Subgroups — #15854473

Order 3328.

604 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₁₀₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#13127259	C₂	1	1	yes	maximal, characteristic
3	#13127132	C₂	1	2	yes	maximal, complemented
4	#13114446	C₂	1	1	yes	maximal, characteristic, complemented
5	#13114412	C₂	1	2	yes	maximal, complemented
6	#13114240	C₂	1	1	yes	maximal, characteristic, complemented
7	#13114225	C₂	1	1	yes	maximal, characteristic, complemented
8	#13114066	C₂	1	2	yes	maximal, complemented
9	#13114003	C₂	1	2	yes	maximal, complemented
10	#13113722	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×C₁₃×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈×D₅₂	C₂²	1	2	yes	—
13	#11719268	C₂²	1	2	yes	—
14	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
16	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
17	#11718831	—	2	2	no	—
18	#11718776	—	2	1	no	—
19	#11718767	—	2	1	no	—
20	#11718682	C₂²	1	2	yes	—
21	#11718680	C₂²	1	2	yes	—
22	#11718528	—	2	2	no	—
23	#11718507	—	2	2	no	—
24	#11718507	C₂²	1	2	yes	—
25	#11718503	C₂²	1	2	yes	—
26	#11718463	—	2	2	no	—
27	#11718441	—	2	2	no	—
28	#11718398	—	2	2	no	—
29	C₄×D₁₀₄	—	2	1	no	—
30	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
31	#11718349	C₂²	1	2	yes	—
32	#11718343	—	2	1	no	—
33	#11718343	C₂²	1	1	yes	characteristic
34	#11718343	C₂²	1	1	yes	characteristic
35	#11718342	C₂²	1	2	yes	—
36	#11718342	C₂²	1	2	yes	—
37	C₁₃×#334	C₂²	1	1	yes	characteristic
38	C₁₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
39	C₁₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₈²×C₁₃	C₂²	1	1	yes	characteristic
41	#11718078	C₂²	1	2	yes	—
42	#11718073	C₂²	1	2	yes	—
43	#11718051	C₂²	1	2	yes	—
44	#11718050	C₂²	1	1	yes	characteristic
45	C₂×Q₈×D₁₃	—	2	2	no	—
46	C₂×C₁₃×SD₁₆	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₁₃	—	2	1	no	—
48	C₄×Q₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₄×D₄×C₁₃	C₂³	1	1	yes	characteristic
50	#88192	—	2	2	no	—
51	C₂×#6105	—	2	2	no	—
52	#88184	—	2	2	no	—
53	#88180	—	2	2	no	—
54	C₂×D₁₀₄	D₄	1	1	yes	characteristic
55	C₂×#6094	—	2	2	no	—
56	#88151	—	2	2	no	—
57	#88151	—	4	2	no	—
58	#88146	—	2	2	no	—
59	#88139	—	4	2	no	—
60	C₄×D₅₂	—	2	2	no	—
61	C₄×D₅₂	C₂³	1	2	yes	—
62	#88123	—	2	2	no	—
63	#88123	C₂³	1	2	yes	—
64	#88123	C₂³	1	2	yes	—
65	C₁₃×#106	D₄	1	1	yes	characteristic
66	C₁₃×C₄⋊C₈	—	2	1	no	—
67	C₁₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
68	C₁₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
69	C₁₃×#103	D₄	1	1	yes	characteristic
70	C₁₃×#102	D₄	1	1	yes	characteristic
71	C₁₃×C₂²⋊C₈	—	2	1	no	—
72	C₄×C₈×C₁₃	—	2	1	no	—
73	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
76	#88075	—	2	2	no	—
77	#88061	—	2	2	no	—
78	#88061	—	2	2	no	—
79	#88061	D₄	1	2	yes	—
80	#88060	—	2	2	no	—
81	#88058	—	2	1	no	—
82	#88058	—	2	1	no	—
83	#88058	—	2	1	no	—
84	#88058	—	2	1	no	—
85	#88058	—	2	2	no	—
86	#88058	D₄	1	1	yes	characteristic
87	#88057	—	2	2	no	—
88	#88056	—	2	2	no	—
89	#88056	—	2	2	no	—
90	#88056	—	4	2	no	—
91	#88050	—	2	2	no	—
92	#88047	—	2	2	no	—
93	C₄⋊#1055	C₂³	1	2	yes	—
94	#19905	—	13	1	no	maximal, Sylow
95	C₂×Q₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×D₄×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₁₃	—	2	1	no	—
98	C₂×C₂²⋊D₁₃	—	4	2	no	—
99	Q₈×D₁₃	—	4	4	no	—
100	Q₈×D₁₃	—	8	2	no	—
101	C₂×D₅₂	C₂×D₄	1	2	yes	—
102	C₂×C₄×D₁₃	—	2	2	no	—
103	C₂×C₄×D₁₃	—	4	2	no	—
104	C₂×Dic₂₆	—	2	2	no	—
105	C₂×Dic₂₆	—	2	2	no	—
106	C₂×Dic₂₆	—	4	2	no	—
107	C₁₃×SD₁₆	—	2	2	no	—
108	C₂×C₈×C₁₃	—	2	1	no	—
109	C₂×C₈×C₁₃	—	2	1	no	—
110	C₂×C₈×C₁₃	—	2	1	no	—
111	C₂×C₈×C₁₃	—	4	2	no	—
112	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
116	C₁₃×C₄⋊C₄	—	2	1	no	—
117	C₁₃×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
118	C₁₃×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
119	C₁₃×C₂²⋊C₄	—	2	1	no	—
120	C₄²×C₁₃	—	2	1	no	—
121	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
122	C₂²⋊Dic₁₃	—	4	2	no	—
123	#6105	—	8	2	no	—
124	#6103	—	2	2	no	—
125	#6103	—	4	2	no	—
126	#6103	—	4	2	no	—
127	C₄⋊Dic₁₃	—	2	2	no	—
128	C₄⋊Dic₁₃	—	2	2	no	—
129	C₄⋊Dic₁₃	—	2	2	no	—
130	C₄⋊Dic₁₃	—	2	2	no	—
131	C₄⋊Dic₁₃	—	4	2	no	—
132	C₄⋊Dic₁₃	C₂×D₄	1	2	yes	—
133	C₄×Dic₁₃	—	4	2	no	—
134	C₂×#1055	—	2	2	no	—
135	D₁₀₄	—	2	2	no	—
136	#6094	—	8	2	no	—
137	Q₈×D₈	—	13	1	no	—
138	#3341	—	13	1	no	—
139	#3329	—	13	2	no	—
140	#1710	—	13	1	no	—
141	#1703	—	13	2	no	—
142	#1676	—	13	1	no	—
143	#1673	—	13	1	no	—
144	#1646	—	13	2	no	—
145	#1637	—	13	2	no	—
146	#1589	—	13	1	no	—
147	C₈×SD₁₆	D₁₃	1	1	yes	characteristic, complemented
148	C₂³×C₁₃	—	2	1	no	—
149	C₂²×D₁₃	—	2	2	no	—
150	Q₈×C₁₃	—	2	1	no	—
151	Q₈×C₁₃	#135	1	2	yes	—
152	D₄×C₁₃	—	2	1	no	—
153	D₄×C₁₃	—	2	1	no	—
154	C₂×C₄×C₁₃	—	2	1	no	—
155	C₂×C₄×C₁₃	—	2	1	no	—
156	C₂×C₄×C₁₃	—	2	1	no	—
157	C₂×C₄×C₁₃	—	4	1	no	—
158	C₂×C₄×C₁₃	—	4	1	no	—
159	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂²⋊D₁₃	—	16	2	no	—
163	C₂×Dic₁₃	—	2	2	no	—
164	C₂×Dic₁₃	—	4	2	no	—
165	C₂×Dic₁₃	—	4	2	no	—
166	D₅₂	—	2	2	no	—
167	D₅₂	—	2	2	no	—
168	C₄×D₁₃	—	4	2	no	—
169	C₄×D₁₃	—	4	2	no	—
170	C₄×D₁₃	—	8	4	no	—
171	Dic₂₆	—	4	2	no	—
172	Dic₂₆	—	4	2	no	—
173	Dic₂₆	—	8	4	no	—
174	Dic₂₆	—	8	2	no	—
175	C₈×C₁₃	—	2	1	no	—
176	C₈×C₁₃	—	2	1	no	—
177	C₈×C₁₃	—	2	1	no	—
178	C₈×C₁₃	—	4	1	no	—
179	C₈×C₁₃	—	4	1	no	—
180	C₈×C₁₃	—	4	1	no	—
181	C₈×C₁₃	#137	1	2	yes	—
182	#1055	—	4	2	no	—
183	#555	—	13	2	no	—
184	D₄×Q₈	—	13	2	no	—
185	#500	—	13	1	no	—
186	#500	—	13	1	no	—
187	#500	—	26	1	no	—
188	#487	—	13	2	no	—
189	#487	—	13	2	no	—
190	#485	—	26	2	no	—
191	#482	—	26	2	no	—
192	#477	—	13	2	no	—
193	#476	—	13	2	no	—
194	#474	—	13	2	no	—
195	#474	—	26	2	no	—
196	#468	—	26	2	no	—
197	#466	—	26	1	no	—
198	#461	—	13	2	no	—
199	#448	—	26	2	no	—
200	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
201	C₄×SD₁₆	C₂×D₁₃	1	1	yes	characteristic
202	C₄×D₈	—	13	1	no	—
203	C₄×D₈	—	26	1	no	—
204	#436	—	13	2	no	—
205	C₈×D₄	C₂×D₁₃	1	1	yes	characteristic
206	#427	—	26	1	no	—
207	#419	—	26	2	no	—
208	#335	—	13	1	no	—
209	#334	C₂×D₁₃	1	1	yes	characteristic
210	#333	—	13	2	no	—
211	#332	—	13	2	no	—
212	Q₈⋊C₈	C₂×D₁₃	1	1	yes	characteristic
213	D₄⋊C₈	—	13	2	no	—
214	D₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
215	C₈²	C₂×D₁₃	1	1	yes	characteristic
216	C₂²×C₁₃	—	4	1	no	—
217	C₂²×C₁₃	—	4	1	no	—
218	C₂²×C₁₃	—	4	1	no	—
219	C₂²×C₁₃	D₄²	1	1	yes	characteristic
220	C₂×D₁₃	—	4	2	no	—
221	C₂×D₁₃	—	4	2	no	—
222	C₂×D₁₃	—	4	2	no	—
223	C₄×C₁₃	—	2	2	no	—
224	C₄×C₁₃	—	2	1	no	—
225	C₄×C₁₃	—	4	1	no	—
226	C₄×C₁₃	—	4	1	no	—
227	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
228	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
229	C₄×C₁₃	#576	1	1	yes	characteristic
230	C₄×C₁₃	#578	1	1	yes	characteristic
231	Dic₁₃	—	4	2	no	—
232	Dic₁₃	—	8	2	no	—
233	Dic₁₃	—	8	2	no	—
234	C₂²×Q₈	—	26	2	no	—
235	C₂×SD₁₆	—	26	2	no	—
236	C₂×SD₁₆	D₅₂	1	1	yes	characteristic
237	C₂×D₈	—	13	1	no	—
238	C₂×M₁₆	—	26	2	no	—
239	C₂²×C₈	—	2	1	no	—
240	C₄⋊Q₈	—	13	2	no	—
241	C₄⋊Q₈	—	13	2	no	—
242	C₄⋊Q₈	—	26	2	no	—
243	#124	—	52	2	no	—
244	C₂²⋊Q₈	—	26	2	no	—
245	C₂²⋊Q₈	—	52	2	no	—
246	#121	—	26	2	no	—
247	#121	—	26	2	no	—
248	C₄×Q₈	C₂²×D₁₃	1	1	yes	characteristic
249	C₄×D₄	—	13	2	no	—
250	C₄×D₄	—	26	2	no	—
251	C₄×D₄	C₂²×D₁₃	1	1	yes	characteristic
252	#117	—	26	2	no	—
253	#117	—	26	2	no	—
254	#107	—	13	1	no	—
255	#107	—	26	2	no	—
256	#107	—	26	1	no	—
257	#107	—	26	1	no	—
258	#107	—	26	1	no	—
259	#107	—	26	1	no	—
260	#107	—	26	2	no	—
261	#106	—	26	2	no	—
262	#106	D₅₂	1	1	yes	characteristic
263	C₄⋊C₈	—	2	1	no	—
264	C₄⋊C₈	—	13	2	no	—
265	C₄⋊C₈	C₂²×D₁₃	1	1	yes	characteristic
266	C₄⋊C₈	C₂²×D₁₃	1	1	yes	characteristic
267	#103	—	26	2	no	—
268	#103	—	26	2	no	—
269	#103	—	26	2	no	—
270	#103	—	26	2	no	—
271	#103	—	52	2	no	—
272	#103	D₅₂	1	1	yes	characteristic
273	#102	—	13	2	no	—
274	#102	—	26	2	no	—
275	#102	—	26	2	no	—
276	#102	D₅₂	1	1	yes	characteristic
277	C₂²⋊C₈	—	2	1	no	—
278	C₂²⋊C₈	—	26	2	no	—
279	C₄×C₈	—	2	1	no	—
280	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
281	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
282	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
283	C₂×C₁₃	—	8	1	no	—
284	C₂×C₁₃	#3276	1	1	yes	characteristic
285	C₂×C₁₃	#3280	1	1	yes	characteristic
286	C₂×C₁₃	#3290	1	1	yes	characteristic
287	D₁₃	—	8	2	no	—
288	C₂×Q₈	—	26	2	no	—
289	C₂×Q₈	—	26	2	no	—
290	C₂×Q₈	—	52	2	no	—
291	C₂×Q₈	—	52	4	no	—
292	C₂×Q₈	—	104	2	no	—
293	C₂×Q₈	C₂×D₅₂	1	1	yes	characteristic
294	C₂×D₄	—	13	2	no	—
295	C₂×D₄	—	52	2	no	—
296	C₂×D₄	C₂×D₅₂	1	1	yes	characteristic
297	C₂²×C₄	—	2	1	no	—
298	C₂²×C₄	—	26	2	no	—
299	C₂²×C₄	—	52	2	no	—
300	SD₁₆	—	2	2	no	—
301	SD₁₆	—	104	2	no	—
302	D₈	—	26	2	no	—
303	M₁₆	—	104	2	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	4	2	no	—
308	C₂×C₈	—	26	2	no	—
309	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
310	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
311	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
312	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	13	2	no	—
315	C₄⋊C₄	—	26	2	no	—
316	C₄⋊C₄	—	26	2	no	—
317	C₄⋊C₄	—	26	2	no	—
318	C₄⋊C₄	—	26	2	no	—
319	C₄⋊C₄	—	52	2	no	—
320	C₄⋊C₄	C₂×D₅₂	1	1	yes	characteristic
321	C₄⋊C₄	C₂×D₅₂	1	1	yes	characteristic
322	C₂²⋊C₄	—	2	1	no	—
323	C₂²⋊C₄	—	26	2	no	—
324	C₂²⋊C₄	—	52	2	no	—
325	C₂²⋊C₄	—	52	2	no	—
326	C₂²⋊C₄	—	52	2	no	—
327	C₄²	—	2	1	no	—
328	C₄²	—	52	2	no	—
329	C₄²	C₂³×D₁₃	1	1	yes	characteristic
330	C₁₃	#19905	1	1	yes	characteristic, complemented, Sylow
331	C₂³	—	2	1	no	—
332	C₂³	—	26	2	no	—
333	Q₈	—	2	1	no	—
334	Q₈	—	52	2	no	—
335	Q₈	—	52	2	no	—
336	Q₈	—	104	4	no	—
337	Q₈	—	104	2	no	—
338	Q₈	#88158	1	2	yes	—
339	D₄	—	2	1	no	—
340	D₄	—	2	1	no	—
341	D₄	—	26	2	no	—
342	D₄	—	26	2	no	—
343	D₄	—	208	2	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	26	2	no	—
350	C₂×C₄	—	52	2	no	—
351	C₂×C₄	—	52	2	no	—
352	C₂×C₄	—	52	2	no	—
353	C₂×C₄	—	52	2	no	—
354	C₂×C₄	—	104	4	no	—
355	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
356	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
357	C₂×C₄	#88256	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	4	1	no	—
362	C₈	—	4	1	no	—
363	C₈	—	4	1	no	—
364	C₈	—	52	2	no	—
365	C₈	#88169	1	2	yes	—
366	C₂²	—	4	1	no	—
367	C₂²	—	4	1	no	—
368	C₂²	—	4	1	no	—
369	C₂²	—	52	2	no	—
370	C₂²	—	52	2	no	—
371	C₂²	—	52	2	no	—
372	C₂²	D₄×D₅₂	1	1	yes	characteristic
373	C₄	—	2	2	no	—
374	C₄	—	2	1	no	—
375	C₄	—	4	1	no	—
376	C₄	—	4	1	no	—
377	C₄	—	52	2	no	—
378	C₄	—	104	2	no	—
379	C₄	—	104	2	no	—
380	C₄	C₂×#88158	1	1	yes	characteristic
381	C₄	#11719457	1	1	yes	characteristic
382	C₄	C₂×#88169	1	1	yes	characteristic
383	C₄	#11719476	1	1	yes	characteristic
384	C₂	—	8	1	no	—
385	C₂	—	104	2	no	—
386	C₂	#13126483	1	1	yes	characteristic
387	C₂	#13126523	1	1	yes	characteristic
388	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×SD₁₆	C₂
socle	C₂²×C₁₃	D₄²
prime radical	#13127259	C₂
hypercenter	C₈×SD₁₆	D₁₃
hypocenter	C₁₃	#19905
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆

Complements

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

N	H ≅ G/N	type
1. Q₈×D₁₀₄	C₂	semidirect
3. #13127132	C₂	semidirect
4. #13114446	C₂	semidirect
5. #13114412	C₂	semidirect
6. #13114240	C₂	semidirect
7. #13114225	C₂	semidirect
8. #13114066	C₂	semidirect
9. #13114003	C₂	semidirect
10. #13113722	C₂	semidirect
11. C₈×C₁₃×SD₁₆	C₂	semidirect
147. C₈×SD₁₆	D₁₃	semidirect
330. C₁₃	#19905	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₁₀₄	2
#13127259	2
#13127132	2
#13114446	2
#13114412	2
#13114240	2
#13114225	2
#13114066	2
#13114003	2
#13113722	2
C₈×C₁₃×SD₁₆	2
#19905	13