Subgroups — #15833116

Order 3328.

658 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13124867	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718685	C₂	1	1	yes	maximal, characteristic, complemented
3	#13123547	C₂	1	1	yes	maximal, characteristic
4	#13113723	C₂	1	4	yes	maximal, complemented
5	#13113675	C₂	1	4	yes	maximal, complemented
6	#13113608	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113236	C₂	1	1	yes	maximal, characteristic
8	#13113120	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#1487	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88161	—	2	1	no	—
12	#11719182	C₂²	1	1	yes	characteristic
13	C₁₃×#431	C₂²	1	1	yes	characteristic
14	C₁₃×#423	C₂²	1	1	yes	characteristic
15	#11718888	C₄	1	2	yes	—
16	#11718790	—	2	1	no	—
17	#11718688	C₄	1	2	yes	—
18	C₄×#6106	C₂²	1	4	yes	—
19	#11718685	—	2	1	no	—
20	#11718685	—	2	1	no	—
21	#11718685	C₂²	1	2	yes	—
22	#11718685	C₂²	1	2	yes	—
23	#11718681	C₄	1	2	yes	—
24	#11718629	C₄	1	2	yes	—
25	#11718586	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊#1055	—	2	1	no	—
27	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
28	C₂×#88043	—	2	1	no	—
29	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
30	#11718499	—	2	1	no	—
31	#11718499	C₂²	1	2	yes	—
32	#11718495	—	2	1	no	—
33	#11718495	C₂²	1	2	yes	—
34	C₁₃×Q₈⋊C₈	C₂²	1	4	yes	—
35	#11718072	C₂²	1	4	yes	—
36	#11718051	C₂²	1	4	yes	—
37	#11718046	C₂²	1	2	yes	—
38	C₈×#1055	C₂²	1	2	yes	—
39	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×#6127	—	2	1	no	—
41	C₂×C₁₃×M₁₆	—	2	1	no	—
42	C₄×Q₈×C₁₃	—	2	1	no	—
43	C₄×Q₈×C₁₃	—	2	1	no	—
44	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
45	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
46	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
47	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₃	—	2	1	no	—
49	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×#6106	C₂×C₄	1	4	yes	—
51	#88184	—	2	1	no	—
52	#88184	C₂×C₄	1	2	yes	—
53	C₂²×#1055	—	2	1	no	—
54	C₂²×#1055	D₄	1	1	yes	characteristic
55	C₂²×#1055	D₄	1	1	yes	characteristic
56	#88161	—	4	4	no	—
57	C₂×#6094	—	2	2	no	—
58	C₂×C₈×D₁₃	—	2	2	no	—
59	#88129	—	2	1	no	—
60	#88128	C₂×C₄	1	1	yes	characteristic
61	C₄×D₅₂	—	2	1	no	—
62	C₄×D₅₂	C₂³	1	2	yes	—
63	#88123	C₂×C₄	1	1	yes	characteristic
64	C₁₃×C₄⋊C₈	—	2	1	no	—
65	C₁₃×C₄⋊C₈	—	2	1	no	—
66	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
67	C₁₃×C₂²⋊C₈	—	2	1	no	—
68	C₄×C₈×C₁₃	C₂³	1	2	yes	—
69	#88075	C₂×C₄	1	4	yes	—
70	C₂²⋊#1055	C₂×C₄	1	2	yes	—
71	#88060	—	2	2	no	—
72	#88060	—	2	2	no	—
73	#88049	C₂×C₄	1	4	yes	—
74	#88048	C₂×C₄	1	4	yes	—
75	C₄⋊#1055	—	2	1	no	—
76	C₄⋊#1055	—	2	2	no	—
77	C₄⋊#1055	—	2	2	no	—
78	C₄⋊#1055	C₂×C₄	1	2	yes	—
79	C₄⋊#1055	C₂³	1	2	yes	—
80	#88043	—	2	2	no	—
81	#88043	—	2	2	no	—
82	C₄×#1055	D₄	1	1	yes	characteristic
83	C₄×#1055	D₄	1	1	yes	characteristic
84	C₄×#1055	C₂³	1	2	yes	—
85	C₄×#1055	C₂³	1	1	yes	characteristic
86	C₄×#1055	C₂³	1	1	yes	characteristic
87	#17929	—	13	1	no	maximal, Sylow
88	C₂×Q₈×C₁₃	—	2	1	no	—
89	C₂×Q₈×C₁₃	—	4	2	no	—
90	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₁₃	—	2	1	no	—
92	C₂²×C₄×C₁₃	—	2	1	no	—
93	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
96	C₂×C₂²⋊D₁₃	—	2	2	no	—
97	#6127	—	4	2	no	—
98	#6127	—	4	2	no	—
99	C₂×D₅₂	—	2	1	no	—
100	C₂×D₅₂	—	2	1	no	—
101	C₂×D₅₂	C₂²×C₄	1	2	yes	—
102	C₂×C₄×D₁₃	—	2	2	no	—
103	C₂×Dic₂₆	—	2	1	no	—
104	C₁₃×M₁₆	—	4	2	no	—
105	C₂×C₈×C₁₃	—	2	2	no	—
106	C₂×C₈×C₁₃	—	2	2	no	—
107	C₁₃×C₄⋊C₄	—	2	2	no	—
108	C₁₃×C₄⋊C₄	—	2	1	no	—
109	C₁₃×C₄⋊C₄	—	2	2	no	—
110	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
111	C₄²×C₁₃	—	2	2	no	—
112	C₄²×C₁₃	—	2	2	no	—
113	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₁₃	#41	1	1	yes	characteristic
116	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
117	#6106	—	4	4	no	—
118	#6103	—	2	2	no	—
119	#6103	—	2	2	no	—
120	C₄⋊Dic₁₃	—	2	1	no	—
121	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
122	C₂×#1055	—	2	2	no	—
123	C₂×#1055	—	2	2	no	—
124	C₂×#1055	—	4	2	no	—
125	C₂×#1055	SD₁₆	1	4	yes	complemented
126	C₂×#1055	C₂²×C₄	1	4	yes	—
127	C₂×#1055	C₂×D₄	1	1	yes	characteristic
128	C₂×#1055	C₂×D₄	1	1	yes	characteristic
129	C₂×#1055	C₂×D₄	1	1	yes	characteristic
130	C₂×#1055	C₂×D₄	1	1	yes	characteristic
131	#6094	—	4	4	no	—
132	C₈×D₁₃	—	4	4	no	—
133	#3103	—	13	1	no	—
134	C₂×#446	—	13	1	no	—
135	#2931	—	13	1	no	—
136	#1590	—	13	4	no	—
137	#1582	—	13	4	no	—
138	#1570	—	13	1	no	—
139	#1487	D₁₃	1	1	yes	characteristic, complemented
140	#1480	—	13	1	no	—
141	#1450	—	13	1	no	—
142	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
143	C₂²×D₁₃	—	2	2	no	—
144	Q₈×C₁₃	—	2	2	no	—
145	Q₈×C₁₃	—	2	2	no	—
146	Q₈×C₁₃	—	4	2	no	—
147	C₂×C₄×C₁₃	—	2	2	no	—
148	C₂×C₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	—	2	2	no	—
150	C₂×C₄×C₁₃	—	4	1	no	—
151	C₂×C₄×C₁₃	—	4	2	no	—
152	C₂×C₄×C₁₃	—	4	2	no	—
153	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
157	C₂×C₄×C₁₃	#135	1	2	yes	—
158	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
161	C₂²⋊D₁₃	—	4	4	no	—
162	C₂×Dic₁₃	—	2	2	no	—
163	D₅₂	—	2	4	no	—
164	D₅₂	—	4	2	no	—
165	D₅₂	—	4	2	no	—
166	C₄×D₁₃	—	4	2	no	—
167	C₄×D₁₃	—	4	2	no	—
168	Dic₂₆	—	4	2	no	—
169	C₈×C₁₃	—	4	2	no	—
170	C₈×C₁₃	—	4	2	no	—
171	#1055	—	2	4	no	—
172	#1055	—	2	1	no	—
173	#1055	—	2	1	no	—
174	#1055	—	4	2	no	—
175	#1055	C₂×SD₁₆	1	2	yes	—
176	#1055	C₂×SD₁₆	1	2	yes	—
177	C₂×#131	—	26	1	no	—
178	#532	—	13	1	no	—
179	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
180	#481	—	13	2	no	—
181	#475	—	13	2	no	—
182	#460	—	13	2	no	—
183	#448	—	13	2	no	—
184	#446	—	13	2	no	—
185	#446	—	13	2	no	—
186	#446	—	26	1	no	—
187	#446	—	26	1	no	—
188	C₄×SD₁₆	—	13	4	no	—
189	#436	—	13	2	no	—
190	#436	—	26	1	no	—
191	C₈×D₄	—	13	2	no	—
192	C₈×D₄	—	26	1	no	—
193	#431	—	13	1	no	—
194	#431	C₂×D₁₃	1	1	yes	characteristic
195	#423	C₂×D₁₃	1	1	yes	characteristic
196	C₂×C₄⋊C₈	—	13	1	no	—
197	C₂×C₄⋊C₈	—	26	1	no	—
198	#408	—	26	1	no	—
199	C₂×#97	—	26	1	no	—
200	C₂×C₄×C₈	—	13	1	no	—
201	#334	—	13	4	no	—
202	Q₈⋊C₈	C₂×D₁₃	1	4	yes	—
203	D₄⋊C₈	—	13	4	no	—
204	#322	—	13	2	no	—
205	C₈²	—	13	2	no	—
206	C₂²×C₁₃	—	2	1	no	—
207	C₂²×C₁₃	—	2	1	no	—
208	C₂²×C₁₃	C₄×SD₁₆	1	2	yes	—
209	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
210	C₂×D₁₃	—	4	2	no	—
211	C₂×D₁₃	—	4	2	no	—
212	C₂×D₁₃	—	4	2	no	—
213	C₄×C₁₃	—	2	1	no	—
214	C₄×C₁₃	—	2	1	no	—
215	C₄×C₁₃	—	2	1	no	—
216	C₄×C₁₃	—	4	2	no	—
217	C₄×C₁₃	—	4	2	no	—
218	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
219	C₄×C₁₃	#568	1	2	yes	—
220	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
221	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
222	Dic₁₃	—	4	2	no	—
223	C₂×#41	—	26	1	no	—
224	C₂²×Q₈	C₄×D₁₃	1	1	yes	characteristic
225	C₂×SD₁₆	—	13	4	no	—
226	#131	—	52	4	no	—
227	C₂×M₁₆	—	2	1	no	—
228	C₂×M₁₆	—	26	2	no	—
229	C₂²×C₈	—	13	1	no	—
230	C₂²×C₈	—	13	1	no	—
231	C₂²×C₈	—	26	2	no	—
232	C₂²×C₈	—	26	1	no	—
233	C₄⋊Q₈	—	13	1	no	—
234	#127	—	13	1	no	—
235	#124	—	26	1	no	—
236	#121	—	13	2	no	—
237	#121	—	26	1	no	—
238	C₄×Q₈	—	2	1	no	—
239	C₄×Q₈	—	2	1	no	—
240	C₄×Q₈	C₄×D₁₃	1	2	yes	—
241	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
242	C₄×D₄	—	13	2	no	—
243	C₄×D₄	—	26	1	no	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
246	C₂×C₄²	—	2	1	no	—
247	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
248	#106	—	13	4	no	—
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	—	2	1	no	—
251	C₄⋊C₈	—	13	2	no	—
252	C₄⋊C₈	—	13	2	no	—
253	C₄⋊C₈	—	26	2	no	—
254	C₄⋊C₈	—	26	2	no	—
255	C₄⋊C₈	—	26	1	no	—
256	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
257	#103	—	13	4	no	—
258	#102	—	13	4	no	—
259	C₂²⋊C₈	—	2	1	no	—
260	C₂²⋊C₈	—	13	2	no	—
261	C₂²⋊C₈	—	26	2	no	—
262	C₂²⋊C₈	—	26	2	no	—
263	#97	—	26	2	no	—
264	#97	—	26	2	no	—
265	C₄×C₈	—	13	1	no	—
266	C₄×C₈	—	13	1	no	—
267	C₄×C₈	—	13	1	no	—
268	C₄×C₈	—	13	2	no	—
269	C₄×C₈	—	13	1	no	—
270	C₄×C₈	C₂²×D₁₃	1	2	yes	—
271	C₂×C₁₃	—	2	2	no	—
272	C₂×C₁₃	#2930	1	1	yes	characteristic
273	C₂×C₁₃	C₂×C₄×SD₁₆	1	1	yes	characteristic
274	C₂×C₁₃	#2955	1	1	yes	characteristic
275	D₁₃	—	8	2	no	—
276	#41	—	52	2	no	—
277	#41	—	52	2	no	—
278	C₂×Q₈	—	2	1	no	—
279	C₂×Q₈	—	4	2	no	—
280	C₂×Q₈	—	26	1	no	—
281	C₂×Q₈	C₂×C₄×D₁₃	1	2	yes	—
282	C₂×D₄	—	13	2	no	—
283	C₂×D₄	—	26	2	no	—
284	C₂×D₄	—	26	1	no	—
285	C₂×D₄	—	26	1	no	—
286	C₂²×C₄	—	2	1	no	—
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	26	2	no	—
289	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
290	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
291	C₂²×C₄	#6129	1	1	yes	characteristic
292	SD₁₆	—	52	4	no	—
293	M₁₆	—	4	2	no	—
294	M₁₆	—	52	4	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	13	4	no	—
298	C₂×C₈	—	13	1	no	—
299	C₂×C₈	—	13	1	no	—
300	C₂×C₈	—	13	4	no	—
301	C₂×C₈	—	13	1	no	—
302	C₂×C₈	—	13	1	no	—
303	C₂×C₈	—	26	2	no	—
304	C₂×C₈	—	26	2	no	—
305	C₂×C₈	—	52	4	no	—
306	C₂×C₈	—	52	2	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	2	1	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	13	2	no	—
311	C₄⋊C₄	—	26	1	no	—
312	C₄⋊C₄	C₂×C₄×D₁₃	1	2	yes	—
313	C₂²⋊C₄	—	26	2	no	—
314	C₂²⋊C₄	—	26	2	no	—
315	C₄²	—	2	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
318	C₄²	D₄×D₁₃	1	1	yes	characteristic
319	C₄²	#6129	1	1	yes	characteristic
320	C₄²	C₂³×D₁₃	1	1	yes	characteristic
321	C₁₃	#17929	1	1	yes	characteristic, complemented, Sylow
322	C₂³	—	26	2	no	—
323	C₂³	#88135	1	1	yes	characteristic
324	Q₈	—	2	2	no	—
325	Q₈	—	2	2	no	—
326	Q₈	—	4	2	no	—
327	Q₈	—	52	2	no	—
328	D₄	—	26	4	no	—
329	D₄	—	52	2	no	—
330	D₄	—	52	4	no	—
331	D₄	—	52	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	26	2	no	—
339	C₂×C₄	—	52	2	no	—
340	C₂×C₄	—	52	2	no	—
341	C₂×C₄	#88135	1	1	yes	characteristic
342	C₂×C₄	#88135	1	1	yes	characteristic
343	C₂×C₄	#88135	1	1	yes	characteristic
344	C₂×C₄	SD₁₆×D₁₃	1	2	yes	—
345	C₂×C₄	#88173	1	2	yes	—
346	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
347	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
348	C₂×C₄	C₂×#6129	1	1	yes	characteristic
349	C₈	—	4	2	no	—
350	C₈	—	4	2	no	—
351	C₈	—	13	2	no	—
352	C₈	—	13	2	no	—
353	C₈	—	26	4	no	—
354	C₈	—	26	1	no	—
355	C₈	—	26	1	no	—
356	C₈	—	52	2	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	52	2	no	—
360	C₂²	—	52	2	no	—
361	C₂²	—	52	2	no	—
362	C₂²	#11718449	1	2	yes	—
363	C₂²	C₂×#88135	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	4	2	no	—
368	C₄	—	4	2	no	—
369	C₄	—	52	2	no	—
370	C₄	#11718449	1	2	yes	—
371	C₄	#11719450	1	2	yes	—
372	C₄	C₂×SD₁₆×D₁₃	1	1	yes	characteristic
373	C₄	C₂×#88173	1	1	yes	characteristic
374	C₂	—	2	2	no	—
375	C₂	—	104	2	no	—
376	C₂	#13123535	1	1	yes	characteristic
377	C₂	C₂×#11718449	1	1	yes	characteristic
378	C₂	#13123735	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1487	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#11719182	C₂²
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1487	D₁₃
hypocenter	C₁₃	#17929
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1487

Complements

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

N	H ≅ G/N	type
2. C₂×#11718685	C₂	semidirect
4. #13113723	C₂	semidirect
5. #13113675	C₂	semidirect
6. #13113608	C₂	semidirect
8. #13113120	C₂	semidirect
9. C₁₃×#1487	C₂	semidirect
125. C₂×#1055	SD₁₆	semidirect
139. #1487	D₁₃	semidirect
321. C₁₃	#17929	semidirect

Maximal subgroups

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

subgroup	index
#13124867	2
C₂×#11718685	2
#13123547	2
#13113723	2
#13113675	2
#13113608	2
#13113236	2
#13113120	2
C₁₃×#1487	2
#17929	13