Subgroups — #15854253

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	#13127811	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₁₀₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#13126503	C₂	1	2	yes	maximal, complemented
4	#13114428	C₂	1	1	yes	maximal, characteristic, complemented
5	#13114302	C₂	1	2	yes	maximal, complemented
6	#13114230	C₂	1	1	yes	maximal, characteristic, complemented
7	#13114196	C₂	1	1	yes	maximal, characteristic, complemented
8	#13114031	C₂	1	2	yes	maximal, complemented
9	#13113941	C₂	1	2	yes	maximal, complemented
10	#13113722	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#88172	—	2	2	no	—
13	#11719282	C₂²	1	2	yes	—
14	Q₈×D₅₂	C₂²	1	2	yes	—
15	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
18	#11718692	C₂²	1	2	yes	—
19	#11718688	C₂²	1	2	yes	—
20	#11718547	—	2	2	no	—
21	#11718507	—	2	2	no	—
22	#11718507	C₂²	1	2	yes	—
23	#11718504	—	2	2	no	—
24	#11718504	C₂²	1	2	yes	—
25	#11718471	—	2	2	no	—
26	#11718467	—	2	2	no	—
27	#11718356	—	2	1	no	—
28	#11718356	C₂²	1	1	yes	characteristic
29	#11718355	C₂²	1	2	yes	—
30	#11718354	C₂²	1	2	yes	—
31	C₄×D₁₀₄	—	2	1	no	—
32	C₄×D₁₀₄	—	2	1	no	—
33	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
34	#11718349	C₂²	1	2	yes	—
35	#11718343	—	2	1	no	—
36	#11718343	C₂²	1	1	yes	characteristic
37	C₁₃×#335	C₂²	1	1	yes	characteristic
38	C₁₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
39	C₁₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
40	C₈²×C₁₃	C₂²	1	1	yes	characteristic
41	#11718078	C₂²	1	2	yes	—
42	#11718077	C₂²	1	2	yes	—
43	#11718051	C₂²	1	2	yes	—
44	#11718050	C₂²	1	1	yes	characteristic
45	C₂×#6131	—	2	2	no	—
46	C₂×Q₈×D₁₃	—	2	2	no	—
47	C₂×C₁₃×Q₁₆	D₄	1	1	yes	characteristic
48	C₄×Q₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₄×Q₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×#6107	—	2	2	no	—
51	C₂×#6106	—	2	2	no	—
52	#88172	—	4	8	no	—
53	C₂×D₁₀₄	—	2	1	no	—
54	C₂×D₁₀₄	—	2	2	no	—
55	C₂×D₁₀₄	—	2	1	no	—
56	C₂×D₁₀₄	D₄	1	1	yes	characteristic
57	C₂×#6095	—	2	2	no	—
58	C₂×#6095	—	2	2	no	—
59	C₂×#6094	—	2	2	no	—
60	#88151	—	2	2	no	—
61	#88151	—	4	2	no	—
62	#88149	—	2	2	no	—
63	#88149	—	4	2	no	—
64	#88128	—	2	2	no	—
65	#88128	C₂³	1	2	yes	—
66	C₄×D₅₂	—	2	2	no	—
67	C₄×D₅₂	—	2	2	no	—
68	C₄×D₅₂	C₂³	1	2	yes	—
69	#88123	—	2	2	no	—
70	#88123	C₂³	1	2	yes	—
71	C₁₃×#107	D₄	1	1	yes	characteristic
72	C₁₃×C₄⋊C₈	—	2	1	no	—
73	C₁₃×C₄⋊C₈	—	2	1	no	—
74	C₁₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
75	C₁₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
76	C₁₃×#103	D₄	1	1	yes	characteristic
77	C₁₃×#103	D₄	1	1	yes	characteristic
78	C₄×C₈×C₁₃	—	2	1	no	—
79	C₄×C₈×C₁₃	—	2	1	no	—
80	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
82	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
83	#88061	—	2	2	no	—
84	#88061	—	2	2	no	—
85	#88061	—	4	2	no	—
86	#88061	D₄	1	2	yes	—
87	#88060	—	2	2	no	—
88	#88058	—	2	1	no	—
89	#88058	—	2	1	no	—
90	#88058	—	2	2	no	—
91	#88058	D₄	1	1	yes	characteristic
92	#88050	—	2	2	no	—
93	#88049	—	2	2	no	—
94	C₄⋊#1055	C₂³	1	2	yes	—
95	#19889	—	13	1	no	maximal, Sylow
96	C₂×Q₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×Q₈×C₁₃	C₂×D₄	1	1	yes	characteristic
98	#6131	—	4	4	no	—
99	#6131	—	8	2	no	—
100	Q₈×D₁₃	—	4	4	no	—
101	Q₈×D₁₃	—	8	2	no	—
102	C₂×D₅₂	—	2	2	no	—
103	C₂×D₅₂	—	2	2	no	—
104	C₂×D₅₂	—	2	2	no	—
105	C₂×D₅₂	—	4	2	no	—
106	C₂×D₅₂	—	4	2	no	—
107	C₂×D₅₂	C₂×D₄	1	2	yes	—
108	C₂×C₄×D₁₃	—	2	2	no	—
109	C₂×C₄×D₁₃	—	4	2	no	—
110	C₂×C₄×D₁₃	—	4	2	no	—
111	C₂×Dic₂₆	—	2	2	no	—
112	C₂×Dic₂₆	—	4	2	no	—
113	C₁₃×Q₁₆	D₈	1	4	yes	complemented
114	C₂×C₈×C₁₃	—	2	1	no	—
115	C₂×C₈×C₁₃	—	2	1	no	—
116	C₂×C₈×C₁₃	—	2	1	no	—
117	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
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×D₄	1	1	yes	characteristic
121	C₁₃×C₄⋊C₄	—	2	1	no	—
122	C₁₃×C₄⋊C₄	—	2	1	no	—
123	C₁₃×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
124	C₁₃×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₁₃	—	2	1	no	—
126	C₄²×C₁₃	—	2	1	no	—
127	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
128	#6107	—	4	4	no	—
129	#6106	—	4	4	no	—
130	#6103	—	2	2	no	—
131	#6103	—	4	2	no	—
132	#6103	—	4	2	no	—
133	C₄⋊Dic₁₃	—	2	2	no	—
134	C₄⋊Dic₁₃	—	2	2	no	—
135	C₄⋊Dic₁₃	—	4	2	no	—
136	C₄⋊Dic₁₃	C₂×D₄	1	2	yes	—
137	C₂×#1055	—	2	2	no	—
138	D₁₀₄	—	2	2	no	—
139	D₁₀₄	—	4	2	no	—
140	D₁₀₄	—	8	1	no	—
141	D₁₀₄	—	8	2	no	—
142	#6095	—	4	4	no	—
143	#6095	—	8	2	no	—
144	#6094	—	4	4	no	—
145	#3391	—	13	1	no	—
146	Q₈×D₈	—	13	1	no	—
147	#3278	—	13	2	no	—
148	#1706	—	13	1	no	—
149	#1685	—	13	2	no	—
150	#1674	—	13	1	no	—
151	#1668	—	13	1	no	—
152	#1641	—	13	2	no	—
153	#1627	—	13	2	no	—
154	#1589	—	13	1	no	—
155	C₈×Q₁₆	D₁₃	1	1	yes	characteristic, complemented
156	C₂²×D₁₃	—	2	2	no	—
157	C₂²×D₁₃	—	4	2	no	—
158	Q₈×C₁₃	—	2	1	no	—
159	Q₈×C₁₃	—	2	1	no	—
160	Q₈×C₁₃	C₂×D₈	1	2	yes	—
161	Q₈×C₁₃	C₂×D₈	1	2	yes	—
162	C₂×C₄×C₁₃	—	2	1	no	—
163	C₂×C₄×C₁₃	—	2	1	no	—
164	C₂×C₄×C₁₃	—	2	1	no	—
165	C₂×C₄×C₁₃	—	2	1	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	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂×Dic₁₃	—	2	2	no	—
170	C₂×Dic₁₃	—	4	2	no	—
171	D₅₂	—	2	2	no	—
172	D₅₂	—	2	2	no	—
173	D₅₂	—	4	2	no	—
174	D₅₂	—	4	2	no	—
175	D₅₂	—	4	2	no	—
176	D₅₂	—	4	2	no	—
177	D₅₂	—	8	4	no	—
178	D₅₂	—	8	2	no	—
179	D₅₂	—	16	2	no	—
180	C₄×D₁₃	—	4	2	no	—
181	C₄×D₁₃	—	4	2	no	—
182	C₄×D₁₃	—	8	4	no	—
183	C₄×D₁₃	—	8	4	no	—
184	Dic₂₆	—	4	2	no	—
185	Dic₂₆	—	4	2	no	—
186	Dic₂₆	—	8	4	no	—
187	C₈×C₁₃	—	2	1	no	—
188	C₈×C₁₃	—	2	1	no	—
189	C₈×C₁₃	—	4	1	no	—
190	C₈×C₁₃	—	4	1	no	—
191	C₈×C₁₃	—	4	1	no	—
192	C₈×C₁₃	C₂×D₈	1	2	yes	—
193	C₈×C₁₃	#137	1	2	yes	complemented
194	#1055	—	4	2	no	—
195	C₂×#137	—	26	2	no	—
196	#550	—	13	2	no	—
197	D₄×Q₈	—	13	2	no	—
198	#500	—	13	1	no	—
199	#500	—	26	1	no	—
200	#493	—	13	1	no	—
201	#493	—	26	1	no	—
202	#492	—	13	2	no	—
203	#486	—	13	2	no	—
204	#474	—	13	2	no	—
205	#474	—	26	2	no	—
206	#468	—	26	2	no	—
207	#462	—	13	2	no	—
208	#460	—	13	2	no	—
209	#459	—	13	2	no	—
210	#459	—	26	2	no	—
211	#452	—	26	2	no	—
212	#448	—	26	2	no	—
213	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
214	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
215	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
216	C₄×D₈	—	13	1	no	—
217	C₄×D₈	—	26	1	no	—
218	C₄×D₈	—	26	1	no	—
219	#436	—	13	2	no	—
220	#335	—	13	1	no	—
221	#335	C₂×D₁₃	1	1	yes	characteristic
222	#332	—	13	2	no	—
223	#331	—	13	2	no	—
224	Q₈⋊C₈	C₂×D₁₃	1	1	yes	characteristic
225	Q₈⋊C₈	C₂×D₁₃	1	1	yes	characteristic
226	D₄⋊C₈	—	13	2	no	—
227	C₈²	C₂×D₁₃	1	1	yes	characteristic
228	C₂²×C₁₃	D₄²	1	1	yes	characteristic
229	C₂×D₁₃	—	4	2	no	—
230	C₂×D₁₃	—	4	2	no	—
231	C₂×D₁₃	—	4	2	no	—
232	C₂×D₁₃	—	8	2	no	—
233	C₂×D₁₃	—	8	2	no	—
234	C₂×D₁₃	—	8	2	no	—
235	C₄×C₁₃	—	2	2	no	—
236	C₄×C₁₃	—	2	2	no	—
237	C₄×C₁₃	—	2	1	no	—
238	C₄×C₁₃	—	4	1	no	—
239	C₄×C₁₃	—	4	1	no	—
240	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
241	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
242	C₄×C₁₃	#576	1	1	yes	characteristic
243	C₄×C₁₃	#577	1	1	yes	characteristic
244	Dic₁₃	—	4	2	no	—
245	Dic₁₃	—	8	2	no	—
246	C₂×#41	—	26	2	no	—
247	C₂²×Q₈	—	26	2	no	—
248	#137	—	52	8	no	—
249	C₂×Q₁₆	—	26	2	no	—
250	C₂×Q₁₆	D₅₂	1	1	yes	characteristic
251	C₂×SD₁₆	—	26	2	no	—
252	C₂×SD₁₆	—	26	2	no	—
253	C₂×SD₁₆	—	26	2	no	—
254	C₂×D₈	—	13	1	no	—
255	C₂×D₈	—	26	2	no	—
256	C₂×D₈	—	26	1	no	—
257	C₂×D₈	—	26	1	no	—
258	C₂×M₁₆	—	26	2	no	—
259	C₄⋊Q₈	—	13	2	no	—
260	C₄⋊Q₈	—	26	2	no	—
261	#127	—	13	2	no	—
262	#127	—	26	2	no	—
263	C₂²⋊Q₈	—	26	2	no	—
264	C₂²⋊Q₈	—	52	2	no	—
265	#121	—	26	2	no	—
266	#121	—	52	2	no	—
267	C₄×Q₈	C₂²×D₁₃	1	1	yes	characteristic
268	C₄×Q₈	C₂²×D₁₃	1	1	yes	characteristic
269	C₄×D₄	—	13	2	no	—
270	C₄×D₄	—	26	2	no	—
271	C₄×D₄	—	26	2	no	—
272	#107	—	13	1	no	—
273	#107	—	26	2	no	—
274	#107	—	26	1	no	—
275	#107	—	26	1	no	—
276	#107	D₅₂	1	1	yes	characteristic
277	C₄⋊C₈	—	2	1	no	—
278	C₄⋊C₈	—	2	1	no	—
279	C₄⋊C₈	—	13	2	no	—
280	C₄⋊C₈	C₂²×D₁₃	1	1	yes	characteristic
281	C₄⋊C₈	C₂²×D₁₃	1	1	yes	characteristic
282	#103	—	26	2	no	—
283	#103	D₅₂	1	1	yes	characteristic
284	#103	D₅₂	1	1	yes	characteristic
285	#102	—	13	2	no	—
286	#102	—	26	2	no	—
287	#102	—	26	2	no	—
288	#102	—	26	2	no	—
289	#102	—	52	2	no	—
290	C₂²⋊C₈	—	26	2	no	—
291	C₄×C₈	—	2	1	no	—
292	C₄×C₈	—	2	1	no	—
293	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
294	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
295	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
296	C₂×C₁₃	#3278	1	1	yes	characteristic
297	C₂×C₁₃	D₄×D₈	1	1	yes	characteristic
298	C₂×C₁₃	#3290	1	1	yes	characteristic
299	D₁₃	—	8	2	no	—
300	D₁₃	—	16	2	no	—
301	#41	—	52	4	no	—
302	#41	—	104	2	no	—
303	C₂×Q₈	—	26	2	no	—
304	C₂×Q₈	—	52	2	no	—
305	C₂×Q₈	—	52	4	no	—
306	C₂×Q₈	—	104	2	no	—
307	C₂×Q₈	C₂×D₅₂	1	1	yes	characteristic
308	C₂×Q₈	C₂×D₅₂	1	1	yes	characteristic
309	C₂×D₄	—	13	2	no	—
310	C₂×D₄	—	26	2	no	—
311	C₂×D₄	—	26	2	no	—
312	C₂×D₄	—	26	2	no	—
313	C₂×D₄	—	52	2	no	—
314	C₂×D₄	—	52	2	no	—
315	C₂²×C₄	—	26	2	no	—
316	C₂²×C₄	—	52	2	no	—
317	C₂²×C₄	—	52	2	no	—
318	Q₁₆	—	52	4	no	—
319	Q₁₆	D₁₀₄	1	4	yes	complemented
320	SD₁₆	—	52	4	no	—
321	SD₁₆	—	52	4	no	—
322	SD₁₆	—	104	2	no	—
323	D₈	—	26	2	no	—
324	D₈	—	52	2	no	—
325	D₈	—	104	1	no	—
326	D₈	—	104	2	no	—
327	M₁₆	—	52	4	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	26	2	no	—
332	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
333	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
334	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
335	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
336	C₄⋊C₄	—	2	1	no	—
337	C₄⋊C₄	—	2	1	no	—
338	C₄⋊C₄	—	13	2	no	—
339	C₄⋊C₄	—	26	2	no	—
340	C₄⋊C₄	—	26	2	no	—
341	C₄⋊C₄	—	52	2	no	—
342	C₄⋊C₄	C₂×D₅₂	1	1	yes	characteristic
343	C₄⋊C₄	C₂×D₅₂	1	1	yes	characteristic
344	C₂²⋊C₄	—	26	2	no	—
345	C₂²⋊C₄	—	52	2	no	—
346	C₂²⋊C₄	—	52	2	no	—
347	C₄²	—	2	1	no	—
348	C₄²	—	2	1	no	—
349	C₄²	C₂³×D₁₃	1	1	yes	characteristic
350	C₁₃	#19889	1	1	yes	characteristic, complemented, Sylow
351	C₂³	—	26	2	no	—
352	C₂³	—	52	2	no	—
353	Q₈	—	2	1	no	—
354	Q₈	—	2	1	no	—
355	Q₈	—	52	2	no	—
356	Q₈	—	52	2	no	—
357	Q₈	—	104	4	no	—
358	Q₈	C₂×D₁₀₄	1	2	yes	—
359	Q₈	C₂×D₁₀₄	1	2	yes	—
360	D₄	—	26	2	no	—
361	D₄	—	26	2	no	—
362	D₄	—	52	2	no	—
363	D₄	—	52	2	no	—
364	D₄	—	52	2	no	—
365	D₄	—	52	2	no	—
366	D₄	—	104	4	no	—
367	D₄	—	104	2	no	—
368	D₄	—	208	2	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	26	2	no	—
374	C₂×C₄	—	52	2	no	—
375	C₂×C₄	—	52	2	no	—
376	C₂×C₄	—	52	2	no	—
377	C₂×C₄	—	104	4	no	—
378	C₂×C₄	—	104	4	no	—
379	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
380	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
381	C₂×C₄	#88256	1	1	yes	characteristic
382	C₈	—	2	1	no	—
383	C₈	—	2	1	no	—
384	C₈	—	4	1	no	—
385	C₈	—	4	1	no	—
386	C₈	—	4	1	no	—
387	C₈	—	52	2	no	—
388	C₈	C₂×D₁₀₄	1	2	yes	—
389	C₈	#88172	1	2	yes	complemented
390	C₂²	—	52	2	no	—
391	C₂²	—	52	2	no	—
392	C₂²	—	52	2	no	—
393	C₂²	—	104	2	no	—
394	C₂²	—	104	2	no	—
395	C₂²	—	104	2	no	—
396	C₂²	D₄×D₅₂	1	1	yes	characteristic
397	C₄	—	2	2	no	—
398	C₄	—	2	2	no	—
399	C₄	—	2	1	no	—
400	C₄	—	4	1	no	—
401	C₄	—	4	1	no	—
402	C₄	—	52	2	no	—
403	C₄	—	104	2	no	—
404	C₄	C₂²×D₁₀₄	1	1	yes	characteristic
405	C₄	#11719457	1	1	yes	characteristic
406	C₄	C₂×#88172	1	1	yes	characteristic
407	C₄	#11719475	1	1	yes	characteristic
408	C₂	—	104	2	no	—
409	C₂	—	208	2	no	—
410	C₂	#13126503	1	1	yes	characteristic
411	C₂	D₄×D₁₀₄	1	1	yes	characteristic
412	C₂	#13126623	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₅₂
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₈×C₁₃×Q₁₆	C₂
socle	C₂²×C₁₃	D₄²
prime radical	#13127811	C₂
hypercenter	C₈×Q₁₆	D₁₃
hypocenter	C₁₃	#19889
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×D₁₀₄	C₂	semidirect
3. #13126503	C₂	semidirect
4. #13114428	C₂	semidirect
5. #13114302	C₂	semidirect
6. #13114230	C₂	semidirect
7. #13114196	C₂	semidirect
8. #13114031	C₂	semidirect
9. #13113941	C₂	semidirect
10. #13113722	C₂	semidirect
11. C₈×C₁₃×Q₁₆	C₂	semidirect
113. C₁₃×Q₁₆	D₈	semidirect
155. C₈×Q₁₆	D₁₃	semidirect
193. C₈×C₁₃	#137	semidirect
319. Q₁₆	D₁₀₄	semidirect
350. C₁₃	#19889	semidirect
389. C₈	#88172	semidirect

Maximal subgroups

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

subgroup	index
#13127811	2
Q₈×D₁₀₄	2
#13126503	2
#13114428	2
#13114302	2
#13114230	2
#13114196	2
#13114031	2
#13113941	2
#13113722	2
C₈×C₁₃×Q₁₆	2
#19889	13