Subgroups — #15921804

Order 3328.

674 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125368	C₂	1	1	yes	maximal, characteristic, complemented
2	#13124689	C₂	1	2	yes	maximal
3	#13123428	C₂	1	1	yes	maximal, characteristic
4	#13116341	C₂	1	2	yes	maximal, complemented
5	#13116185	C₂	1	2	yes	maximal, complemented
6	#13115617	C₂	1	1	yes	maximal, characteristic, complemented
7	#13115070	C₂	1	2	yes	maximal
8	#13115000	C₂	1	1	yes	maximal, characteristic, complemented
9	#13114892	C₂	1	1	yes	maximal, characteristic
10	#13114673	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#1945	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#88155	C₂²	1	1	yes	characteristic
13	#11719228	C₂²	1	2	yes	—
14	#11719224	C₂²	1	2	yes	—
15	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
16	C₁₃×#427	C₂²	1	1	yes	characteristic
17	C₁₃×#424	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	#11718794	C₂²	1	2	yes	—
20	#11718782	C₂²	1	2	yes	—
21	#11718774	C₂²	1	1	yes	characteristic
22	C₂×#88058	C₂²	1	2	yes	—
23	C₂×#88057	C₂²	1	1	yes	characteristic
24	C₂×#88057	C₂²	1	1	yes	characteristic
25	#11718762	C₂²	1	1	yes	characteristic
26	#11718656	C₂²	1	2	yes	—
27	#11718656	C₂²	1	2	yes	—
28	#11718597	C₂²	1	2	yes	—
29	C₂×#88045	C₂²	1	1	yes	characteristic
30	#11718542	—	2	2	no	—
31	#11718537	—	2	2	no	—
32	#11718521	—	2	1	no	—
33	D₁₃×#106	—	2	1	no	—
34	#11718519	—	2	1	no	—
35	#11718516	—	2	1	no	—
36	#11718511	—	2	2	no	—
37	#11718508	—	2	2	no	—
38	#11718507	—	2	2	no	—
39	#11718490	—	2	2	no	—
40	C₁₃×#339	C₂²	1	2	yes	—
41	#11718146	C₂²	1	2	yes	—
42	#11718146	C₂²	1	2	yes	—
43	#11718144	C₂²	1	1	yes	characteristic
44	#11718115	C₂²	1	1	yes	characteristic
45	C₂×#6127	C₂³	1	1	yes	characteristic
46	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
47	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₁₃×#117	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	#88180	C₂³	1	1	yes	characteristic
52	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
53	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
54	C₂×#6099	C₂³	1	1	yes	characteristic
55	C₂×#6099	C₂³	1	1	yes	characteristic
56	#88155	—	2	2	no	—
57	#88155	Q₈	1	4	yes	complemented
58	C₂×#6094	—	2	1	no	—
59	C₂×C₈×D₁₃	—	2	1	no	—
60	#88146	—	2	2	no	—
61	C₄⋊C₄×D₁₃	—	2	2	no	—
62	#88144	—	2	2	no	—
63	#88142	—	2	2	no	—
64	#88135	—	4	2	no	—
65	#88132	—	4	2	no	—
66	C₄×D₅₂	—	2	2	no	—
67	#88124	—	2	2	no	—
68	#88123	—	2	2	no	—
69	C₄×Dic₂₆	—	2	2	no	—
70	C₁₃×#108	—	2	2	no	—
71	C₁₃×#107	D₄	1	2	yes	—
72	C₁₃×#106	D₄	1	1	yes	characteristic
73	C₁₃×#106	D₄	1	1	yes	characteristic
74	C₁₃×C₄⋊C₈	—	2	2	no	—
75	C₁₃×C₄⋊C₈	—	2	2	no	—
76	C₄×C₈×C₁₃	—	2	2	no	—
77	#88071	—	4	2	no	—
78	#88061	—	2	2	no	—
79	#88061	D₄	1	2	yes	—
80	#88058	—	2	2	no	—
81	#88058	—	2	2	no	—
82	#88058	—	2	2	no	—
83	#88057	—	2	1	no	—
84	#88057	—	2	1	no	—
85	#88057	—	2	1	no	—
86	#88057	—	2	1	no	—
87	#88057	—	2	2	no	—
88	#88056	—	2	2	no	—
89	#88056	D₄	1	2	yes	—
90	#88055	—	2	1	no	—
91	C₈×Dic₁₃	—	2	1	no	—
92	#88048	—	2	2	no	—
93	#88048	—	2	2	no	—
94	#88047	—	2	2	no	—
95	#88047	—	2	2	no	—
96	#88045	Q₈	1	4	yes	complemented
97	#26080	—	13	1	no	maximal, Sylow
98	C₂²×C₄×C₁₃	—	2	1	no	—
99	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
100	C₂×C₂²⋊D₁₃	—	2	1	no	—
101	C₂²×Dic₁₃	—	2	2	no	—
102	C₂²×Dic₁₃	—	2	2	no	—
103	#6127	—	2	1	no	—
104	#6127	—	2	1	no	—
105	#6127	C₂×Q₈	1	2	yes	—
106	#6127	C₂×Q₈	1	2	yes	—
107	C₂×D₅₂	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×D₁₃	—	2	1	no	—
109	C₂×C₄×D₁₃	—	4	2	no	—
110	C₂×Dic₂₆	—	4	2	no	—
111	C₂×Dic₂₆	C₂×D₄	1	1	yes	characteristic
112	C₁₃×M₁₆	—	2	2	no	—
113	C₁₃×M₁₆	—	2	2	no	—
114	C₂×C₈×C₁₃	—	2	1	no	—
115	C₂×C₈×C₁₃	—	2	1	no	—
116	C₂×C₈×C₁₃	—	2	2	no	—
117	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
118	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
121	C₂×C₈×C₁₃	#41	1	2	yes	—
122	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
123	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
124	C₁₃×C₂²⋊C₄	—	2	2	no	—
125	C₄²×C₁₃	—	2	2	no	—
126	C₄²×C₁₃	—	2	1	no	—
127	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
128	C₂²⋊Dic₁₃	—	2	1	no	—
129	#6103	—	4	2	no	—
130	C₄⋊Dic₁₃	—	2	2	no	—
131	C₄⋊Dic₁₃	—	2	2	no	—
132	C₄⋊Dic₁₃	—	2	2	no	—
133	C₄⋊Dic₁₃	—	2	2	no	—
134	C₄⋊Dic₁₃	—	4	2	no	—
135	C₄⋊Dic₁₃	C₂×D₄	1	2	yes	—
136	#6101	—	4	2	no	—
137	#6101	—	4	2	no	—
138	#6101	—	4	2	no	—
139	C₄×Dic₁₃	—	2	1	no	—
140	C₄×Dic₁₃	—	4	2	no	—
141	#6099	—	2	1	no	—
142	#6099	—	2	1	no	—
143	#6099	C₂×Q₈	1	2	yes	—
144	#6099	C₂×Q₈	1	2	yes	—
145	C₂×#1055	—	2	1	no	—
146	C₂×#1055	—	2	1	no	—
147	#6094	—	2	4	no	—
148	C₈×D₁₃	—	2	4	no	—
149	#3163	—	13	1	no	—
150	#3083	—	13	2	no	—
151	#2912	—	13	1	no	—
152	#2089	—	13	2	no	—
153	#2061	—	13	2	no	—
154	#1951	—	13	1	no	—
155	#1945	D₁₃	1	1	yes	characteristic, complemented
156	#1860	—	13	2	no	—
157	#1846	—	13	1	no	—
158	#1815	—	13	1	no	—
159	#1764	—	13	1	no	—
160	C₂³×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
161	C₂²×D₁₃	—	2	1	no	—
162	C₂×C₄×C₁₃	—	2	2	no	—
163	C₂×C₄×C₁₃	—	2	2	no	—
164	C₂×C₄×C₁₃	—	2	2	no	—
165	C₂×C₄×C₁₃	—	4	2	no	—
166	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
169	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
170	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
171	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
172	C₂²⋊D₁₃	—	2	2	no	—
173	C₂²⋊D₁₃	—	2	2	no	—
174	C₂×Dic₁₃	—	2	1	no	—
175	C₂×Dic₁₃	—	2	2	no	—
176	C₂×Dic₁₃	—	4	2	no	—
177	C₂×Dic₁₃	—	4	2	no	—
178	C₂×Dic₁₃	—	8	2	no	—
179	C₂×Dic₁₃	—	8	2	no	—
180	D₅₂	—	2	1	no	—
181	D₅₂	C₄⋊Q₈	1	2	yes	—
182	C₄×D₁₃	—	2	2	no	—
183	C₄×D₁₃	—	2	2	no	—
184	C₄×D₁₃	—	16	2	no	—
185	Dic₂₆	—	2	1	no	—
186	Dic₂₆	—	16	2	no	—
187	Dic₂₆	C₄⋊Q₈	1	2	yes	—
188	C₈×C₁₃	—	2	2	no	—
189	C₈×C₁₃	—	4	2	no	—
190	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
191	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
192	#1055	—	2	2	no	—
193	#1055	—	2	2	no	—
194	C₂×#131	—	13	1	no	—
195	#544	—	13	2	no	—
196	#520	—	13	2	no	—
197	#501	—	26	2	no	—
198	#499	—	26	1	no	—
199	#498	—	26	1	no	—
200	#479	—	26	2	no	—
201	#478	—	26	2	no	—
202	#477	—	26	2	no	—
203	#474	—	26	2	no	—
204	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
205	#428	—	13	2	no	—
206	#428	—	13	2	no	—
207	#428	—	13	2	no	—
208	#428	—	26	2	no	—
209	#427	—	26	1	no	—
210	#427	C₂×D₁₃	1	1	yes	characteristic
211	C₂×#107	—	13	2	no	—
212	C₂×#106	—	13	1	no	—
213	C₂×#106	—	13	1	no	—
214	C₂×#106	—	26	1	no	—
215	#424	C₂×D₁₃	1	2	yes	—
216	C₂×#104	—	13	1	no	—
217	#417	—	13	2	no	—
218	#416	—	13	1	no	—
219	#405	—	13	1	no	—
220	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
221	#382	—	13	2	no	—
222	#343	—	13	2	no	—
223	#343	—	13	2	no	—
224	#341	—	13	1	no	—
225	#339	—	13	1	no	—
226	#339	C₂×D₁₃	1	2	yes	—
227	C₂²×C₁₃	—	2	2	no	—
228	C₂²×C₁₃	#544	1	1	yes	characteristic
229	C₂²×C₁₃	D₄²	1	1	yes	characteristic
230	C₂²×C₁₃	#554	1	1	yes	characteristic
231	C₂×D₁₃	—	2	2	no	—
232	C₂×D₁₃	—	4	2	no	—
233	C₄×C₁₃	—	4	2	no	—
234	C₄×C₁₃	—	4	2	no	—
235	C₄×C₁₃	—	4	2	no	—
236	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
237	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
238	C₄×C₁₃	#547	1	1	yes	characteristic
239	C₄×C₁₃	D₄×Q₈	1	1	yes	characteristic
240	Dic₁₃	—	2	2	no	—
241	Dic₁₃	—	4	2	no	—
242	Dic₁₃	—	8	2	no	—
243	Dic₁₃	—	8	2	no	—
244	C₂×#41	—	13	1	no	—
245	#131	—	13	4	no	—
246	#131	—	26	2	no	—
247	C₂×M₁₆	—	13	1	no	—
248	C₂×M₁₆	—	13	1	no	—
249	C₂×M₁₆	—	26	1	no	—
250	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
251	C₂²×C₈	—	26	1	no	—
252	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
253	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
254	C₄⋊Q₈	—	26	2	no	—
255	C₄⋊Q₈	—	26	2	no	—
256	#125	—	26	2	no	—
257	#125	—	26	2	no	—
258	C₂²⋊Q₈	—	52	2	no	—
259	C₄×Q₈	—	26	2	no	—
260	C₄×D₄	—	26	2	no	—
261	C₄×D₄	—	52	2	no	—
262	#117	—	13	1	no	—
263	#117	—	26	2	no	—
264	#117	C₂²×D₁₃	1	2	yes	—
265	C₂×C₄⋊C₄	—	13	2	no	—
266	C₂×C₄⋊C₄	—	13	2	no	—
267	C₂×C₄⋊C₄	—	26	2	no	—
268	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
269	#108	—	2	2	no	—
270	#107	—	26	2	no	—
271	#107	—	26	2	no	—
272	#107	—	26	2	no	—
273	#107	—	26	2	no	—
274	#107	—	26	2	no	—
275	#107	D₅₂	1	2	yes	—
276	#106	—	26	2	no	—
277	#106	—	26	1	no	—
278	#106	—	26	1	no	—
279	#106	—	26	1	no	—
280	#106	—	26	1	no	—
281	#106	—	26	2	no	—
282	#106	—	26	2	no	—
283	#106	D₅₂	1	1	yes	characteristic
284	#106	D₅₂	1	1	yes	characteristic
285	C₄⋊C₈	—	2	2	no	—
286	C₄⋊C₈	—	2	2	no	—
287	#104	—	13	4	no	—
288	#103	—	13	2	no	—
289	#103	—	26	2	no	—
290	#102	—	13	2	no	—
291	#102	—	26	2	no	—
292	#97	—	26	1	no	—
293	C₄×C₈	—	2	2	no	—
294	C₄×C₈	—	26	1	no	—
295	#95	—	52	2	no	—
296	C₂×C₁₃	—	2	1	no	—
297	C₂×C₁₃	—	2	1	no	—
298	C₂×C₁₃	#2440	1	1	yes	characteristic
299	C₂×C₁₃	#3290	1	2	yes	—
300	D₁₃	—	4	2	no	—
301	#41	—	13	2	no	—
302	#41	—	13	2	no	—
303	#41	—	26	1	no	—
304	#41	—	26	1	no	—
305	C₂×Q₈	—	13	1	no	—
306	C₂×Q₈	—	52	2	no	—
307	C₂×D₄	—	13	1	no	—
308	C₂×D₄	—	26	1	no	—
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	26	1	no	—
311	C₂²×C₄	—	26	2	no	—
312	C₂²×C₄	—	26	2	no	—
313	C₂²×C₄	—	52	2	no	—
314	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
315	M₁₆	—	2	2	no	—
316	M₁₆	—	2	2	no	—
317	M₁₆	—	13	2	no	—
318	M₁₆	—	13	2	no	—
319	M₁₆	—	26	1	no	—
320	M₁₆	—	26	1	no	—
321	M₁₆	—	26	4	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	26	1	no	—
326	C₂×C₈	—	26	1	no	—
327	C₂×C₈	—	26	4	no	—
328	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
329	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
330	C₂×C₈	D₄×D₁₃	1	2	yes	—
331	C₂×C₈	#6129	1	2	yes	—
332	C₂×C₈	Q₈×D₁₃	1	2	yes	—
333	C₄⋊C₄	—	13	2	no	—
334	C₄⋊C₄	—	26	2	no	—
335	C₄⋊C₄	—	26	2	no	—
336	C₄⋊C₄	—	26	2	no	—
337	C₄⋊C₄	—	26	2	no	—
338	C₄⋊C₄	—	52	2	no	—
339	C₄⋊C₄	—	52	2	no	—
340	C₄⋊C₄	—	52	2	no	—
341	C₄⋊C₄	—	52	2	no	—
342	C₄⋊C₄	C₂×D₅₂	1	2	yes	—
343	C₄⋊C₄	C₂×D₅₂	1	2	yes	—
344	C₂²⋊C₄	—	2	2	no	—
345	C₂²⋊C₄	—	26	1	no	—
346	C₂²⋊C₄	—	52	2	no	—
347	C₄²	—	2	2	no	—
348	C₄²	—	2	1	no	—
349	C₄²	—	26	1	no	—
350	C₄²	—	52	2	no	—
351	C₄²	Q₈×D₁₃	1	2	yes	—
352	C₁₃	#26080	1	1	yes	characteristic, complemented, Sylow
353	C₂³	—	26	1	no	—
354	C₂³	#88256	1	1	yes	characteristic
355	Q₈	—	13	2	no	—
356	Q₈	—	26	1	no	—
357	Q₈	—	208	2	no	—
358	D₄	—	13	2	no	—
359	D₄	—	26	2	no	—
360	D₄	—	26	2	no	—
361	D₄	—	26	1	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	26	1	no	—
367	C₂×C₄	—	26	2	no	—
368	C₂×C₄	—	26	2	no	—
369	C₂×C₄	—	26	2	no	—
370	C₂×C₄	—	52	2	no	—
371	C₂×C₄	—	52	2	no	—
372	C₂×C₄	—	104	2	no	—
373	C₂×C₄	—	104	2	no	—
374	C₂×C₄	—	208	2	no	—
375	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
376	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
377	C₂×C₄	C₂×#6129	1	1	yes	characteristic
378	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
379	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
380	C₂×C₄	#88257	1	1	yes	characteristic
381	C₈	—	2	2	no	—
382	C₈	—	4	2	no	—
383	C₈	—	26	2	no	—
384	C₈	—	26	2	no	—
385	C₈	#88151	1	2	yes	—
386	C₈	#88151	1	2	yes	—
387	C₂²	—	2	2	no	—
388	C₂²	—	26	2	no	—
389	C₂²	—	52	2	no	—
390	C₂²	D₄×D₅₂	1	1	yes	characteristic
391	C₂²	#11719340	1	1	yes	characteristic
392	C₂²	#11719394	1	1	yes	characteristic
393	C₄	—	4	2	no	—
394	C₄	—	4	2	no	—
395	C₄	—	4	2	no	—
396	C₄	—	26	2	no	—
397	C₄	—	52	2	no	—
398	C₄	—	104	2	no	—
399	C₄	—	104	2	no	—
400	C₄	C₂×#88151	1	1	yes	characteristic
401	C₄	Q₈×D₅₂	1	1	yes	characteristic
402	C₄	#11719316	1	1	yes	characteristic
403	C₄	D₁₃×C₄⋊Q₈	1	1	yes	characteristic
404	C₂	—	2	1	no	—
405	C₂	—	2	1	no	—
406	C₂	—	52	2	no	—
407	C₂	#13118536	1	1	yes	characteristic
408	C₂	#13126623	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₅₂
commutator, cosocle	C₂²×C₄×C₁₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#1945	C₂
socle	C₂²×C₁₃	D₄²
prime radical	C₂×#6127	C₂³
hypercenter	#1945	D₁₃
hypocenter	C₁₃	#26080
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1945

Complements

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

N	H ≅ G/N	type
1. #13125368	C₂	semidirect
4. #13116341	C₂	semidirect
5. #13116185	C₂	semidirect
6. #13115617	C₂	semidirect
8. #13115000	C₂	semidirect
11. C₁₃×#1945	C₂	semidirect
57. #88155	Q₈	semidirect
96. #88045	Q₈	semidirect
155. #1945	D₁₃	semidirect
352. C₁₃	#26080	semidirect

Maximal subgroups

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

subgroup	index
#13125368	2
#13124689	2
#13123428	2
#13116341	2
#13116185	2
#13115617	2
#13115070	2
#13115000	2
#13114892	2
#13114673	2
C₁₃×#1945	2
#26080	13