Subgroups — #15832776

Order 3328.

646 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123952	C₂	1	2	yes	maximal, complemented
2	C₄×#88173	C₂	1	1	yes	maximal, characteristic
3	#13113751	C₂	1	1	yes	maximal, characteristic, complemented
4	#13113749	C₂	1	1	yes	maximal, characteristic, complemented
5	#13113745	C₂	1	1	yes	maximal, characteristic, complemented
6	#13113726	C₂	1	2	yes	maximal, complemented
7	#13113723	C₂	1	2	yes	maximal, complemented
8	D₁₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
9	#13113222	C₂	1	2	yes	maximal, complemented
10	#13113121	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#1593	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#88173	C₄	1	1	yes	characteristic
13	C₄×#6131	C₂²	1	2	yes	—
14	C₁₃×#446	C₂²	1	2	yes	—
15	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
16	Q₁₆×Dic₁₃	C₄	1	1	yes	characteristic
17	C₄×#6106	C₂²	1	2	yes	—
18	#11718685	C₂²	1	2	yes	—
19	#11718541	C₄	1	1	yes	characteristic
20	#11718533	C₄	1	1	yes	characteristic
21	#11718499	C₂²	1	2	yes	—
22	#11718498	—	2	2	no	—
23	#11718497	—	2	2	no	—
24	D₁₃×C₄⋊C₈	C₂²	1	2	yes	—
25	#11718465	C₄	1	2	yes	—
26	#11718449	C₄	1	2	yes	—
27	#11718367	—	2	2	no	—
28	#11718366	—	2	2	no	—
29	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
30	#11718349	C₂²	1	2	yes	—
31	#11718346	C₂²	1	1	yes	characteristic
32	#11718346	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
34	C₁₃×#334	C₂²	1	1	yes	characteristic
35	C₁₃×Q₈⋊C₈	C₂²	1	2	yes	—
36	C₁₃×#322	C₂²	1	1	yes	characteristic
37	Q₈⋊#1055	C₂²	1	2	yes	—
38	#11718075	C₂²	1	2	yes	—
39	#11718072	C₂²	1	2	yes	—
40	#11718051	C₂²	1	2	yes	—
41	#11718049	C₂²	1	1	yes	characteristic
42	#11718047	C₂²	1	1	yes	characteristic
43	C₈×#1055	C₂²	1	1	yes	characteristic
44	C₈×#1055	C₂²	1	1	yes	characteristic
45	C₂×#6131	C₂×C₄	1	2	yes	—
46	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
47	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
48	Q₈×Dic₁₃	C₂×C₄	1	2	yes	—
49	C₂×#6106	C₂×C₄	1	2	yes	—
50	#88173	—	2	4	no	—
51	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
52	C₂×#6094	—	2	2	no	—
53	C₂×#6094	—	2	2	no	—
54	C₂×C₈×D₁₃	—	2	2	no	—
55	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
57	#88147	—	2	2	no	—
58	#88147	C₂×C₄	1	2	yes	—
59	#88146	—	2	2	no	—
60	#88146	C₂×C₄	1	2	yes	—
61	C₄×D₅₂	—	2	2	no	—
62	C₄×D₅₂	C₂³	1	2	yes	—
63	C₄²×D₁₃	—	2	2	no	—
64	C₄²×D₁₃	C₂³	1	1	yes	characteristic
65	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
66	C₁₃×C₄⋊C₈	—	2	2	no	—
67	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
68	C₁₃×#103	C₂×C₄	1	2	yes	—
69	C₁₃×#97	—	2	2	no	—
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	#88075	C₂×C₄	1	2	yes	—
74	#88061	C₂×C₄	1	2	yes	—
75	#88060	—	2	1	no	—
76	#88060	—	2	2	no	—
77	#88060	—	2	2	no	—
78	#88060	D₄	1	2	yes	—
79	#88058	C₂×C₄	1	1	yes	characteristic
80	#88054	—	2	2	no	—
81	#88054	—	2	1	no	—
82	#88054	D₄	1	2	yes	—
83	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
85	#88049	C₂×C₄	1	2	yes	—
86	#88048	C₂×C₄	1	2	yes	—
87	C₄⋊#1055	—	2	2	no	—
88	C₄⋊#1055	C₂³	1	2	yes	—
89	#88043	—	2	2	no	—
90	C₄×#1055	C₂³	1	1	yes	characteristic
91	C₄×#1055	C₂³	1	1	yes	characteristic
92	C₄×#1055	C₂³	1	1	yes	characteristic
93	#17899	—	13	1	no	maximal, Sylow
94	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
95	#6131	—	2	4	no	—
96	#6131	—	2	4	no	—
97	C₂×D₅₂	—	2	2	no	—
98	C₂×D₅₂	C₂²×C₄	1	2	yes	—
99	C₂×C₄×D₁₃	—	2	2	no	—
100	C₂×C₄×D₁₃	—	2	2	no	—
101	C₂×C₄×D₁₃	—	2	2	no	—
102	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
105	C₁₃×Q₁₆	—	2	2	no	—
106	C₂×C₈×C₁₃	—	2	2	no	—
107	C₂×C₈×C₁₃	—	2	1	no	—
108	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
111	C₁₃×C₄⋊C₄	—	2	2	no	—
112	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
113	C₄²×C₁₃	—	2	2	no	—
114	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
115	#6106	—	2	4	no	—
116	#6103	—	2	2	no	—
117	#6103	—	2	2	no	—
118	#6103	—	2	2	no	—
119	C₄⋊Dic₁₃	—	2	2	no	—
120	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
121	C₄×Dic₁₃	—	2	2	no	—
122	C₄×Dic₁₃	—	2	2	no	—
123	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
124	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
125	C₄×Dic₁₃	#41	1	1	yes	characteristic
126	C₂×#1055	—	2	2	no	—
127	C₂×#1055	—	2	1	no	—
128	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
129	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
130	C₂×#1055	C₂×D₄	1	2	yes	—
131	D₁₀₄	—	2	2	no	—
132	#6094	—	8	2	no	—
133	#6094	—	8	2	no	—
134	C₈×D₁₃	—	2	2	no	—
135	C₈×D₁₃	—	8	2	no	—
136	C₈×D₁₃	M₁₆	1	4	yes	complemented
137	#2989	—	13	2	no	—
138	C₄×#135	—	13	1	no	—
139	#1594	—	13	1	no	—
140	#1594	—	13	1	no	—
141	#1593	—	13	1	no	—
142	#1593	D₁₃	1	1	yes	characteristic, complemented
143	#1590	—	13	2	no	—
144	#1590	—	13	2	no	—
145	C₂×#334	—	13	1	no	—
146	#1477	—	13	2	no	—
147	#1450	—	13	1	no	—
148	C₂²×D₁₃	—	2	2	no	—
149	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
150	Q₈×C₁₃	—	2	2	no	—
151	Q₈×C₁₃	—	2	2	no	—
152	C₂×C₄×C₁₃	—	2	2	no	—
153	C₂×C₄×C₁₃	—	2	2	no	—
154	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
157	C₂×Dic₁₃	—	2	2	no	—
158	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
159	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
160	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
161	D₅₂	—	2	2	no	—
162	D₅₂	—	2	2	no	—
163	D₅₂	—	4	4	no	—
164	C₄×D₁₃	—	2	2	no	—
165	C₄×D₁₃	—	4	1	no	—
166	C₄×D₁₃	—	4	2	no	—
167	C₄×D₁₃	—	4	2	no	—
168	C₄×D₁₃	—	4	4	no	—
169	C₄×D₁₃	—	8	2	no	—
170	C₄×D₁₃	C₂×M₁₆	1	2	yes	—
171	C₄×D₁₃	#131	1	2	yes	—
172	C₈×C₁₃	—	2	1	no	—
173	C₈×C₁₃	—	2	2	no	—
174	C₈×C₁₃	—	4	1	no	—
175	C₈×C₁₃	—	4	2	no	—
176	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
177	#1055	—	2	1	no	—
178	#1055	—	2	2	no	—
179	#1055	—	4	1	no	—
180	#1055	—	4	2	no	—
181	#1055	C₂×M₁₆	1	2	yes	—
182	C₂×#135	—	13	1	no	—
183	C₄×#41	—	13	2	no	—
184	#446	—	13	2	no	—
185	#446	C₂×D₁₃	1	2	yes	—
186	C₄×Q₁₆	—	13	1	no	—
187	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
188	C₄×SD₁₆	—	13	2	no	—
189	C₄×SD₁₆	—	13	2	no	—
190	C₄×D₈	—	13	1	no	—
191	C₄×D₈	—	13	1	no	—
192	#436	—	13	2	no	—
193	#436	—	13	2	no	—
194	#435	—	26	2	no	—
195	#435	—	26	2	no	—
196	#432	—	26	2	no	—
197	#431	—	13	1	no	—
198	#431	—	13	1	no	—
199	#427	—	13	1	no	—
200	#423	—	26	2	no	—
201	C₂×C₄⋊C₈	—	13	2	no	—
202	#416	—	13	2	no	—
203	C₂×C₄×C₈	—	13	1	no	—
204	#334	—	13	2	no	—
205	#334	—	13	1	no	—
206	#334	C₂×D₁₃	1	1	yes	characteristic
207	Q₈⋊C₈	—	13	2	no	—
208	Q₈⋊C₈	C₂×D₁₃	1	2	yes	—
209	D₄⋊C₈	—	13	2	no	—
210	D₄⋊C₈	—	13	2	no	—
211	#322	—	13	1	no	—
212	#322	C₂×D₁₃	1	1	yes	characteristic
213	C₈²	—	13	1	no	—
214	C₈²	—	13	1	no	—
215	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
216	C₂×D₁₃	—	2	1	no	—
217	C₂×D₁₃	—	2	1	no	—
218	C₂×D₁₃	—	4	2	no	—
219	C₂×D₁₃	—	4	2	no	—
220	C₂×D₁₃	—	4	2	no	—
221	C₂×D₁₃	#436	1	2	yes	—
222	C₄×C₁₃	—	2	1	no	—
223	C₄×C₁₃	—	4	2	no	—
224	C₄×C₁₃	—	4	2	no	—
225	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
226	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
227	C₄×C₁₃	#577	1	2	yes	—
228	Dic₁₃	—	2	1	no	—
229	Dic₁₃	—	2	1	no	—
230	Dic₁₃	—	4	2	no	—
231	Dic₁₃	#436	1	2	yes	—
232	C₂×#41	—	13	2	no	—
233	#135	—	26	4	no	—
234	C₂×Q₁₆	C₄×D₁₃	1	1	yes	characteristic
235	C₂×SD₁₆	—	13	2	no	—
236	C₂×D₈	—	13	1	no	—
237	C₂×M₁₆	—	26	2	no	—
238	C₂×M₁₆	—	26	2	no	—
239	C₂²×C₈	—	13	1	no	—
240	C₂²×C₈	—	13	1	no	—
241	C₂²×C₈	—	26	2	no	—
242	C₄×Q₈	—	13	2	no	—
243	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
244	C₄×D₄	—	13	2	no	—
245	C₄×D₄	—	13	2	no	—
246	C₄×D₄	—	26	2	no	—
247	C₄×D₄	—	26	2	no	—
248	#117	—	13	2	no	—
249	#117	—	26	2	no	—
250	C₂×C₄²	—	13	1	no	—
251	C₂×C₄²	—	26	2	no	—
252	#107	—	13	1	no	—
253	#107	C₄×D₁₃	1	1	yes	characteristic
254	#106	—	13	2	no	—
255	C₄⋊C₈	—	2	2	no	—
256	C₄⋊C₈	—	13	2	no	—
257	C₄⋊C₈	—	13	2	no	—
258	C₄⋊C₈	—	26	2	no	—
259	C₄⋊C₈	—	26	2	no	—
260	C₄⋊C₈	—	26	1	no	—
261	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
262	#103	—	13	2	no	—
263	#103	C₄×D₁₃	1	2	yes	—
264	#102	—	13	2	no	—
265	#102	—	13	2	no	—
266	C₂²⋊C₈	—	13	2	no	—
267	C₂²⋊C₈	—	26	2	no	—
268	C₂²⋊C₈	—	26	1	no	—
269	C₂²⋊C₈	—	26	2	no	—
270	#97	—	2	2	no	—
271	#97	—	26	2	no	—
272	C₄×C₈	—	13	1	no	—
273	C₄×C₈	—	13	1	no	—
274	C₄×C₈	—	13	1	no	—
275	C₄×C₈	—	13	1	no	—
276	C₄×C₈	—	13	1	no	—
277	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
278	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
280	C₂×C₁₃	C₂×#436	1	1	yes	characteristic
281	C₂×C₁₃	#2952	1	1	yes	characteristic
282	C₂×C₁₃	#2957	1	1	yes	characteristic
283	D₁₃	—	2	2	no	—
284	D₁₃	—	8	2	no	—
285	#41	—	26	4	no	—
286	#41	—	26	4	no	—
287	C₂×Q₈	C₂×C₄×D₁₃	1	2	yes	—
288	C₂×D₄	—	13	2	no	—
289	C₂×D₄	—	26	2	no	—
290	C₂²×C₄	—	13	1	no	—
291	C₂²×C₄	—	13	1	no	—
292	C₂²×C₄	—	13	1	no	—
293	C₂²×C₄	—	26	2	no	—
294	C₂²×C₄	—	26	2	no	—
295	C₂²×C₄	—	26	2	no	—
296	Q₁₆	—	2	2	no	—
297	SD₁₆	—	26	4	no	—
298	D₈	—	26	2	no	—
299	M₁₆	—	104	2	no	—
300	M₁₆	—	104	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	13	1	no	—
304	C₂×C₈	—	13	4	no	—
305	C₂×C₈	—	13	1	no	—
306	C₂×C₈	—	13	2	no	—
307	C₂×C₈	—	26	2	no	—
308	C₂×C₈	—	26	2	no	—
309	C₂×C₈	—	26	1	no	—
310	C₂×C₈	—	104	2	no	—
311	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
312	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
313	C₂×C₈	D₄×D₁₃	1	2	yes	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	13	2	no	—
316	C₄⋊C₄	—	26	2	no	—
317	C₄⋊C₄	C₂×C₄×D₁₃	1	2	yes	—
318	C₂²⋊C₄	—	26	2	no	—
319	C₂²⋊C₄	—	26	2	no	—
320	C₂²⋊C₄	—	26	2	no	—
321	C₄²	—	2	2	no	—
322	C₄²	—	13	1	no	—
323	C₄²	—	13	1	no	—
324	C₄²	—	13	1	no	—
325	C₄²	—	26	2	no	—
326	C₄²	—	26	2	no	—
327	C₄²	C₂³×D₁₃	1	1	yes	characteristic
328	C₁₃	#17899	1	1	yes	characteristic, complemented, Sylow
329	C₂³	—	13	1	no	—
330	C₂³	—	26	2	no	—
331	Q₈	—	2	2	no	—
332	Q₈	—	2	2	no	—
333	D₄	—	26	2	no	—
334	D₄	—	26	2	no	—
335	D₄	—	52	4	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	13	1	no	—
339	C₂×C₄	—	13	2	no	—
340	C₂×C₄	—	13	2	no	—
341	C₂×C₄	—	13	1	no	—
342	C₂×C₄	—	13	1	no	—
343	C₂×C₄	—	26	2	no	—
344	C₂×C₄	—	26	2	no	—
345	C₂×C₄	—	52	1	no	—
346	C₂×C₄	—	52	4	no	—
347	C₂×C₄	—	52	2	no	—
348	C₂×C₄	—	52	2	no	—
349	C₂×C₄	—	104	2	no	—
350	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
351	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
352	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
353	C₈	—	2	1	no	—
354	C₈	—	2	2	no	—
355	C₈	—	4	1	no	—
356	C₈	—	4	2	no	—
357	C₈	—	13	2	no	—
358	C₈	—	26	1	no	—
359	C₈	—	26	2	no	—
360	C₈	—	52	1	no	—
361	C₈	—	52	2	no	—
362	C₈	M₁₆×D₁₃	1	2	yes	—
363	C₂²	—	13	2	no	—
364	C₂²	—	26	1	no	—
365	C₂²	—	26	1	no	—
366	C₂²	—	52	2	no	—
367	C₂²	—	52	2	no	—
368	C₂²	—	52	2	no	—
369	C₂²	C₄×D₄×D₁₃	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	4	2	no	—
372	C₄	—	4	2	no	—
373	C₄	—	13	2	no	—
374	C₄	—	26	1	no	—
375	C₄	—	26	1	no	—
376	C₄	—	52	2	no	—
377	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
378	C₄	D₁₃×#131	1	1	yes	characteristic
379	C₄	#11719483	1	2	yes	—
380	C₂	—	26	2	no	—
381	C₂	—	104	2	no	—
382	C₂	D₁₃×#436	1	1	yes	characteristic
383	C₂	#13123716	1	1	yes	characteristic
384	C₂	#13123750	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×#1593	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂×#88173	C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1593	D₁₃
hypocenter	C₁₃	#17899
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1593

Complements

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

N	H ≅ G/N	type
1. #13123952	C₂	semidirect
3. #13113751	C₂	semidirect
4. #13113749	C₂	semidirect
5. #13113745	C₂	semidirect
6. #13113726	C₂	semidirect
7. #13113723	C₂	semidirect
8. D₁₃×#334	C₂	semidirect
9. #13113222	C₂	semidirect
10. #13113121	C₂	semidirect
11. C₁₃×#1593	C₂	semidirect
136. C₈×D₁₃	M₁₆	semidirect
142. #1593	D₁₃	semidirect
328. C₁₃	#17899	semidirect

Maximal subgroups

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

subgroup	index
#13123952	2
C₄×#88173	2
#13113751	2
#13113749	2
#13113745	2
#13113726	2
#13113723	2
D₁₃×#334	2
#13113222	2
#13113121	2
C₁₃×#1593	2
#17899	13