Subgroups — #15921813

Order 3328.

690 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13124694	C₂	1	2	yes	maximal
2	#13123428	C₂	1	1	yes	maximal, characteristic
3	#13116349	C₂	1	2	yes	maximal, complemented
4	#13116178	C₂	1	2	yes	maximal, complemented
5	#13115616	C₂	1	1	yes	maximal, characteristic, complemented
6	#13115585	C₂	1	1	yes	maximal, characteristic, complemented
7	#13115059	C₂	1	2	yes	maximal
8	#13114991	C₂	1	1	yes	maximal, characteristic, complemented
9	#13114896	C₂	1	1	yes	maximal, characteristic
10	#13114675	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#3163	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#544	C₂²	1	2	yes	—
13	C₂×#88155	C₂²	1	1	yes	characteristic
14	#11719228	C₂²	1	2	yes	—
15	C₁₃×#501	—	2	2	no	—
16	C₁₃×#499	—	2	1	no	—
17	C₁₃×#498	—	2	1	no	—
18	C₁₃×#428	C₂²	1	2	yes	—
19	C₁₃×#427	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
21	C₁₃×#405	C₂²	1	1	yes	characteristic
22	C₂×#88065	C₂²	1	1	yes	characteristic
23	C₂×#88059	C₂²	1	1	yes	characteristic
24	C₂×#88057	C₂²	1	1	yes	characteristic
25	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
26	#11718755	—	2	2	no	—
27	#11718745	—	2	2	no	—
28	#11718727	—	2	2	no	—
29	#11718724	—	2	2	no	—
30	#11718663	C₂²	1	1	yes	characteristic
31	#11718660	C₂²	1	2	yes	—
32	#11718656	C₂²	1	2	yes	—
33	#11718656	C₂²	1	2	yes	—
34	#11718637	C₂²	1	2	yes	—
35	#11718627	C₂²	1	2	yes	—
36	C₂×#88048	C₂²	1	2	yes	—
37	#11718542	—	2	2	no	—
38	#11718521	—	2	1	no	—
39	D₁₃×#106	—	2	1	no	—
40	#11718150	C₂²	1	1	yes	characteristic
41	#11718146	C₂²	1	2	yes	—
42	#11718124	C₂²	1	1	yes	characteristic
43	#11718124	C₂²	1	2	yes	—
44	#11718122	C₂²	1	2	yes	—
45	C₂×#6127	C₂³	1	1	yes	characteristic
46	C₂×C₁₃×M₁₆	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
48	C₁₃×C₄⋊Q₈	—	2	2	no	—
49	C₁₃×C₄⋊Q₈	—	2	2	no	—
50	C₁₃×#125	—	2	2	no	—
51	C₁₃×#125	—	2	2	no	—
52	C₁₃×C₂²⋊Q₈	—	4	2	no	—
53	C₁₃×#117	C₂³	1	1	yes	characteristic
54	C₁₃×#117	C₂³	1	2	yes	—
55	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
56	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
57	C₂×#6101	—	4	2	no	—
58	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
59	C₂×#6099	C₂³	1	2	yes	—
60	C₂×#6099	C₂³	1	1	yes	characteristic
61	C₂²×#1055	C₂³	1	1	yes	characteristic
62	#88155	—	2	2	no	—
63	#88155	Q₈	1	4	yes	complemented
64	C₂×#6094	—	2	1	no	—
65	C₂×C₈×D₁₃	—	2	1	no	—
66	#88146	—	2	2	no	—
67	C₄⋊C₄×D₁₃	—	2	2	no	—
68	#88135	—	4	2	no	—
69	C₄×D₅₂	—	2	2	no	—
70	C₄×Dic₂₆	—	2	2	no	—
71	C₁₃×#107	—	2	2	no	—
72	C₁₃×#107	D₄	1	2	yes	—
73	C₁₃×#106	—	2	1	no	—
74	C₁₃×#106	—	2	1	no	—
75	C₁₃×#106	—	2	2	no	—
76	C₁₃×#106	D₄	1	1	yes	characteristic
77	C₁₃×#106	D₄	1	1	yes	characteristic
78	C₁₃×#97	—	2	1	no	—
79	C₄×C₈×C₁₃	—	2	1	no	—
80	#88065	Q₈	1	4	yes	complemented
81	#88059	—	2	2	no	—
82	#88057	—	2	1	no	—
83	#88057	—	2	1	no	—
84	C₈×Dic₁₃	—	2	2	no	—
85	#88050	—	2	2	no	—
86	#88050	D₄	1	2	yes	—
87	#88049	—	2	2	no	—
88	#88049	D₄	1	2	yes	—
89	#88048	—	2	2	no	—
90	#88048	—	2	2	no	—
91	#88048	—	2	2	no	—
92	#88048	—	2	2	no	—
93	#88047	—	2	2	no	—
94	#88047	—	2	2	no	—
95	C₄⋊#1055	—	2	2	no	—
96	C₄⋊#1055	—	2	2	no	—
97	#26081	—	13	1	no	maximal, Sylow
98	C₂×Q₈×C₁₃	—	4	2	no	—
99	C₂²×C₄×C₁₃	—	2	2	no	—
100	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
101	C₂×C₂²⋊D₁₃	—	2	1	no	—
102	C₂²×Dic₁₃	—	2	1	no	—
103	C₂²×Dic₁₃	—	2	2	no	—
104	#6127	—	2	1	no	—
105	#6127	—	2	1	no	—
106	#6127	C₂×Q₈	1	2	yes	—
107	#6127	C₂×Q₈	1	2	yes	—
108	C₂×D₅₂	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×D₁₃	—	2	1	no	—
110	C₂×C₄×D₁₃	—	4	2	no	—
111	C₂×Dic₂₆	C₂×D₄	1	1	yes	characteristic
112	C₁₃×M₁₆	—	2	1	no	—
113	C₁₃×M₁₆	C₂×Q₈	1	2	yes	—
114	C₂×C₈×C₁₃	—	2	1	no	—
115	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
118	C₂×C₈×C₁₃	#41	1	2	yes	—
119	C₁₃×C₄⋊C₄	—	2	2	no	—
120	C₁₃×C₄⋊C₄	—	2	2	no	—
121	C₁₃×C₄⋊C₄	—	4	2	no	—
122	C₁₃×C₄⋊C₄	—	4	2	no	—
123	C₁₃×C₄⋊C₄	—	4	2	no	—
124	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
125	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
126	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
127	C₁₃×C₂²⋊C₄	—	2	2	no	—
128	C₁₃×C₂²⋊C₄	—	2	1	no	—
129	C₄²×C₁₃	—	2	1	no	—
130	C₄²×C₁₃	—	2	2	no	—
131	#6103	—	4	2	no	—
132	C₄⋊Dic₁₃	—	2	2	no	—
133	C₄⋊Dic₁₃	—	2	2	no	—
134	#6101	—	4	2	no	—
135	#6101	—	8	4	no	—
136	C₄×Dic₁₃	—	2	1	no	—
137	C₄×Dic₁₃	—	4	2	no	—
138	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
139	#6099	—	2	1	no	—
140	#6099	—	2	2	no	—
141	#6099	—	2	2	no	—
142	#6099	C₂×Q₈	1	2	yes	—
143	C₂×#1055	—	2	1	no	—
144	C₂×#1055	—	2	1	no	—
145	C₂×#1055	—	2	1	no	—
146	C₂×#1055	—	2	2	no	—
147	C₂×#1055	C₂×D₄	1	2	yes	—
148	#6094	—	2	4	no	—
149	C₈×D₁₃	—	2	4	no	—
150	#3163	D₁₃	1	1	yes	characteristic, complemented
151	#3084	—	13	2	no	—
152	#2912	—	13	1	no	—
153	#2090	—	13	2	no	—
154	#2060	—	13	2	no	—
155	#1951	—	13	1	no	—
156	#1945	—	13	1	no	—
157	#1859	—	13	2	no	—
158	#1844	—	13	1	no	—
159	#1815	—	13	1	no	—
160	#1764	—	13	1	no	—
161	C₂³×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂²×D₁₃	—	2	1	no	—
163	Q₈×C₁₃	—	16	2	no	—
164	C₂×C₄×C₁₃	—	2	2	no	—
165	C₂×C₄×C₁₃	—	2	2	no	—
166	C₂×C₄×C₁₃	—	2	2	no	—
167	C₂×C₄×C₁₃	—	4	2	no	—
168	C₂×C₄×C₁₃	—	8	2	no	—
169	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
172	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
173	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
174	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
175	C₂²⋊D₁₃	—	2	2	no	—
176	C₂²⋊D₁₃	—	2	2	no	—
177	C₂×Dic₁₃	—	2	4	no	—
178	C₂×Dic₁₃	—	2	1	no	—
179	C₂×Dic₁₃	—	2	2	no	—
180	C₂×Dic₁₃	—	4	2	no	—
181	C₂×Dic₁₃	—	8	2	no	—
182	D₅₂	—	2	1	no	—
183	D₅₂	C₄⋊Q₈	1	2	yes	—
184	C₄×D₁₃	—	2	2	no	—
185	C₄×D₁₃	—	2	2	no	—
186	C₄×D₁₃	—	16	2	no	—
187	Dic₂₆	—	2	1	no	—
188	Dic₂₆	C₄⋊Q₈	1	2	yes	—
189	C₈×C₁₃	—	2	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	#1055	—	4	2	no	—
195	C₂×#131	—	13	1	no	—
196	#544	C₂×D₁₃	1	2	yes	—
197	#520	—	13	2	no	—
198	#501	—	2	2	no	—
199	#499	—	2	1	no	—
200	#498	—	2	1	no	—
201	#479	—	26	2	no	—
202	#478	—	26	2	no	—
203	#476	—	26	2	no	—
204	#475	—	26	2	no	—
205	C₂×#108	—	13	1	no	—
206	#428	—	13	2	no	—
207	#428	—	13	2	no	—
208	#428	—	26	2	no	—
209	#428	C₂×D₁₃	1	2	yes	—
210	#427	—	26	1	no	—
211	#427	C₂×D₁₃	1	1	yes	characteristic
212	C₂×#106	—	13	1	no	—
213	C₂×#106	—	13	2	no	—
214	C₂×#106	—	26	1	no	—
215	C₂×#106	C₂×D₁₃	1	1	yes	characteristic
216	#424	—	13	2	no	—
217	C₂×#104	—	13	1	no	—
218	#417	—	13	2	no	—
219	#416	—	13	1	no	—
220	#405	C₂×D₁₃	1	1	yes	characteristic
221	C₂×C₄×C₈	—	13	1	no	—
222	#384	—	13	2	no	—
223	#343	—	13	2	no	—
224	#343	—	13	2	no	—
225	#341	—	13	1	no	—
226	#339	—	13	1	no	—
227	#339	—	13	2	no	—
228	C₂²×C₁₃	—	2	2	no	—
229	C₂²×C₁₃	#544	1	1	yes	characteristic
230	C₂²×C₁₃	D₄²	1	1	yes	characteristic
231	C₂²×C₁₃	#554	1	1	yes	characteristic
232	C₂×D₁₃	—	2	2	no	—
233	C₂×D₁₃	—	4	2	no	—
234	C₄×C₁₃	—	4	2	no	—
235	C₄×C₁₃	—	4	2	no	—
236	C₄×C₁₃	—	4	2	no	—
237	C₄×C₁₃	—	8	2	no	—
238	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
239	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
240	C₄×C₁₃	#547	1	1	yes	characteristic
241	C₄×C₁₃	D₄×Q₈	1	1	yes	characteristic
242	Dic₁₃	—	2	4	no	—
243	Dic₁₃	—	2	2	no	—
244	Dic₁₃	—	8	2	no	—
245	C₂×#41	—	13	1	no	—
246	#131	—	13	4	no	—
247	#131	—	26	2	no	—
248	C₂×M₁₆	—	13	1	no	—
249	C₂×M₁₆	—	13	2	no	—
250	C₂×M₁₆	—	26	1	no	—
251	C₂×M₁₆	C₂²×D₁₃	1	1	yes	characteristic
252	C₂²×C₈	—	13	1	no	—
253	C₂²×C₈	—	26	1	no	—
254	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
255	C₄⋊Q₈	—	2	2	no	—
256	C₄⋊Q₈	—	2	2	no	—
257	#125	—	2	2	no	—
258	#125	—	2	2	no	—
259	C₂²⋊Q₈	—	4	2	no	—
260	C₄×Q₈	—	26	2	no	—
261	C₄×D₄	—	26	2	no	—
262	C₄×D₄	—	52	2	no	—
263	#117	—	26	2	no	—
264	#117	C₂²×D₁₃	1	1	yes	characteristic
265	#117	C₂²×D₁₃	1	2	yes	—
266	C₂×C₄⋊C₄	—	13	2	no	—
267	C₂×C₄⋊C₄	—	26	2	no	—
268	C₂×C₄⋊C₄	—	52	2	no	—
269	C₂×C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
270	C₂×C₄²	—	13	1	no	—
271	#108	—	26	2	no	—
272	#107	—	2	2	no	—
273	#107	—	26	2	no	—
274	#107	—	26	2	no	—
275	#107	C₂²⋊D₁₃	1	2	yes	—
276	#106	—	2	1	no	—
277	#106	—	2	1	no	—
278	#106	—	2	2	no	—
279	#106	—	26	2	no	—
280	#106	—	26	2	no	—
281	#106	—	26	1	no	—
282	#106	—	26	1	no	—
283	#106	—	26	2	no	—
284	#106	—	26	2	no	—
285	#106	C₂²⋊D₁₃	1	1	yes	characteristic
286	#106	C₂²⋊D₁₃	1	1	yes	characteristic
287	C₄⋊C₈	—	26	2	no	—
288	C₄⋊C₈	—	26	2	no	—
289	#104	—	13	4	no	—
290	#103	—	13	2	no	—
291	#103	—	26	2	no	—
292	#102	—	13	2	no	—
293	#102	—	26	2	no	—
294	#97	—	2	1	no	—
295	C₄×C₈	—	2	1	no	—
296	C₄×C₈	—	26	2	no	—
297	C₂×C₁₃	—	2	1	no	—
298	C₂×C₁₃	—	2	1	no	—
299	C₂×C₁₃	#2440	1	1	yes	characteristic
300	C₂×C₁₃	#3291	1	2	yes	—
301	D₁₃	—	4	2	no	—
302	#41	—	13	2	no	—
303	#41	—	13	2	no	—
304	#41	—	26	1	no	—
305	#41	—	26	1	no	—
306	C₂×Q₈	—	4	2	no	—
307	C₂×Q₈	—	13	1	no	—
308	C₂×D₄	—	13	1	no	—
309	C₂×D₄	—	26	1	no	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	26	1	no	—
312	C₂²×C₄	—	26	1	no	—
313	C₂²×C₄	—	26	2	no	—
314	C₂²×C₄	—	52	2	no	—
315	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
316	M₁₆	—	2	1	no	—
317	M₁₆	—	13	2	no	—
318	M₁₆	—	26	1	no	—
319	M₁₆	—	26	4	no	—
320	M₁₆	—	26	2	no	—
321	M₁₆	—	26	2	no	—
322	M₁₆	Q₈×D₁₃	1	2	yes	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	13	2	no	—
325	C₂×C₈	—	26	1	no	—
326	C₂×C₈	—	26	1	no	—
327	C₂×C₈	—	26	1	no	—
328	C₂×C₈	—	26	4	no	—
329	C₂×C₈	—	26	2	no	—
330	C₂×C₈	Q₈×D₁₃	1	2	yes	—
331	C₂×C₈	#6131	1	2	yes	—
332	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
333	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	4	2	no	—
337	C₄⋊C₄	—	4	2	no	—
338	C₄⋊C₄	—	4	2	no	—
339	C₄⋊C₄	—	26	2	no	—
340	C₄⋊C₄	—	26	2	no	—
341	C₄⋊C₄	—	52	2	no	—
342	C₄⋊C₄	—	104	4	no	—
343	C₄⋊C₄	D₄×D₁₃	1	2	yes	—
344	C₄⋊C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
345	C₄⋊C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
346	C₂²⋊C₄	—	2	2	no	—
347	C₂²⋊C₄	—	2	1	no	—
348	C₂²⋊C₄	—	52	2	no	—
349	C₄²	—	2	2	no	—
350	C₄²	—	2	1	no	—
351	C₄²	—	13	2	no	—
352	C₄²	—	26	1	no	—
353	C₄²	—	52	2	no	—
354	C₁₃	#26081	1	1	yes	characteristic, complemented, Sylow
355	C₂³	—	26	1	no	—
356	C₂³	#88251	1	1	yes	characteristic
357	Q₈	—	13	2	no	—
358	Q₈	—	16	2	no	—
359	Q₈	—	26	1	no	—
360	D₄	—	13	2	no	—
361	D₄	—	26	2	no	—
362	D₄	—	26	2	no	—
363	D₄	—	26	1	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	8	2	no	—
369	C₂×C₄	—	26	4	no	—
370	C₂×C₄	—	26	2	no	—
371	C₂×C₄	—	26	1	no	—
372	C₂×C₄	—	26	2	no	—
373	C₂×C₄	—	26	2	no	—
374	C₂×C₄	—	52	2	no	—
375	C₂×C₄	—	104	2	no	—
376	C₂×C₄	—	208	2	no	—
377	C₂×C₄	C₂×D₄×D₁₃	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	C₂×#6131	1	1	yes	characteristic
381	C₂×C₄	#88254	1	1	yes	characteristic
382	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
383	C₈	—	2	2	no	—
384	C₈	—	26	2	no	—
385	C₈	—	26	2	no	—
386	C₈	—	52	2	no	—
387	C₈	#88200	1	2	yes	—
388	C₈	#88200	1	2	yes	—
389	C₂²	—	2	2	no	—
390	C₂²	—	26	2	no	—
391	C₂²	—	52	2	no	—
392	C₂²	#11719434	1	1	yes	characteristic
393	C₂²	#11719435	1	1	yes	characteristic
394	C₂²	D₄×C₂²⋊D₁₃	1	1	yes	characteristic
395	C₄	—	4	2	no	—
396	C₄	—	4	2	no	—
397	C₄	—	4	2	no	—
398	C₄	—	8	2	no	—
399	C₄	—	26	4	no	—
400	C₄	—	26	2	no	—
401	C₄	—	104	2	no	—
402	C₄	D₁₃×C₄⋊Q₈	1	1	yes	characteristic
403	C₄	#11719431	1	1	yes	characteristic
404	C₄	C₂×#88200	1	1	yes	characteristic
405	C₄	Q₈×C₂²⋊D₁₃	1	1	yes	characteristic
406	C₂	—	2	1	no	—
407	C₂	—	2	1	no	—
408	C₂	—	52	2	no	—
409	C₂	#13118534	1	1	yes	characteristic
410	C₂	#13126627	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₁₃
commutator, cosocle	C₂²×C₄×C₁₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#3163	C₂
socle	C₂²×C₁₃	D₄²
prime radical	C₂×#6127	C₂³
hypercenter	#3163	D₁₃
hypocenter	C₁₃	#26081
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #3163

Complements

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

N	H ≅ G/N	type
3. #13116349	C₂	semidirect
4. #13116178	C₂	semidirect
5. #13115616	C₂	semidirect
6. #13115585	C₂	semidirect
8. #13114991	C₂	semidirect
11. C₁₃×#3163	C₂	semidirect
63. #88155	Q₈	semidirect
80. #88065	Q₈	semidirect
150. #3163	D₁₃	semidirect
354. C₁₃	#26081	semidirect

Maximal subgroups

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

subgroup	index
#13124694	2
#13123428	2
#13116349	2
#13116178	2
#13115616	2
#13115585	2
#13115059	2
#13114991	2
#13114896	2
#13114675	2
C₁₃×#3163	2
#26081	13