×D₅₂	—	2	1	no	—
66	C₄×D₅₂	C₂³	1	2	yes	—
67	#88123	C₂×C₄	1	1	yes	characteristic
68	C₁₃×C₄⋊C₈	—	2	2	no	—
69	C₁₃×C₄⋊C₈	—	2	2	no	—
70	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
71	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
72	C₁₃×C₂²⋊C₈	—	2	1	no	—
73	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
74	C₁₃×#97	—	2	2	no	—
75	C₁₃×#97	—	2	1	no	—
76	C₄×C₈×C₁₃	—	2	2	no	—
77	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
79	#88075	C₂×C₄	1	2	yes	—
80	#88075	C₂×C₄	1	2	yes	—
81	C₂²⋊#1055	—	2	1	no	—
82	C₂²⋊#1055	—	2	1	no	—
83	#88060	—	2	2	no	—
84	#88060	—	2	2	no	—
85	#88049	C₂×C₄	1	2	yes	—
86	#88049	C₂×C₄	1	2	yes	—
87	#88048	C₂×C₄	1	2	yes	—
88	#88048	C₂×C₄	1	2	yes	—
89	C₄⋊#1055	—	2	1	no	—
90	C₄⋊#1055	C₂³	1	2	yes	—
91	#88043	—	2	1	no	—
92	C₄×#1055	C₂×C₄	1	1	yes	characteristic
93	C₄×#1055	C₂×C₄	1	1	yes	characteristic
94	C₄×#1055	C₂³	1	1	yes	characteristic
95	C₄×#1055	C₂³	1	1	yes	characteristic
96	C₄×#1055	C₂³	1	1	yes	characteristic
97	C₄×#1055	C₂³	1	1	yes	characteristic
98	#17990	—	13	1	no	maximal, Sylow
99	C₂×Q₈×C₁₃	—	2	1	no	—
100	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
104	C₂×C₂²⋊D₁₃	—	2	2	no	—
105	#6127	—	8	2	no	—
106	C₂×D₅₂	—	2	1	no	—
107	C₂×D₅₂	—	2	1	no	—
108	C₂×D₅₂	C₂²×C₄	1	2	yes	—
109	C₂×C₄×D₁₃	—	2	2	no	—
110	C₂×Dic₂₆	—	2	1	no	—
111	C₂×C₈×C₁₃	—	2	2	no	—
112	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
114	C₁₃×C₄⋊C₄	—	2	1	no	—
115	C₁₃×C₄⋊C₄	—	2	1	no	—
116	C₁₃×C₄⋊C₄	—	2	1	no	—
117	C₁₃×C₄⋊C₄	—	2	2	no	—
118	C₁₃×C₄⋊C₄	—	2	2	no	—
119	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
120	C₁₃×C₂²⋊C₄	—	2	2	no	—
121	C₄²×C₁₃	—	2	2	no	—
122	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₁₃	#41	1	1	yes	characteristic
125	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
126	#6106	—	4	2	no	—
127	#6106	—	4	2	no	—
128	#6103	—	2	2	no	—
129	#6103	—	2	2	no	—
130	C₄⋊Dic₁₃	—	2	1	no	—
131	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
132	C₂×#1055	—	2	2	no	—
133	C₂×#1055	—	2	1	no	—
134	C₂×#1055	—	2	1	no	—
135	C₂×#1055	—	4	1	no	—
136	C₂×#1055	—	4	1	no	—
137	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
138	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
139	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
140	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
141	#6094	—	8	2	no	—
142	C₈×D₁₃	—	8	2	no	—
143	#3167	—	13	1	no	—
144	#2985	D₁₃	1	1	yes	characteristic, complemented
145	#2981	—	13	1	no	—
146	#1588	—	13	2	no	—
147	#1585	—	13	2	no	—
148	#1583	—	13	2	no	—
149	#1582	—	13	2	no	—
150	#1568	—	13	1	no	—
151	#1497	—	13	1	no	—
152	#1496	—	13	1	no	—
153	#1452	—	13	1	no	—
154	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
155	C₂²×D₁₃	—	2	2	no	—
156	Q₈×C₁₃	—	2	2	no	—
157	Q₈×C₁₃	—	2	2	no	—
158	Q₈×C₁₃	—	8	1	no	—
159	C₂×C₄×C₁₃	—	2	1	no	—
160	C₂×C₄×C₁₃	—	2	1	no	—
161	C₂×C₄×C₁₃	—	2	2	no	—
162	C₂×C₄×C₁₃	—	2	2	no	—
163	C₂×C₄×C₁₃	—	2	2	no	—
164	C₂×C₄×C₁₃	—	4	1	no	—
165	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
166	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
167	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
168	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
169	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
171	C₂²⋊D₁₃	—	8	2	no	—
172	C₂×Dic₁₃	—	2	2	no	—
173	D₅₂	—	2	2	no	—
174	D₅₂	—	2	2	no	—
175	D₅₂	—	8	1	no	—
176	D₅₂	—	8	1	no	—
177	C₄×D₁₃	—	4	2	no	—
178	C₄×D₁₃	—	4	2	no	—
179	Dic₂₆	—	8	1	no	—
180	C₈×C₁₃	—	2	2	no	—
181	C₈×C₁₃	—	2	2	no	—
182	C₈×C₁₃	—	4	2	no	—
183	#1055	—	2	1	no	—
184	#1055	—	2	1	no	—
185	#1055	—	2	1	no	—
186	#1055	—	2	1	no	—
187	#1055	—	4	1	no	—
188	#1055	—	4	1	no	—
189	#1055	—	4	2	no	—
190	#533	C₂×D₁₃	1	1	yes	characteristic
191	#532	—	13	1	no	—
192	#498	—	13	1	no	—
193	#492	—	13	1	no	—
194	#487	—	13	1	no	—
195	#486	—	13	1	no	—
196	#465	—	13	2	no	—
197	#465	—	13	2	no	—
198	#446	—	2	2	no	—
199	#446	C₂×D₁₃	1	2	yes	—
200	C₈×Q₈	C₂×D₁₃	1	2	yes	—
201	C₄×SD₁₆	—	13	2	no	—
202	C₄×SD₁₆	—	13	2	no	—
203	#436	—	13	2	no	—
204	#435	—	26	2	no	—
205	C₈×D₄	—	13	2	no	—
206	#433	—	2	2	no	—
207	#432	—	13	1	no	—
208	#432	C₂×D₁₃	1	1	yes	characteristic
209	#431	—	13	1	no	—
210	#431	C₂×D₁₃	1	1	yes	characteristic
211	#408	—	26	2	no	—
212	C₂×C₄×C₈	—	13	1	no	—
213	#335	—	13	2	no	—
214	#334	—	13	2	no	—
215	Q₈⋊C₈	—	13	2	no	—
216	Q₈⋊C₈	—	13	2	no	—
217	D₄⋊C₈	—	13	2	no	—
218	D₄⋊C₈	—	13	2	no	—
219	#322	—	13	1	no	—
220	#322	—	13	1	no	—
221	#322	—	13	1	no	—
222	#322	—	13	1	no	—
223	C₂²×C₁₃	—	2	1	no	—
224	C₂²×C₁₃	—	2	1	no	—
225	C₂²×C₁₃	—	2	1	no	—
226	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
227	C₂×D₁₃	—	4	2	no	—
228	C₂×D₁₃	—	4	2	no	—
229	C₂×D₁₃	—	4	2	no	—
230	C₄×C₁₃	—	2	1	no	—
231	C₄×C₁₃	—	2	1	no	—
232	C₄×C₁₃	—	2	1	no	—
233	C₄×C₁₃	—	2	1	no	—
234	C₄×C₁₃	—	4	2	no	—
235	C₄×C₁₃	—	4	2	no	—
236	C₄×C₁₃	#568	1	1	yes	characteristic
237	C₄×C₁₃	#568	1	1	yes	characteristic
238	C₄×C₁₃	#577	1	2	yes	—
239	Dic₁₃	—	4	2	no	—
240	C₂×#41	—	26	1	no	—
241	C₂×SD₁₆	—	13	2	no	—
242	C₂×SD₁₆	—	13	2	no	—
243	C₂×M₁₆	—	26	2	no	—
244	C₂²×C₈	—	13	1	no	—
245	C₂²×C₈	—	13	1	no	—
246	C₂²×C₈	—	26	2	no	—
247	C₄⋊Q₈	—	13	1	no	—
248	C₄⋊Q₈	C₄×D₁₃	1	1	yes	characteristic
249	C₄⋊Q₈	C₄×D₁₃	1	1	yes	characteristic
250	#127	—	13	1	no	—
251	#125	—	2	1	no	—
252	#124	—	26	1	no	—
253	C₂²⋊Q₈	—	2	1	no	—
254	C₂²⋊Q₈	C₄×D₁₃	1	2	yes	—
255	#121	—	13	2	no	—
256	#121	—	26	1	no	—
257	C₄×Q₈	—	2	1	no	—
258	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
259	C₄×D₄	—	13	2	no	—
260	C₄×D₄	—	26	1	no	—
261	#117	—	2	1	no	—
262	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
263	#106	—	13	2	no	—
264	#106	—	13	2	no	—
265	C₄⋊C₈	—	2	2	no	—
266	C₄⋊C₈	—	2	2	no	—
267	C₄⋊C₈	—	13	2	no	—
268	C₄⋊C₈	—	26	1	no	—
269	C₄⋊C₈	C₂²⋊D₁₃	1	2	yes	—
270	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
271	#103	—	13	2	no	—
272	#103	—	13	2	no	—
273	#102	—	13	2	no	—
274	#102	—	13	2	no	—
275	C₂²⋊C₈	—	2	1	no	—
276	C₂²⋊C₈	—	26	1	no	—
277	C₂²⋊C₈	—	26	1	no	—
278	C₂²⋊C₈	—	26	2	no	—
279	C₂²⋊C₈	—	26	2	no	—
280	C₂²⋊C₈	C₂²⋊D₁₃	1	2	yes	—
281	#97	—	2	2	no	—
282	#97	—	2	1	no	—
283	#97	—	26	1	no	—
284	C₄×C₈	—	2	2	no	—
285	C₄×C₈	—	13	1	no	—
286	C₄×C₈	—	13	1	no	—
287	C₄×C₈	—	13	1	no	—
288	C₄×C₈	—	13	1	no	—
289	C₄×C₈	—	13	1	no	—
290	C₄×C₈	—	13	1	no	—
291	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
292	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
293	C₂×C₁₃	—	4	1	no	—
294	C₂×C₁₃	#2932	1	1	yes	characteristic
295	C₂×C₁₃	#2949	1	1	yes	characteristic
296	C₂×C₁₃	#2957	1	1	yes	characteristic
297	D₁₃	—	8	2	no	—
298	#41	—	104	2	no	—
299	C₂×Q₈	—	2	1	no	—
300	C₂×Q₈	—	26	1	no	—
301	C₂×Q₈	C₂×C₄×D₁₃	1	2	yes	—
302	C₂×D₄	—	13	2	no	—
303	C₂×D₄	—	26	2	no	—
304	C₂×D₄	—	26	1	no	—
305	C₂×D₄	—	26	1	no	—
306	C₂²×C₄	—	26	2	no	—
307	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
308	C₂²×C₄	#6127	1	1	yes	characteristic
309	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
310	SD₁₆	—	52	2	no	—
311	SD₁₆	—	52	2	no	—
312	M₁₆	—	104	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	13	1	no	—
315	C₂×C₈	—	13	1	no	—
316	C₂×C₈	—	13	1	no	—
317	C₂×C₈	—	13	1	no	—
318	C₂×C₈	—	26	1	no	—
319	C₂×C₈	—	26	1	no	—
320	C₂×C₈	—	26	2	no	—
321	C₂×C₈	—	52	1	no	—
322	C₂×C₈	—	52	1	no	—
323	C₂×C₈	—	104	2	no	—
324	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
325	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
326	C₄⋊C₄	—	2	1	no	—
327	C₄⋊C₄	—	2	1	no	—
328	C₄⋊C₄	—	2	1	no	—
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	13	2	no	—
332	C₄⋊C₄	—	26	1	no	—
333	C₄⋊C₄	C₂×C₄×D₁₃	1	2	yes	—
334	C₂²⋊C₄	—	2	2	no	—
335	C₂²⋊C₄	—	26	2	no	—
336	C₂²⋊C₄	—	26	2	no	—
337	C₄²	—	2	2	no	—
338	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
339	C₄²	#6127	1	1	yes	characteristic
340	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
341	C₄²	C₂³×D₁₃	1	1	yes	characteristic
342	C₁₃	#17990	1	1	yes	characteristic, complemented, Sylow
343	C₂³	—	26	2	no	—
344	C₂³	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
345	Q₈	—	2	2	no	—
346	Q₈	—	2	2	no	—
347	Q₈	—	8	1	no	—
348	Q₈	—	104	1	no	—
349	D₄	—	26	2	no	—
350	D₄	—	26	2	no	—
351	D₄	—	104	1	no	—
352	D₄	—	104	1	no	—
353	D₄	—	104	2	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	4	1	no	—
360	C₂×C₄	—	26	2	no	—
361	C₂×C₄	—	52	2	no	—
362	C₂×C₄	—	52	2	no	—
363	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
364	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
365	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
366	C₂×C₄	C₂²×C₄×D₁₃	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	2	no	—
370	C₈	—	2	2	no	—
371	C₈	—	4	2	no	—
372	C₈	—	26	1	no	—
373	C₈	—	26	1	no	—
374	C₈	—	26	1	no	—
375	C₈	—	26	1	no	—
376	C₈	—	52	1	no	—
377	C₈	—	52	1	no	—
378	C₈	—	52	2	no	—
379	C₂²	—	2	1	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	1	no	—
382	C₂²	—	52	2	no	—
383	C₂²	—	52	2	no	—
384	C₂²	—	52	2	no	—
385	C₂²	C₂×C₄×C₂²⋊D₁₃	1	1	yes	characteristic
386	C₄	—	2	1	no	—
387	C₄	—	2	1	no	—
388	C₄	—	2	1	no	—
389	C₄	—	2	1	no	—
390	C₄	—	4	2	no	—
391	C₄	—	4	2	no	—
392	C₄	—	52	2	no	—
393	C₄	#11719455	1	1	yes	characteristic
394	C₄	#11719455	1	1	yes	characteristic
395	C₄	#11719542	1	2	yes	—
396	C₂	—	4	1	no	—
397	C₂	—	104	2	no	—
398	C₂	#13123552	1	1	yes	characteristic
399	C₂	#13123695	1	1	yes	characteristic
400	C₂	#13123751	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#2985	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#11719182	C₂²
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#2985	D₁₃
hypocenter	C₁₃	#17990
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #2985

Complements

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

N	H ≅ G/N	type
3. #13113714	C₂	semidirect
4. #13113691	C₂	semidirect
5. #13113679	C₂	semidirect
6. #13113670	C₂	semidirect
7. #13113597	C₂	semidirect
8. #13113309	C₂	semidirect
10. #13113129	C₂	semidirect
11. C₁₃×#2985	C₂	semidirect
144. #2985	D₁₃	semidirect
342. C₁₃	#17990	semidirect

Maximal subgroups

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

subgroup	index
#13125396	2
#13123911	2
#13113714	2
#13113691	2
#13113679	2
#13113670	2
#13113597	2
#13113309	2
#13113304	2
#13113129	2
C₁₃×#2985	2
#17990	13