Subgroups — #15834412

Order 3328.

630 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#6094	C₂	1	2	yes	maximal
2	C₄×Q₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
3	#13113746	C₂	1	1	yes	maximal, characteristic, complemented
4	#13113744	C₂	1	1	yes	maximal, characteristic, complemented
5	#13113707	C₂	1	2	yes	maximal, complemented
6	#13113704	C₂	1	2	yes	maximal, complemented
7	#13113655	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113590	C₂	1	1	yes	maximal, characteristic
9	#13113263	C₂	1	2	yes	maximal
10	#13113122	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#1579	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₁₃	C₄	1	1	yes	characteristic
13	C₄×Q₈×D₁₃	C₂²	1	2	yes	—
14	C₈×Q₈×C₁₃	C₂²	1	2	yes	—
15	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
16	Q₁₆×Dic₁₃	C₄	1	1	yes	characteristic
17	C₄×#6107	C₂²	1	2	yes	—
18	Q₈×#1055	C₂²	1	2	yes	—
19	D₁₃×#107	C₄	1	1	yes	characteristic
20	#11718534	C₄	1	1	yes	characteristic
21	#11718498	—	2	2	no	—
22	#11718498	C₂²	1	2	yes	—
23	#11718492	—	2	2	no	—
24	#11718492	C₂²	1	2	yes	—
25	D₁₃×#103	C₄	1	2	yes	—
26	#11718451	C₄	1	2	yes	—
27	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
28	C₄×#6094	—	2	2	no	—
29	C₄×#6094	C₂²	1	1	yes	characteristic
30	#11718346	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
32	#11718340	—	2	2	no	—
33	#11718340	C₂²	1	2	yes	—
34	C₁₃×#334	C₂²	1	1	yes	characteristic
35	C₁₃×Q₈⋊C₈	C₂²	1	2	yes	—
36	C₁₃×#322	C₂²	1	1	yes	characteristic
37	Q₈⋊#1055	C₂²	1	2	yes	—
38	#11718076	C₂²	1	2	yes	—
39	#11718072	C₂²	1	2	yes	—
40	#11718049	C₂²	1	1	yes	characteristic
41	#11718048	C₂²	1	2	yes	—
42	#11718047	C₂²	1	1	yes	characteristic
43	#11718047	C₂²	1	1	yes	characteristic
44	#11718046	C₂²	1	1	yes	characteristic
45	C₂×Q₈×D₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
47	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
48	Q₈×Dic₁₃	C₂×C₄	1	2	yes	—
49	C₂×#6107	C₂×C₄	1	2	yes	—
50	Q₁₆×D₁₃	—	2	4	no	—
51	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
52	C₂×#6094	—	2	2	no	—
53	C₂×#6094	D₄	1	2	yes	—
54	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
56	C₄⋊C₄×D₁₃	—	2	2	no	—
57	C₄⋊C₄×D₁₃	C₂×C₄	1	2	yes	—
58	#88141	—	2	2	no	—
59	#88141	C₂×C₄	1	2	yes	—
60	C₄²×D₁₃	—	2	2	no	—
61	C₄²×D₁₃	C₂³	1	1	yes	characteristic
62	C₄×Dic₂₆	—	2	2	no	—
63	C₄×Dic₂₆	C₂³	1	2	yes	—
64	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
65	C₁₃×C₄⋊C₈	—	2	2	no	—
66	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
67	C₁₃×#103	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₁₃	—	2	2	no	—
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	#88075	C₂×C₄	1	2	yes	—
73	#88060	—	2	1	no	—
74	#88058	C₂×C₄	1	1	yes	characteristic
75	#88056	C₂×C₄	1	2	yes	—
76	#88055	—	2	2	no	—
77	#88055	—	2	2	no	—
78	#88055	D₄	1	2	yes	—
79	#88054	—	2	2	no	—
80	#88054	—	2	2	no	—
81	#88054	—	2	1	no	—
82	#88054	—	2	2	no	—
83	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
85	#88050	C₂×C₄	1	2	yes	—
86	#88047	C₂×C₄	1	2	yes	—
87	C₄⋊#1055	—	2	2	no	—
88	C₄⋊#1055	C₂³	1	2	yes	—
89	C₄×#1055	—	2	2	no	—
90	C₄×#1055	C₂³	1	1	yes	characteristic
91	C₄×#1055	C₂³	1	1	yes	characteristic
92	C₄×#1055	C₂³	1	1	yes	characteristic
93	#18042	—	13	1	no	maximal, Sylow
94	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
95	Q₈×D₁₃	—	2	4	no	—
96	Q₈×D₁₃	—	4	2	no	—
97	C₂×C₄×D₁₃	—	2	2	no	—
98	C₂×C₄×D₁₃	—	2	2	no	—
99	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
102	C₂×Dic₂₆	—	2	2	no	—
103	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
104	C₁₃×Q₁₆	M₁₆	1	4	yes	complemented
105	C₂×C₈×C₁₃	—	2	2	no	—
106	C₂×C₈×C₁₃	—	2	1	no	—
107	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
110	C₁₃×C₄⋊C₄	—	2	2	no	—
111	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
112	C₄²×C₁₃	—	2	2	no	—
113	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
114	#6107	—	2	4	no	—
115	C₄⋊Dic₁₃	—	2	2	no	—
116	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
117	#6101	—	2	2	no	—
118	#6101	—	2	2	no	—
119	#6101	—	2	2	no	—
120	C₄×Dic₁₃	—	2	2	no	—
121	C₄×Dic₁₃	—	2	2	no	—
122	C₄×Dic₁₃	—	2	2	no	—
123	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
124	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
125	C₄×Dic₁₃	#41	1	1	yes	characteristic
126	C₂×#1055	—	2	2	no	—
127	C₂×#1055	—	2	1	no	—
128	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
129	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
130	C₂×#1055	C₂×D₄	1	2	yes	—
131	Dic₅₂	—	2	2	no	—
132	#6094	—	2	4	no	—
133	#6094	—	4	4	no	—
134	C₈×D₁₃	—	2	2	no	—
135	C₈×D₁₃	—	4	1	no	—
136	Q₈×M₁₆	—	13	2	no	—
137	C₂×C₄×Q₁₆	—	13	1	no	—
138	#1593	—	13	1	no	—
139	#1593	—	13	1	no	—
140	#1587	—	13	2	no	—
141	#1587	—	13	2	no	—
142	#1579	—	13	1	no	—
143	#1579	D₁₃	1	1	yes	characteristic, complemented
144	#1566	—	13	1	no	—
145	#1487	—	13	2	no	—
146	#1451	—	13	1	no	—
147	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
148	Q₈×C₁₃	—	2	2	no	—
149	Q₈×C₁₃	C₂×M₁₆	1	4	yes	—
150	C₂×C₄×C₁₃	—	2	2	no	—
151	C₂×C₄×C₁₃	—	2	2	no	—
152	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
155	C₂×Dic₁₃	—	2	2	no	—
156	C₂×Dic₁₃	—	2	2	no	—
157	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
158	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
159	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
160	C₄×D₁₃	—	2	1	no	—
161	C₄×D₁₃	—	2	1	no	—
162	C₄×D₁₃	—	4	1	no	—
163	C₄×D₁₃	—	4	4	no	—
164	C₄×D₁₃	—	8	2	no	—
165	C₄×D₁₃	#137	1	4	yes	—
166	Dic₂₆	—	2	2	no	—
167	Dic₂₆	—	2	2	no	—
168	Dic₂₆	—	4	4	no	—
169	C₈×C₁₃	—	2	1	no	—
170	C₈×C₁₃	—	2	2	no	—
171	C₈×C₁₃	—	4	1	no	—
172	C₈×C₁₃	—	4	2	no	—
173	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
174	#1055	—	2	1	no	—
175	#1055	—	2	1	no	—
176	#1055	—	2	2	no	—
177	#1055	—	4	1	no	—
178	#1055	—	4	2	no	—
179	C₂²×Q₁₆	—	13	1	no	—
180	C₂×C₄×Q₈	—	13	2	no	—
181	#446	—	13	2	no	—
182	#446	—	13	2	no	—
183	#446	—	26	2	no	—
184	#446	—	26	2	no	—
185	C₈×Q₈	—	13	2	no	—
186	C₈×Q₈	C₂×D₁₃	1	2	yes	—
187	C₄×Q₁₆	—	13	1	no	—
188	C₄×Q₁₆	—	13	2	no	—
189	C₄×Q₁₆	—	13	1	no	—
190	C₄×Q₁₆	—	13	2	no	—
191	C₄×Q₁₆	—	13	1	no	—
192	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
193	#431	—	13	1	no	—
194	C₂×#107	—	13	1	no	—
195	#423	—	13	2	no	—
196	#423	—	26	2	no	—
197	C₂×#103	—	13	2	no	—
198	C₄×M₁₆	—	13	1	no	—
199	C₄×M₁₆	—	26	2	no	—
200	C₂×C₄×C₈	—	13	1	no	—
201	#334	—	13	2	no	—
202	#334	—	13	1	no	—
203	#334	C₂×D₁₃	1	1	yes	characteristic
204	Q₈⋊C₈	—	13	2	no	—
205	Q₈⋊C₈	—	13	2	no	—
206	Q₈⋊C₈	—	13	2	no	—
207	Q₈⋊C₈	C₂×D₁₃	1	2	yes	—
208	#322	—	13	1	no	—
209	#322	—	13	1	no	—
210	#322	—	13	1	no	—
211	#322	C₂×D₁₃	1	1	yes	characteristic
212	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
213	C₂×D₁₃	—	2	1	no	—
214	C₂×D₁₃	—	2	1	no	—
215	C₂×D₁₃	#441	1	2	yes	—
216	C₄×C₁₃	—	2	4	no	—
217	C₄×C₁₃	—	2	1	no	—
218	C₄×C₁₃	—	4	2	no	—
219	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
220	C₄×C₁₃	#568	1	1	yes	characteristic
221	C₄×C₁₃	C₂×#137	1	2	yes	—
222	Dic₁₃	—	2	1	no	—
223	Dic₁₃	—	2	1	no	—
224	Dic₁₃	—	4	2	no	—
225	Dic₁₃	—	4	2	no	—
226	Dic₁₃	#441	1	2	yes	—
227	C₂²×Q₈	—	13	2	no	—
228	C₂×Q₁₆	—	13	2	no	—
229	C₂×Q₁₆	—	13	1	no	—
230	C₂×Q₁₆	—	26	4	no	—
231	C₂×Q₁₆	C₄×D₁₃	1	1	yes	characteristic
232	C₂×M₁₆	—	13	2	no	—
233	C₂×M₁₆	—	26	2	no	—
234	C₂²×C₈	—	13	1	no	—
235	C₂²×C₈	—	13	1	no	—
236	C₄×Q₈	—	13	2	no	—
237	C₄×Q₈	—	13	2	no	—
238	C₄×Q₈	—	13	2	no	—
239	C₄×Q₈	—	26	2	no	—
240	C₄×Q₈	—	26	2	no	—
241	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
242	C₂×C₄⋊C₄	—	13	2	no	—
243	C₂×C₄⋊C₄	—	26	2	no	—
244	C₂×C₄²	—	13	1	no	—
245	C₂×C₄²	—	26	2	no	—
246	#107	—	13	1	no	—
247	#107	—	13	2	no	—
248	#107	C₄×D₁₃	1	1	yes	characteristic
249	C₄⋊C₈	—	2	2	no	—
250	C₄⋊C₈	—	13	2	no	—
251	C₄⋊C₈	—	26	2	no	—
252	C₄⋊C₈	—	26	2	no	—
253	C₄⋊C₈	—	26	1	no	—
254	C₄⋊C₈	—	26	2	no	—
255	C₄⋊C₈	—	26	2	no	—
256	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
257	#103	—	13	2	no	—
258	#103	—	13	2	no	—
259	#103	—	13	2	no	—
260	#103	C₄×D₁₃	1	2	yes	—
261	C₂²⋊C₈	—	26	1	no	—
262	#97	—	13	2	no	—
263	#97	—	26	2	no	—
264	#97	—	26	2	no	—
265	C₄×C₈	—	2	2	no	—
266	C₄×C₈	—	13	1	no	—
267	C₄×C₈	—	13	1	no	—
268	C₄×C₈	—	13	1	no	—
269	C₄×C₈	—	13	1	no	—
270	C₄×C₈	—	13	1	no	—
271	C₄×C₈	—	26	2	no	—
272	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
274	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
275	C₂×C₁₃	D₄×M₁₆	1	1	yes	characteristic
276	C₂×C₁₃	C₂×#441	1	1	yes	characteristic
277	C₂×C₁₃	#2957	1	1	yes	characteristic
278	D₁₃	—	2	2	no	—
279	C₂×Q₈	—	13	2	no	—
280	C₂×Q₈	—	26	2	no	—
281	C₂×Q₈	—	26	4	no	—
282	C₂×Q₈	—	52	2	no	—
283	C₂×Q₈	C₂×C₄×D₁₃	1	2	yes	—
284	C₂²×C₄	—	13	1	no	—
285	C₂²×C₄	—	13	1	no	—
286	C₂²×C₄	—	13	1	no	—
287	C₂²×C₄	—	26	2	no	—
288	C₂²×C₄	—	26	2	no	—
289	Q₁₆	—	26	4	no	—
290	Q₁₆	—	26	2	no	—
291	Q₁₆	#6094	1	4	yes	complemented
292	M₁₆	—	26	4	no	—
293	M₁₆	—	52	4	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	13	1	no	—
297	C₂×C₈	—	13	1	no	—
298	C₂×C₈	—	13	2	no	—
299	C₂×C₈	—	26	2	no	—
300	C₂×C₈	—	26	2	no	—
301	C₂×C₈	—	26	1	no	—
302	C₂×C₈	—	52	1	no	—
303	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
304	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
305	C₂×C₈	D₄×D₁₃	1	2	yes	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	13	2	no	—
308	C₄⋊C₄	—	26	2	no	—
309	C₄⋊C₄	—	26	2	no	—
310	C₄⋊C₄	—	26	2	no	—
311	C₄⋊C₄	—	26	2	no	—
312	C₄⋊C₄	C₂×C₄×D₁₃	1	2	yes	—
313	C₄²	—	2	2	no	—
314	C₄²	—	13	1	no	—
315	C₄²	—	13	1	no	—
316	C₄²	—	13	1	no	—
317	C₄²	—	26	2	no	—
318	C₄²	—	26	2	no	—
319	C₄²	—	26	2	no	—
320	C₄²	C₂³×D₁₃	1	1	yes	characteristic
321	C₁₃	#18042	1	1	yes	characteristic, complemented, Sylow
322	C₂³	—	13	1	no	—
323	Q₈	—	2	2	no	—
324	Q₈	—	26	2	no	—
325	Q₈	—	26	2	no	—
326	Q₈	—	52	4	no	—
327	Q₈	C₂×#6094	1	4	yes	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	13	1	no	—
331	C₂×C₄	—	13	4	no	—
332	C₂×C₄	—	13	1	no	—
333	C₂×C₄	—	13	1	no	—
334	C₂×C₄	—	26	1	no	—
335	C₂×C₄	—	26	1	no	—
336	C₂×C₄	—	26	2	no	—
337	C₂×C₄	—	26	2	no	—
338	C₂×C₄	—	52	1	no	—
339	C₂×C₄	—	52	4	no	—
340	C₂×C₄	—	104	2	no	—
341	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
342	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
343	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
344	C₈	—	2	1	no	—
345	C₈	—	2	2	no	—
346	C₈	—	4	1	no	—
347	C₈	—	4	2	no	—
348	C₈	—	26	1	no	—
349	C₈	—	26	1	no	—
350	C₈	—	26	2	no	—
351	C₈	—	52	1	no	—
352	C₈	—	52	2	no	—
353	C₈	C₂×#6094	1	2	yes	—
354	C₂²	—	13	2	no	—
355	C₂²	—	26	1	no	—
356	C₂²	—	26	1	no	—
357	C₂²	C₄×D₄×D₁₃	1	1	yes	characteristic
358	C₄	—	2	1	no	—
359	C₄	—	2	4	no	—
360	C₄	—	4	2	no	—
361	C₄	—	13	2	no	—
362	C₄	—	26	1	no	—
363	C₄	—	26	1	no	—
364	C₄	—	52	2	no	—
365	C₄	—	52	2	no	—
366	C₄	C₂²×#6094	1	1	yes	characteristic
367	C₄	#11719455	1	1	yes	characteristic
368	C₄	D₁₃×#137	1	2	yes	—
369	C₂	—	26	2	no	—
370	C₂	D₄×#6094	1	1	yes	characteristic
371	C₂	D₁₃×#441	1	1	yes	characteristic
372	C₂	#13123750	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1579	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂²×D₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1579	D₁₃
hypocenter	C₁₃	#18042
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1579

Complements

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

N	H ≅ G/N	type
3. #13113746	C₂	semidirect
4. #13113744	C₂	semidirect
5. #13113707	C₂	semidirect
6. #13113704	C₂	semidirect
7. #13113655	C₂	semidirect
11. C₁₃×#1579	C₂	semidirect
104. C₁₃×Q₁₆	M₁₆	semidirect
143. #1579	D₁₃	semidirect
291. Q₁₆	#6094	semidirect
321. C₁₃	#18042	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#6094	2
C₄×Q₁₆×D₁₃	2
#13113746	2
#13113744	2
#13113707	2
#13113704	2
#13113655	2
#13113590	2
#13113263	2
#13113122	2
C₁₃×#1579	2
#18042	13