Subgroups — #15846060

Order 3328.

714 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125966	C₂	1	1	yes	maximal, characteristic, complemented
2	#13125756	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718355	C₂	1	1	yes	maximal, characteristic, complemented
4	#13123549	C₂	1	1	yes	maximal, characteristic, complemented
5	#13114370	C₂	1	4	yes	maximal, complemented
6	#13114246	C₂	1	4	yes	maximal, complemented
7	#13113516	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113503	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#88155	—	2	1	no	—
11	#11719182	C₂²	1	1	yes	characteristic
12	C₂×#88128	C₂²	1	1	yes	characteristic
13	C₂×#88123	C₂²	1	1	yes	characteristic
14	C₁₃×#431	C₂²	1	1	yes	characteristic
15	C₁₃×#431	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×#88061	—	2	1	no	—
18	C₂×#88061	—	2	1	no	—
19	#11718771	—	2	1	no	—
20	#11718585	C₂²	1	1	yes	characteristic
21	#11718504	C₂²	1	4	yes	—
22	#11718503	C₂²	1	4	yes	complemented
23	#11718396	—	2	2	no	—
24	#11718393	—	2	2	no	—
25	#11718356	—	2	1	no	—
26	#11718355	—	2	1	no	—
27	#11718355	—	2	1	no	—
28	#11718355	—	2	1	no	—
29	#11718355	—	2	1	no	—
30	#11718355	C₂²	1	2	yes	—
31	#11718355	C₂²	1	2	yes	—
32	#11718349	—	2	1	no	—
33	#11718349	C₂²	1	2	yes	complemented
34	C₈×D₅₂	—	2	1	no	—
35	C₈×D₅₂	C₂²	1	2	yes	complemented
36	#11718341	—	2	1	no	—
37	C₁₃×#335	C₂²	1	2	yes	complemented
38	C₁₃×#334	C₂²	1	2	yes	complemented
39	#11718077	C₂²	1	4	yes	—
40	#11718073	C₂²	1	4	yes	complemented
41	C₂×#6127	—	2	1	no	—
42	C₂²×D₅₂	—	2	1	no	—
43	C₂²×D₅₂	—	2	1	no	—
44	C₂²×D₅₂	—	2	1	no	—
45	C₂²×Dic₂₆	—	2	1	no	—
46	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	#88184	—	2	1	no	—
50	#88184	D₄	1	2	yes	—
51	C₂×C₄⋊Dic₁₃	—	2	1	no	—
52	C₂×C₄⋊Dic₁₃	—	2	1	no	—
53	C₂×#6099	—	2	1	no	—
54	C₂×D₁₀₄	—	2	4	no	—
55	#88155	—	4	4	no	—
56	C₂×#6094	—	2	2	no	—
57	C₂×C₈×D₁₃	—	2	2	no	—
58	#88129	—	2	1	no	—
59	#88128	—	2	1	no	—
60	#88128	—	2	1	no	—
61	#88128	—	2	1	no	—
62	#88128	D₄	1	1	yes	characteristic
63	#88128	C₂³	1	2	yes	—
64	C₄×D₅₂	—	2	1	no	—
65	C₄×D₅₂	C₂³	1	2	yes	—
66	#88123	—	2	1	no	—
67	#88123	—	2	1	no	—
68	#88123	—	2	1	no	—
69	#88123	D₄	1	1	yes	characteristic
70	#88123	C₂³	1	2	yes	—
71	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
72	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
73	C₁₃×C₂²⋊C₈	—	2	1	no	—
74	C₁₃×C₂²⋊C₈	—	2	1	no	—
75	C₄×C₈×C₁₃	—	2	1	no	—
76	C₄×C₈×C₁₃	—	2	1	no	—
77	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
78	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
79	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
81	#88061	—	2	4	no	—
82	#88061	—	2	2	no	—
83	#88061	—	2	2	no	—
84	#88061	—	2	2	no	—
85	#88061	—	2	2	no	—
86	#88061	—	2	4	no	—
87	#88060	—	2	2	no	—
88	#88060	—	2	2	no	—
89	#88057	—	2	4	no	—
90	C₄⋊#1055	—	2	1	no	—
91	C₄⋊#1055	C₂³	1	2	yes	—
92	#19196	—	13	1	no	maximal, Sylow
93	C₂³×D₁₃	—	4	1	no	—
94	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
97	C₂×C₂²⋊D₁₃	—	2	2	no	—
98	C₂²×Dic₁₃	—	4	1	no	—
99	#6127	—	4	2	no	—
100	#6127	—	4	2	no	—
101	C₂×D₅₂	—	2	1	no	—
102	C₂×D₅₂	—	2	1	no	—
103	C₂×D₅₂	—	2	2	no	—
104	C₂×D₅₂	—	2	2	no	—
105	C₂×D₅₂	—	2	2	no	—
106	C₂×D₅₂	—	4	2	no	—
107	C₂×D₅₂	—	4	2	no	—
108	C₂×D₅₂	—	4	1	no	—
109	C₂×D₅₂	—	4	1	no	—
110	C₂×D₅₂	—	4	1	no	—
111	C₂×D₅₂	—	8	2	no	—
112	C₂×D₅₂	—	8	2	no	—
113	C₂×D₅₂	C₂×D₄	1	2	yes	—
114	C₂×C₄×D₁₃	—	2	2	no	—
115	C₂×Dic₂₆	—	2	1	no	—
116	C₂×Dic₂₆	—	2	2	no	—
117	C₂×Dic₂₆	—	4	1	no	—
118	C₂×Dic₂₆	—	8	2	no	—
119	C₂×C₈×C₁₃	—	2	2	no	—
120	C₂×C₈×C₁₃	—	2	2	no	—
121	C₂×C₈×C₁₃	—	2	2	no	—
122	C₂×C₈×C₁₃	—	2	2	no	—
123	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₁₃	#41	1	1	yes	characteristic
130	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
131	#6103	—	2	2	no	—
132	#6103	—	2	2	no	—
133	C₄⋊Dic₁₃	—	2	1	no	—
134	C₄⋊Dic₁₃	—	2	2	no	—
135	C₄⋊Dic₁₃	—	2	2	no	—
136	C₄⋊Dic₁₃	—	4	1	no	—
137	C₄⋊Dic₁₃	—	4	1	no	—
138	C₄⋊Dic₁₃	C₂×D₄	1	2	yes	—
139	#6099	—	4	2	no	—
140	C₂×#1055	—	2	2	no	—
141	D₁₀₄	—	8	4	no	—
142	#6094	—	4	4	no	—
143	C₈×D₁₃	—	4	4	no	—
144	#3227	—	13	1	no	—
145	#3205	—	13	1	no	—
146	C₂×#486	—	13	1	no	—
147	#2931	—	13	1	no	—
148	#1697	—	13	4	no	—
149	#1677	—	13	4	no	—
150	#1564	D₁₃	1	1	yes	characteristic, complemented
151	#1548	—	13	1	no	—
152	#1545	—	13	1	no	—
153	C₂³×C₁₃	#121	1	1	yes	characteristic
154	C₂²×D₁₃	—	2	2	no	—
155	C₂²×D₁₃	—	4	2	no	—
156	C₂²×D₁₃	—	8	2	no	—
157	C₂²×D₁₃	—	8	2	no	—
158	C₂²×D₁₃	—	8	2	no	—
159	C₂×C₄×C₁₃	—	2	2	no	—
160	C₂×C₄×C₁₃	—	2	2	no	—
161	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
162	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
163	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
164	C₂×C₄×C₁₃	#135	1	2	yes	—
165	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
166	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
169	C₂²⋊D₁₃	—	4	4	no	—
170	C₂×Dic₁₃	—	2	2	no	—
171	C₂×Dic₁₃	—	4	2	no	—
172	C₂×Dic₁₃	—	8	2	no	—
173	D₅₂	—	2	4	no	—
174	D₅₂	—	4	2	no	—
175	D₅₂	—	4	2	no	—
176	D₅₂	—	4	4	no	—
177	D₅₂	—	4	2	no	—
178	D₅₂	—	4	2	no	—
179	D₅₂	—	8	2	no	—
180	D₅₂	—	8	2	no	—
181	D₅₂	—	8	2	no	—
182	D₅₂	—	8	2	no	—
183	C₄×D₁₃	—	4	2	no	—
184	C₄×D₁₃	—	4	2	no	—
185	Dic₂₆	—	4	2	no	—
186	Dic₂₆	—	8	2	no	—
187	Dic₂₆	—	8	2	no	—
188	C₈×C₁₃	—	2	1	no	—
189	C₈×C₁₃	—	2	1	no	—
190	C₈×C₁₃	—	2	1	no	—
191	C₈×C₁₃	—	2	1	no	—
192	C₈×C₁₃	—	4	2	no	—
193	C₈×C₁₃	—	4	2	no	—
194	#1055	—	4	2	no	—
195	C₂×#131	—	26	1	no	—
196	#532	—	13	1	no	—
197	C₂×C₄⋊Q₈	—	13	1	no	—
198	C₂×#127	—	13	1	no	—
199	#498	—	26	1	no	—
200	#493	—	26	1	no	—
201	#486	—	13	2	no	—
202	#486	—	13	2	no	—
203	#486	—	26	1	no	—
204	#486	—	26	1	no	—
205	#486	—	26	1	no	—
206	#486	—	26	1	no	—
207	#481	—	26	2	no	—
208	#476	—	13	4	no	—
209	#459	—	13	4	no	—
210	#447	—	26	2	no	—
211	#436	—	13	2	no	—
212	#436	—	26	1	no	—
213	C₈×D₄	—	13	2	no	—
214	C₈×D₄	—	26	1	no	—
215	#431	C₂×D₁₃	1	1	yes	characteristic
216	#431	C₂×D₁₃	1	1	yes	characteristic
217	#423	—	13	1	no	—
218	C₂×#102	—	26	1	no	—
219	C₂×#102	—	26	1	no	—
220	#408	—	26	1	no	—
221	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
222	#335	C₂×D₁₃	1	2	yes	complemented
223	#334	C₂×D₁₃	1	2	yes	complemented
224	#333	—	13	4	no	—
225	#331	—	13	4	no	—
226	C₂²×C₁₃	—	2	1	no	—
227	C₂²×C₁₃	—	2	1	no	—
228	C₂²×C₁₃	#463	1	2	yes	—
229	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
230	C₂×D₁₃	—	4	2	no	—
231	C₂×D₁₃	—	4	2	no	—
232	C₂×D₁₃	—	4	2	no	—
233	C₂×D₁₃	—	8	2	no	—
234	C₂×D₁₃	—	8	2	no	—
235	C₂×D₁₃	—	8	2	no	—
236	C₂×D₁₃	—	16	4	no	—
237	C₄×C₁₃	—	2	1	no	—
238	C₄×C₁₃	—	2	1	no	—
239	C₄×C₁₃	#463	1	2	yes	—
240	C₄×C₁₃	#472	1	2	yes	—
241	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
242	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
243	C₄×C₁₃	#577	1	2	yes	—
244	Dic₁₃	—	4	2	no	—
245	Dic₁₃	—	8	2	no	—
246	C₂×#41	—	26	1	no	—
247	C₂²×Q₈	—	26	1	no	—
248	C₂²×D₄	—	26	1	no	—
249	C₂²×D₄	—	26	1	no	—
250	C₂²×D₄	—	26	1	no	—
251	C₂×D₈	—	26	4	no	—
252	#131	—	52	4	no	—
253	C₂×M₁₆	—	26	1	no	—
254	C₂×M₁₆	—	26	2	no	—
255	C₂²×C₈	—	26	2	no	—
256	C₂²×C₈	D₅₂	1	1	yes	characteristic
257	C₂²×C₈	D₅₂	1	1	yes	characteristic
258	C₄⋊Q₈	—	13	1	no	—
259	C₄⋊Q₈	—	13	2	no	—
260	C₄⋊Q₈	—	26	1	no	—
261	C₄⋊Q₈	—	26	1	no	—
262	C₄⋊Q₈	—	26	1	no	—
263	#127	—	13	1	no	—
264	#127	—	13	2	no	—
265	#127	—	26	1	no	—
266	#127	—	26	1	no	—
267	#127	—	26	1	no	—
268	#124	—	26	1	no	—
269	#121	—	13	2	no	—
270	#121	—	26	1	no	—
271	C₄×D₄	—	13	2	no	—
272	C₄×D₄	—	26	1	no	—
273	C₂×C₄⋊C₄	—	26	1	no	—
274	C₂×C₄⋊C₄	—	26	1	no	—
275	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
276	#106	—	26	4	no	—
277	C₄⋊C₈	—	13	2	no	—
278	C₄⋊C₈	—	26	1	no	—
279	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
280	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
281	#102	—	26	4	no	—
282	#102	—	26	4	no	—
283	#102	—	26	2	no	—
284	#102	—	26	2	no	—
285	#102	—	26	2	no	—
286	#102	—	26	2	no	—
287	C₂²⋊C₈	—	2	1	no	—
288	C₂²⋊C₈	—	2	1	no	—
289	C₂²⋊C₈	—	26	2	no	—
290	C₂²⋊C₈	—	26	2	no	—
291	C₄×C₈	—	2	1	no	—
292	C₄×C₈	—	2	1	no	—
293	C₄×C₈	D₅₂	1	1	yes	characteristic
294	C₄×C₈	D₅₂	1	1	yes	characteristic
295	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
296	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
297	C₂×C₁₃	—	2	2	no	—
298	C₂×C₁₃	C₂×#463	1	1	yes	characteristic
299	C₂×C₁₃	#3057	1	1	yes	characteristic
300	C₂×C₁₃	C₂×#472	1	1	yes	characteristic
301	D₁₃	—	8	2	no	—
302	D₁₃	—	16	2	no	—
303	C₂⁴	—	52	1	no	—
304	#41	—	52	2	no	—
305	#41	—	52	2	no	—
306	C₂×Q₈	—	26	1	no	—
307	C₂×Q₈	—	26	2	no	—
308	C₂×Q₈	—	52	1	no	—
309	C₂×Q₈	—	104	2	no	—
310	C₂×D₄	—	13	2	no	—
311	C₂×D₄	—	26	2	no	—
312	C₂×D₄	—	26	1	no	—
313	C₂×D₄	—	26	1	no	—
314	C₂×D₄	—	26	2	no	—
315	C₂×D₄	—	26	2	no	—
316	C₂×D₄	—	26	2	no	—
317	C₂×D₄	—	52	1	no	—
318	C₂×D₄	—	52	1	no	—
319	C₂×D₄	—	52	1	no	—
320	C₂×D₄	—	52	2	no	—
321	C₂×D₄	—	52	2	no	—
322	C₂×D₄	—	104	2	no	—
323	C₂×D₄	—	104	2	no	—
324	C₂²×C₄	—	26	2	no	—
325	C₂²×C₄	—	52	1	no	—
326	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
327	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
328	C₂²×C₄	#6131	1	1	yes	characteristic
329	D₈	—	104	4	no	—
330	M₁₆	—	52	4	no	—
331	M₁₆	—	52	2	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	26	2	no	—
337	C₂×C₈	—	52	4	no	—
338	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
339	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
340	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
341	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
342	C₄⋊C₄	—	13	2	no	—
343	C₄⋊C₄	—	26	1	no	—
344	C₄⋊C₄	—	26	2	no	—
345	C₄⋊C₄	—	26	2	no	—
346	C₄⋊C₄	—	52	1	no	—
347	C₄⋊C₄	—	52	1	no	—
348	C₂²⋊C₄	—	26	2	no	—
349	C₂²⋊C₄	—	26	2	no	—
350	C₄²	C₂×D₅₂	1	1	yes	characteristic
351	C₄²	D₄×D₁₃	1	1	yes	characteristic
352	C₄²	#6131	1	1	yes	characteristic
353	C₄²	C₂³×D₁₃	1	1	yes	characteristic
354	C₁₃	#19196	1	1	yes	characteristic, complemented, Sylow
355	C₂³	—	26	2	no	—
356	C₂³	—	52	2	no	—
357	C₂³	—	104	2	no	—
358	C₂³	—	104	2	no	—
359	C₂³	—	104	2	no	—
360	C₂³	#88149	1	1	yes	characteristic
361	Q₈	—	52	2	no	—
362	Q₈	—	104	2	no	—
363	Q₈	—	104	2	no	—
364	D₄	—	26	4	no	—
365	D₄	—	52	4	no	—
366	D₄	—	52	2	no	—
367	D₄	—	52	2	no	—
368	D₄	—	52	4	no	—
369	D₄	—	52	2	no	—
370	D₄	—	52	2	no	—
371	D₄	—	104	2	no	—
372	D₄	—	104	2	no	—
373	D₄	—	104	2	no	—
374	D₄	—	104	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	26	2	no	—
378	C₂×C₄	—	52	2	no	—
379	C₂×C₄	—	52	2	no	—
380	C₂×C₄	—	52	2	no	—
381	C₂×C₄	—	104	2	no	—
382	C₂×C₄	#88149	1	1	yes	characteristic
383	C₂×C₄	#88149	1	1	yes	characteristic
384	C₂×C₄	#88149	1	1	yes	characteristic
385	C₂×C₄	#88158	1	2	yes	—
386	C₂×C₄	#88162	1	2	yes	—
387	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
388	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
389	C₂×C₄	C₂×#6131	1	1	yes	characteristic
390	C₈	—	2	1	no	—
391	C₈	—	2	1	no	—
392	C₈	—	2	1	no	—
393	C₈	—	2	1	no	—
394	C₈	—	4	2	no	—
395	C₈	—	4	2	no	—
396	C₈	—	52	2	no	—
397	C₂²	—	2	1	no	—
398	C₂²	—	2	1	no	—
399	C₂²	—	52	2	no	—
400	C₂²	—	52	2	no	—
401	C₂²	—	52	2	no	—
402	C₂²	—	104	2	no	—
403	C₂²	—	104	2	no	—
404	C₂²	—	104	2	no	—
405	C₂²	—	208	4	no	—
406	C₂²	#11718505	1	2	yes	—
407	C₂²	C₂×#88149	1	1	yes	characteristic
408	C₄	—	2	1	no	—
409	C₄	—	2	1	no	—
410	C₄	—	52	2	no	—
411	C₄	—	104	2	no	—
412	C₄	#11718505	1	2	yes	—
413	C₄	#11718558	1	2	yes	—
414	C₄	C₂×#88158	1	1	yes	characteristic
415	C₄	C₂×#88162	1	1	yes	characteristic
416	C₄	#11719475	1	2	yes	—
417	C₂	—	2	2	no	—
418	C₂	—	104	2	no	—
419	C₂	—	208	2	no	—
420	C₂	C₂×#11718505	1	1	yes	characteristic
421	C₂	#13124476	1	1	yes	characteristic
422	C₂	C₂×#11718558	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88149
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1564	C₂
socle	C₂²×C₁₃	C₂×#121
prime radical	#13125756	C₂
hypercenter	#1564	D₁₃
hypocenter	C₁₃	#19196
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
1. #13125966	C₂	semidirect
3. C₂×#11718355	C₂	semidirect
4. #13123549	C₂	semidirect
5. #13114370	C₂	semidirect
6. #13114246	C₂	semidirect
7. #13113516	C₂	semidirect
8. #13113503	C₂	semidirect
9. C₁₃×#1564	C₂	semidirect
22. #11718503	C₂²	semidirect
33. #11718349	C₂²	semidirect
35. C₈×D₅₂	C₂²	semidirect
37. C₁₃×#335	C₂²	semidirect
38. C₁₃×#334	C₂²	semidirect
40. #11718073	C₂²	semidirect
150. #1564	D₁₃	semidirect
222. #335	C₂×D₁₃	semidirect
223. #334	C₂×D₁₃	semidirect
354. C₁₃	#19196	semidirect

Maximal subgroups

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

subgroup	index
#13125966	2
#13125756	2
C₂×#11718355	2
#13123549	2
#13114370	2
#13114246	2
#13113516	2
#13113503	2
C₁₃×#1564	2
#19196	13