Subgroups — #15846119

Order 3328.

710 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125774	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718746	C₂	1	1	yes	maximal, characteristic
3	#13123771	C₂	1	1	yes	maximal, characteristic, complemented
4	#13114310	C₂	1	4	yes	maximal, complemented
5	#13114186	C₂	1	4	yes	maximal, complemented
6	#13113582	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113508	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113503	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#3229	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#533	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂²×#6106	—	2	1	no	—
13	C₂²×#6106	—	2	1	no	—
14	C₂×#88128	C₂²	1	1	yes	characteristic
15	C₁₃×#498	—	2	1	no	—
16	C₁₃×#487	—	2	1	no	—
17	C₁₃×#483	—	2	2	no	—
18	C₁₃×#475	C₂²	1	4	yes	complemented
19	C₁₃×#431	C₂²	1	1	yes	characteristic
20	#11718749	—	2	1	no	—
21	#11718746	—	2	1	no	—
22	#11718746	—	2	1	no	—
23	#11718746	—	2	1	no	—
24	#11718746	C₂²	1	2	yes	—
25	#11718746	C₂²	1	2	yes	—
26	#11718707	—	2	2	no	—
27	#11718688	C₂²	1	4	yes	—
28	#11718685	—	2	1	no	—
29	#11718685	C₂²	1	2	yes	complemented
30	Q₈×#1055	—	2	1	no	—
31	Q₈×#1055	C₂²	1	2	yes	complemented
32	#11718660	—	2	1	no	—
33	#11718658	—	2	1	no	—
34	#11718586	C₂²	1	1	yes	characteristic
35	#11718585	C₂²	1	1	yes	characteristic
36	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
37	#11718139	C₂²	1	4	yes	complemented
38	#11718077	C₂²	1	4	yes	—
39	#11718072	C₂²	1	2	yes	complemented
40	#11718071	C₂²	1	2	yes	complemented
41	C₂²×Q₈×C₁₃	—	2	1	no	—
42	C₂²×D₅₂	—	2	1	no	—
43	C₂²×D₅₂	—	2	1	no	—
44	C₂²×D₅₂	—	2	1	no	—
45	C₁₃×C₄⋊Q₈	—	2	1	no	—
46	C₁₃×C₄⋊Q₈	—	2	1	no	—
47	C₁₃×C₄⋊Q₈	—	2	1	no	—
48	C₁₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
49	C₁₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
50	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
51	C₁₃×#125	—	2	1	no	—
52	C₁₃×C₂²⋊Q₈	—	2	1	no	—
53	C₁₃×C₂²⋊Q₈	D₄	1	2	yes	—
54	C₄×Q₈×C₁₃	—	2	1	no	—
55	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
56	C₁₃×#117	—	2	1	no	—
57	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
58	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
59	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
60	C₂×#6106	—	2	4	no	—
61	C₂×#6106	—	2	2	no	—
62	C₂×#6106	—	2	2	no	—
63	C₂×#6106	—	2	2	no	—
64	C₂×#6106	—	2	2	no	—
65	C₂×#6106	—	4	4	no	—
66	C₂×#6106	—	4	4	no	—
67	C₂×#6099	—	2	1	no	—
68	C₂²×#1055	D₄	1	1	yes	characteristic
69	C₂²×#1055	D₄	1	1	yes	characteristic
70	#88128	—	2	1	no	—
71	#88128	—	2	1	no	—
72	#88128	—	2	1	no	—
73	#88128	C₂³	1	2	yes	—
74	C₁₃×#106	—	2	4	no	—
75	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
76	C₁₃×#103	—	2	4	no	—
77	C₁₃×C₂²⋊C₈	—	2	1	no	—
78	C₄×C₈×C₁₃	—	2	1	no	—
79	C₂²⋊#1055	—	2	1	no	—
80	#88049	—	2	4	no	—
81	C₄⋊#1055	—	2	1	no	—
82	C₄⋊#1055	—	2	2	no	—
83	C₄⋊#1055	—	2	2	no	—
84	C₄⋊#1055	C₂³	1	2	yes	—
85	C₄⋊#1055	C₂³	1	2	yes	—
86	#88043	—	2	2	no	—
87	C₄×#1055	—	2	1	no	—
88	C₄×#1055	—	2	2	no	—
89	C₄×#1055	D₄	1	1	yes	characteristic
90	C₄×#1055	D₄	1	1	yes	characteristic
91	C₄×#1055	C₂³	1	1	yes	characteristic
92	C₄×#1055	C₂³	1	1	yes	characteristic
93	#19202	—	13	1	no	maximal, Sylow
94	C₂³×D₁₃	—	4	1	no	—
95	C₂×Q₈×C₁₃	—	2	1	no	—
96	C₂×Q₈×C₁₃	—	2	2	no	—
97	C₂×Q₈×C₁₃	—	4	1	no	—
98	C₂×Q₈×C₁₃	—	4	2	no	—
99	C₂×Q₈×C₁₃	—	4	2	no	—
100	C₂×Q₈×C₁₃	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₁₃	—	4	1	no	—
102	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
105	C₂×D₅₂	—	2	2	no	—
106	C₂×D₅₂	—	2	2	no	—
107	C₂×D₅₂	—	2	2	no	—
108	C₂×D₅₂	—	4	2	no	—
109	C₂×D₅₂	—	4	2	no	—
110	C₂×D₅₂	—	4	1	no	—
111	C₂×D₅₂	—	4	1	no	—
112	C₂×D₅₂	—	4	1	no	—
113	C₂×D₅₂	—	8	2	no	—
114	C₂×D₅₂	—	8	2	no	—
115	C₂×C₈×C₁₃	—	2	2	no	—
116	C₁₃×C₄⋊C₄	—	2	1	no	—
117	C₁₃×C₄⋊C₄	—	2	1	no	—
118	C₁₃×C₄⋊C₄	—	2	1	no	—
119	C₁₃×C₄⋊C₄	—	2	2	no	—
120	C₁₃×C₄⋊C₄	—	2	2	no	—
121	C₁₃×C₄⋊C₄	—	2	2	no	—
122	C₁₃×C₄⋊C₄	—	2	2	no	—
123	C₁₃×C₄⋊C₄	—	4	1	no	—
124	C₁₃×C₄⋊C₄	—	4	1	no	—
125	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
126	C₁₃×C₂²⋊C₄	—	2	2	no	—
127	C₄²×C₁₃	—	2	2	no	—
128	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₁₃	#41	1	1	yes	characteristic
131	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
132	#6106	—	4	4	no	—
133	#6106	—	4	4	no	—
134	#6106	—	4	4	no	—
135	#6106	—	4	4	no	—
136	#6106	—	8	4	no	—
137	#6099	—	4	2	no	—
138	C₂×#1055	—	2	2	no	—
139	C₂×#1055	—	2	2	no	—
140	C₂×#1055	—	2	2	no	—
141	C₂×#1055	SD₁₆	1	4	yes	complemented
142	C₂×#1055	C₂×D₄	1	1	yes	characteristic
143	C₂×#1055	C₂×D₄	1	1	yes	characteristic
144	C₂×#1055	C₂×D₄	1	1	yes	characteristic
145	C₂×#1055	C₂×D₄	1	1	yes	characteristic
146	#3229	D₁₃	1	1	yes	characteristic, complemented
147	#3208	—	13	1	no	—
148	C₂×#492	—	13	1	no	—
149	#2960	—	13	1	no	—
150	#1686	—	13	4	no	—
151	#1666	—	13	4	no	—
152	#1564	—	13	1	no	—
153	#1546	—	13	1	no	—
154	#1545	—	13	1	no	—
155	C₂³×C₁₃	#121	1	1	yes	characteristic
156	C₂²×D₁₃	—	4	2	no	—
157	C₂²×D₁₃	—	8	2	no	—
158	C₂²×D₁₃	—	8	2	no	—
159	C₂²×D₁₃	—	8	2	no	—
160	Q₈×C₁₃	—	2	4	no	—
161	Q₈×C₁₃	—	4	2	no	—
162	Q₈×C₁₃	—	4	2	no	—
163	Q₈×C₁₃	—	4	2	no	—
164	Q₈×C₁₃	—	8	2	no	—
165	C₂×C₄×C₁₃	—	2	2	no	—
166	C₂×C₄×C₁₃	—	2	2	no	—
167	C₂×C₄×C₁₃	—	2	2	no	—
168	C₂×C₄×C₁₃	—	2	2	no	—
169	C₂×C₄×C₁₃	—	4	2	no	—
170	C₂×C₄×C₁₃	—	8	2	no	—
171	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
172	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
173	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
174	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
175	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
176	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
179	D₅₂	—	4	2	no	—
180	D₅₂	—	4	2	no	—
181	D₅₂	—	4	4	no	—
182	D₅₂	—	8	2	no	—
183	D₅₂	—	8	2	no	—
184	D₅₂	—	8	2	no	—
185	D₅₂	—	8	2	no	—
186	C₈×C₁₃	—	4	2	no	—
187	#1055	—	2	1	no	—
188	#1055	—	2	1	no	—
189	#1055	—	4	2	no	—
190	#1055	—	4	2	no	—
191	#1055	C₂×SD₁₆	1	2	yes	—
192	#1055	C₂×SD₁₆	1	2	yes	—
193	C₂²×SD₁₆	—	26	1	no	—
194	C₂²×SD₁₆	—	26	1	no	—
195	#533	C₂×D₁₃	1	1	yes	characteristic
196	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
197	C₂×#127	—	13	1	no	—
198	#498	—	2	1	no	—
199	#492	—	13	2	no	—
200	#492	—	13	2	no	—
201	#492	—	26	1	no	—
202	#492	—	26	1	no	—
203	#492	—	26	1	no	—
204	#487	—	2	1	no	—
205	#486	—	26	1	no	—
206	#483	—	2	2	no	—
207	#475	C₂×D₁₃	1	4	yes	complemented
208	#460	—	13	4	no	—
209	#450	—	26	2	no	—
210	#446	—	13	2	no	—
211	#446	—	26	1	no	—
212	C₈×Q₈	—	13	2	no	—
213	C₈×Q₈	—	26	1	no	—
214	#431	—	13	1	no	—
215	#431	C₂×D₁₃	1	1	yes	characteristic
216	#424	—	26	1	no	—
217	#423	—	13	1	no	—
218	#405	—	26	1	no	—
219	C₂×C₄×C₈	—	13	1	no	—
220	#335	—	13	2	no	—
221	#334	—	13	2	no	—
222	#332	—	13	4	no	—
223	#331	—	13	4	no	—
224	C₂²×C₁₃	—	2	1	no	—
225	C₂²×C₁₃	—	2	1	no	—
226	C₂²×C₁₃	#460	1	2	yes	—
227	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
228	C₂×D₁₃	—	8	2	no	—
229	C₂×D₁₃	—	8	2	no	—
230	C₂×D₁₃	—	8	2	no	—
231	C₂×D₁₃	—	16	4	no	—
232	C₄×C₁₃	—	2	1	no	—
233	C₄×C₁₃	—	2	1	no	—
234	C₄×C₁₃	—	4	2	no	—
235	C₄×C₁₃	—	4	2	no	—
236	C₄×C₁₃	—	8	2	no	—
237	C₄×C₁₃	#460	1	2	yes	—
238	C₄×C₁₃	#471	1	2	yes	—
239	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
240	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
241	C₄×C₁₃	#577	1	2	yes	—
242	C₂²×Q₈	—	2	1	no	—
243	C₂²×D₄	—	26	1	no	—
244	C₂²×D₄	—	26	1	no	—
245	C₂²×D₄	—	26	1	no	—
246	C₂×SD₁₆	—	26	2	no	—
247	C₂×SD₁₆	—	26	2	no	—
248	C₂×SD₁₆	—	26	2	no	—
249	C₂×SD₁₆	—	26	2	no	—
250	C₂×SD₁₆	—	26	4	no	—
251	C₂×SD₁₆	—	52	4	no	—
252	C₂×SD₁₆	—	52	4	no	—
253	C₂×M₁₆	—	26	1	no	—
254	C₂²×C₈	—	13	1	no	—
255	C₂²×C₈	—	13	1	no	—
256	C₄⋊Q₈	—	2	1	no	—
257	C₄⋊Q₈	—	2	1	no	—
258	C₄⋊Q₈	—	2	1	no	—
259	C₄⋊Q₈	C₂²⋊D₁₃	1	1	yes	characteristic
260	C₄⋊Q₈	C₂²⋊D₁₃	1	1	yes	characteristic
261	C₄⋊Q₈	C₂²×D₁₃	1	2	yes	—
262	#127	—	13	2	no	—
263	#127	—	26	1	no	—
264	#127	—	26	1	no	—
265	#127	—	26	1	no	—
266	#125	—	2	1	no	—
267	C₂²⋊Q₈	—	2	1	no	—
268	C₂²⋊Q₈	C₂²⋊D₁₃	1	2	yes	—
269	C₄×Q₈	—	2	1	no	—
270	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
271	#117	—	2	1	no	—
272	C₂×C₄⋊C₄	—	2	1	no	—
273	C₂×C₄⋊C₄	—	2	1	no	—
274	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
275	#106	—	2	4	no	—
276	C₄⋊C₈	—	13	2	no	—
277	C₄⋊C₈	—	13	2	no	—
278	C₄⋊C₈	—	26	2	no	—
279	C₄⋊C₈	—	26	2	no	—
280	C₄⋊C₈	—	26	1	no	—
281	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
282	#103	—	2	4	no	—
283	#102	—	26	4	no	—
284	C₂²⋊C₈	—	2	1	no	—
285	C₂²⋊C₈	—	26	1	no	—
286	#97	—	26	2	no	—
287	C₄×C₈	—	2	1	no	—
288	C₄×C₈	—	13	1	no	—
289	C₄×C₈	—	13	1	no	—
290	C₄×C₈	—	13	1	no	—
291	C₄×C₈	—	13	1	no	—
292	C₄×C₈	—	26	2	no	—
293	C₄×C₈	—	26	1	no	—
294	C₂×C₁₃	—	2	2	no	—
295	C₂×C₁₃	C₂×#460	1	1	yes	characteristic
296	C₂×C₁₃	#3057	1	1	yes	characteristic
297	C₂×C₁₃	C₂×#471	1	1	yes	characteristic
298	D₁₃	—	16	2	no	—
299	C₂⁴	—	52	1	no	—
300	C₂×Q₈	—	2	1	no	—
301	C₂×Q₈	—	2	2	no	—
302	C₂×Q₈	—	4	1	no	—
303	C₂×Q₈	—	4	2	no	—
304	C₂×Q₈	—	4	2	no	—
305	C₂×Q₈	C₂×C₂²⋊D₁₃	1	2	yes	—
306	C₂×D₄	—	26	2	no	—
307	C₂×D₄	—	26	2	no	—
308	C₂×D₄	—	26	2	no	—
309	C₂×D₄	—	52	2	no	—
310	C₂×D₄	—	52	2	no	—
311	C₂×D₄	—	52	1	no	—
312	C₂×D₄	—	52	1	no	—
313	C₂×D₄	—	52	1	no	—
314	C₂×D₄	—	104	2	no	—
315	C₂×D₄	—	104	2	no	—
316	C₂²×C₄	—	4	1	no	—
317	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
318	C₂²×C₄	#6129	1	1	yes	characteristic
319	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
320	SD₁₆	—	52	4	no	—
321	SD₁₆	—	52	4	no	—
322	SD₁₆	—	52	4	no	—
323	SD₁₆	—	52	4	no	—
324	SD₁₆	—	104	4	no	—
325	M₁₆	—	52	2	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	13	4	no	—
328	C₂×C₈	—	13	1	no	—
329	C₂×C₈	—	13	1	no	—
330	C₂×C₈	—	13	1	no	—
331	C₂×C₈	—	13	1	no	—
332	C₂×C₈	—	26	2	no	—
333	C₂×C₈	—	26	2	no	—
334	C₂×C₈	—	26	2	no	—
335	C₄⋊C₄	—	2	1	no	—
336	C₄⋊C₄	—	2	1	no	—
337	C₄⋊C₄	—	2	1	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	4	1	no	—
343	C₄⋊C₄	—	4	1	no	—
344	C₄⋊C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
345	C₂²⋊C₄	—	2	2	no	—
346	C₄²	—	2	2	no	—
347	C₄²	D₄×D₁₃	1	1	yes	characteristic
348	C₄²	#6129	1	1	yes	characteristic
349	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
350	C₄²	C₂³×D₁₃	1	1	yes	characteristic
351	C₁₃	#19202	1	1	yes	characteristic, complemented, Sylow
352	C₂³	—	52	2	no	—
353	C₂³	—	104	2	no	—
354	C₂³	—	104	2	no	—
355	C₂³	—	104	2	no	—
356	C₂³	#88193	1	1	yes	characteristic
357	Q₈	—	2	4	no	—
358	Q₈	—	4	2	no	—
359	Q₈	—	4	2	no	—
360	Q₈	—	4	2	no	—
361	Q₈	—	8	2	no	—
362	D₄	—	52	2	no	—
363	D₄	—	52	2	no	—
364	D₄	—	52	4	no	—
365	D₄	—	104	2	no	—
366	D₄	—	104	2	no	—
367	D₄	—	104	2	no	—
368	D₄	—	104	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	8	2	no	—
375	C₂×C₄	SD₁₆×D₁₃	1	2	yes	—
376	C₂×C₄	#88168	1	2	yes	—
377	C₂×C₄	#88193	1	1	yes	characteristic
378	C₂×C₄	#88193	1	1	yes	characteristic
379	C₂×C₄	#88193	1	1	yes	characteristic
380	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
381	C₂×C₄	C₂×#6129	1	1	yes	characteristic
382	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
383	C₈	—	4	2	no	—
384	C₈	—	13	2	no	—
385	C₈	—	13	2	no	—
386	C₈	—	26	1	no	—
387	C₈	—	26	1	no	—
388	C₈	—	52	2	no	—
389	C₈	—	52	2	no	—
390	C₂²	—	2	1	no	—
391	C₂²	—	2	1	no	—
392	C₂²	—	104	2	no	—
393	C₂²	—	104	2	no	—
394	C₂²	—	104	2	no	—
395	C₂²	—	208	4	no	—
396	C₂²	#11718824	1	2	yes	—
397	C₂²	C₂×#88193	1	1	yes	characteristic
398	C₄	—	2	1	no	—
399	C₄	—	2	1	no	—
400	C₄	—	4	2	no	—
401	C₄	—	4	2	no	—
402	C₄	—	8	2	no	—
403	C₄	#11718824	1	2	yes	—
404	C₄	#11718864	1	2	yes	—
405	C₄	C₂×SD₁₆×D₁₃	1	1	yes	characteristic
406	C₄	C₂×#88168	1	1	yes	characteristic
407	C₄	#11719542	1	2	yes	—
408	C₂	—	2	2	no	—
409	C₂	—	208	2	no	—
410	C₂	C₂×#11718824	1	1	yes	characteristic
411	C₂	#13124477	1	1	yes	characteristic
412	C₂	C₂×#11718864	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88193
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#3229	C₂
socle	C₂²×C₁₃	C₂×#121
prime radical	C₂×#88128	C₂²
hypercenter	#3229	D₁₃
hypocenter	C₁₃	#19202
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3229

Complements

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

N	H ≅ G/N	type
3. #13123771	C₂	semidirect
4. #13114310	C₂	semidirect
5. #13114186	C₂	semidirect
6. #13113582	C₂	semidirect
7. #13113508	C₂	semidirect
9. C₁₃×#3229	C₂	semidirect
18. C₁₃×#475	C₂²	semidirect
29. #11718685	C₂²	semidirect
31. Q₈×#1055	C₂²	semidirect
37. #11718139	C₂²	semidirect
39. #11718072	C₂²	semidirect
40. #11718071	C₂²	semidirect
141. C₂×#1055	SD₁₆	semidirect
146. #3229	D₁₃	semidirect
207. #475	C₂×D₁₃	semidirect
351. C₁₃	#19202	semidirect

Maximal subgroups

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

subgroup	index
#13125774	2
C₂×#11718746	2
#13123771	2
#13114310	2
#13114186	2
#13113582	2
#13113508	2
#13113503	2
C₁₃×#3229	2
#19202	13