Subgroups — #15832751

Order 3328.

662 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123921	C₂	1	1	yes	maximal, characteristic, complemented
2	#13123876	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#6106	C₂	1	1	yes	maximal, characteristic
4	#13113741	C₂	1	4	yes	maximal, complemented
5	#13113723	C₂	1	4	yes	maximal, complemented
6	#13113582	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113228	C₂	1	1	yes	maximal, characteristic
8	#13113120	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#1479	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×#6106	C₄	1	1	yes	characteristic
12	C₂×C₄×D₅₂	C₂²	1	1	yes	characteristic
13	C₁₃×#431	C₂²	1	1	yes	characteristic
14	C₁₃×#431	C₂²	1	1	yes	characteristic
15	C₂×#88075	C₄	1	1	yes	characteristic
16	C₄×#6106	C₄	1	2	yes	—
17	C₄×#6106	C₄	1	2	yes	—
18	C₄×#6106	C₂²	1	2	yes	—
19	C₄×#6106	C₂²	1	2	yes	—
20	#11718685	—	2	2	no	—
21	#11718685	C₂²	1	4	yes	complemented
22	C₂×#88049	C₄	1	1	yes	characteristic
23	C₂×#88048	C₄	1	1	yes	characteristic
24	#11718587	—	2	2	no	—
25	#11718586	C₂²	1	1	yes	characteristic
26	#11718586	C₂²	1	1	yes	characteristic
27	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
28	#11718499	C₂²	1	4	yes	—
29	#11718387	—	2	2	no	—
30	#11718349	—	2	2	no	—
31	C₁₃×Q₈⋊C₈	C₂²	1	4	yes	complemented
32	#11718072	C₂²	1	2	yes	complemented
33	#11718071	C₂²	1	2	yes	complemented
34	#11718051	C₂²	1	4	yes	—
35	#11718046	C₂²	1	2	yes	complemented
36	C₈×#1055	C₂²	1	2	yes	complemented
37	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×D₅₂	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×D₁₃	—	2	1	no	—
40	C₄×Q₈×C₁₃	—	2	1	no	—
41	C₄×Q₈×C₁₃	—	2	1	no	—
42	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
43	C₄×Q₈×C₁₃	C₂³	1	2	yes	—
44	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
45	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	—	2	1	no	—
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×#6106	—	2	4	no	—
49	C₂×#6106	C₂×C₄	1	2	yes	—
50	C₂×#6106	C₂×C₄	1	2	yes	—
51	C₂×#6103	—	2	1	no	—
52	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
54	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
55	C₂×#6094	—	2	4	no	—
56	C₄×D₅₂	—	2	1	no	—
57	C₄×D₅₂	—	2	1	no	—
58	C₄×D₅₂	C₂×C₄	1	2	yes	—
59	C₄×D₅₂	C₂³	1	2	yes	—
60	C₁₃×C₄⋊C₈	—	2	1	no	—
61	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
62	C₁₃×C₂²⋊C₈	—	2	1	no	—
63	C₁₃×C₂²⋊C₈	—	2	1	no	—
64	C₄×C₈×C₁₃	—	2	1	no	—
65	C₄×C₈×C₁₃	C₂³	1	2	yes	—
66	#88075	C₂×C₄	1	2	yes	—
67	#88075	C₂×C₄	1	2	yes	—
68	C₂²⋊#1055	—	2	1	no	—
69	C₂²⋊#1055	D₄	1	2	yes	—
70	#88060	—	2	4	no	—
71	#88049	C₂×C₄	1	2	yes	—
72	#88049	C₂×C₄	1	2	yes	—
73	#88048	C₂×C₄	1	2	yes	—
74	#88048	C₂×C₄	1	2	yes	—
75	C₄⋊#1055	—	2	4	no	—
76	C₄⋊#1055	D₄	1	2	yes	—
77	C₄⋊#1055	C₂³	1	2	yes	—
78	#88043	—	2	4	no	—
79	C₄×#1055	—	2	1	no	—
80	C₄×#1055	C₂×C₄	1	1	yes	characteristic
81	C₄×#1055	C₂×C₄	1	1	yes	characteristic
82	C₄×#1055	C₂³	1	1	yes	characteristic
83	C₄×#1055	C₂³	1	1	yes	characteristic
84	C₄×#1055	C₂³	1	2	yes	—
85	#17896	—	13	1	no	maximal, Sylow
86	C₂³×D₁₃	—	2	1	no	—
87	C₂×Q₈×C₁₃	—	2	1	no	—
88	C₂×Q₈×C₁₃	—	2	2	no	—
89	C₂×Q₈×C₁₃	—	2	2	no	—
90	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₁₃	—	2	1	no	—
92	C₂²×C₄×C₁₃	—	2	1	no	—
93	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
96	C₂²×Dic₁₃	—	2	1	no	—
97	C₂×D₅₂	—	2	2	no	—
98	C₂×D₅₂	—	2	2	no	—
99	C₂×D₅₂	—	2	1	no	—
100	C₂×D₅₂	C₂²×C₄	1	2	yes	—
101	C₂×C₄×D₁₃	—	2	2	no	—
102	C₂×C₄×D₁₃	—	2	2	no	—
103	C₂×C₄×D₁₃	—	8	2	no	—
104	C₂×C₈×C₁₃	—	2	2	no	—
105	C₂×C₈×C₁₃	—	2	2	no	—
106	C₁₃×C₄⋊C₄	—	2	2	no	—
107	C₁₃×C₄⋊C₄	—	2	1	no	—
108	C₁₃×C₄⋊C₄	—	2	2	no	—
109	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
110	C₄²×C₁₃	—	2	2	no	—
111	C₄²×C₁₃	—	2	2	no	—
112	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₁₃	#41	1	1	yes	characteristic
115	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
116	#6106	—	2	4	no	—
117	#6106	—	2	4	no	—
118	#6103	—	2	2	no	—
119	#6103	—	2	2	no	—
120	C₄⋊Dic₁₃	—	2	1	no	—
121	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
122	C₂×#1055	—	2	2	no	—
123	C₂×#1055	—	2	2	no	—
124	C₂×#1055	M₁₆	1	4	yes	complemented
125	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
126	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
127	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
128	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
129	C₂×#1055	C₂×D₄	1	4	yes	—
130	#6094	—	8	4	no	—
131	#2983	—	13	1	no	—
132	#2975	—	13	1	no	—
133	C₂×C₄×SD₁₆	—	13	1	no	—
134	#1592	—	13	4	no	—
135	#1590	—	13	4	no	—
136	#1564	—	13	1	no	—
137	#1479	D₁₃	1	1	yes	characteristic, complemented
138	#1478	—	13	1	no	—
139	#1450	—	13	1	no	—
140	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
141	C₂²×D₁₃	—	2	2	no	—
142	C₂²×D₁₃	—	4	2	no	—
143	C₂²×D₁₃	—	8	1	no	—
144	C₂²×D₁₃	—	8	1	no	—
145	Q₈×C₁₃	—	2	2	no	—
146	Q₈×C₁₃	—	2	2	no	—
147	Q₈×C₁₃	—	4	2	no	—
148	C₂×C₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	—	2	2	no	—
150	C₂×C₄×C₁₃	—	2	2	no	—
151	C₂×C₄×C₁₃	—	4	1	no	—
152	C₂×C₄×C₁₃	—	4	2	no	—
153	C₂×C₄×C₁₃	—	8	1	no	—
154	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
158	C₂×C₄×C₁₃	#131	1	2	yes	—
159	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
162	C₂×Dic₁₃	—	2	2	no	—
163	C₂×Dic₁₃	—	8	1	no	—
164	D₅₂	—	2	2	no	—
165	D₅₂	—	2	2	no	—
166	D₅₂	—	4	2	no	—
167	C₄×D₁₃	—	4	4	no	—
168	C₄×D₁₃	—	8	2	no	—
169	C₈×C₁₃	—	4	2	no	—
170	C₈×C₁₃	—	4	2	no	—
171	#1055	—	2	1	no	—
172	#1055	—	2	1	no	—
173	#1055	—	2	4	no	—
174	#1055	—	4	2	no	—
175	#1055	C₂×M₁₆	1	2	yes	—
176	#1055	C₂×M₁₆	1	2	yes	—
177	C₂²×SD₁₆	—	13	1	no	—
178	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
179	C₂×C₄×D₄	—	13	1	no	—
180	#446	—	13	4	no	—
181	#446	—	26	2	no	—
182	C₄×SD₁₆	—	13	2	no	—
183	C₄×SD₁₆	—	13	2	no	—
184	C₄×SD₁₆	—	13	2	no	—
185	C₄×SD₁₆	—	13	2	no	—
186	#436	—	13	4	no	—
187	#436	—	26	2	no	—
188	#432	—	26	2	no	—
189	#431	—	13	1	no	—
190	#431	—	13	1	no	—
191	#431	C₂×D₁₃	1	1	yes	characteristic
192	#431	C₂×D₁₃	1	1	yes	characteristic
193	C₂×#106	—	13	1	no	—
194	C₂×#103	—	13	1	no	—
195	C₂×#102	—	13	1	no	—
196	#407	—	26	2	no	—
197	C₂×C₄×C₈	—	13	1	no	—
198	#335	—	13	2	no	—
199	#334	—	13	2	no	—
200	Q₈⋊C₈	C₂×D₁₃	1	4	yes	complemented
201	D₄⋊C₈	—	13	4	no	—
202	#322	—	13	2	no	—
203	C₈²	—	13	2	no	—
204	C₂²×C₁₃	—	2	1	no	—
205	C₂²×C₁₃	—	2	1	no	—
206	C₂²×C₁₃	#436	1	2	yes	—
207	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
208	C₂×D₁₃	—	4	2	no	—
209	C₂×D₁₃	—	4	2	no	—
210	C₂×D₁₃	—	4	2	no	—
211	C₂×D₁₃	—	8	4	no	—
212	C₄×C₁₃	—	2	1	no	—
213	C₄×C₁₃	—	2	1	no	—
214	C₄×C₁₃	—	2	1	no	—
215	C₄×C₁₃	—	4	2	no	—
216	C₄×C₁₃	—	4	2	no	—
217	C₄×C₁₃	#436	1	2	yes	—
218	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
219	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
220	C₄×C₁₃	#577	1	2	yes	—
221	Dic₁₃	—	4	2	no	—
222	C₂²×Q₈	C₄×D₁₃	1	1	yes	characteristic
223	C₂²×D₄	—	13	1	no	—
224	C₂³×C₄	—	26	1	no	—
225	C₂×SD₁₆	—	13	2	no	—
226	C₂×SD₁₆	—	13	2	no	—
227	C₂×SD₁₆	—	26	4	no	—
228	C₂×M₁₆	—	26	4	no	—
229	C₂²×C₈	—	13	1	no	—
230	C₂²×C₈	—	13	1	no	—
231	C₄×Q₈	—	2	1	no	—
232	C₄×Q₈	—	2	1	no	—
233	C₄×Q₈	C₄×D₁₃	1	2	yes	—
234	C₄×Q₈	C₂²×D₁₃	1	2	yes	—
235	C₄×D₄	—	13	2	no	—
236	C₄×D₄	—	13	2	no	—
237	C₄×D₄	—	26	1	no	—
238	C₄×D₄	—	26	1	no	—
239	C₂×C₄⋊C₄	—	2	1	no	—
240	C₂×C₄⋊C₄	—	13	1	no	—
241	C₂×C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
242	C₂×C₂²⋊C₄	—	26	1	no	—
243	C₂×C₄²	—	2	1	no	—
244	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
245	#106	—	13	2	no	—
246	#106	—	13	2	no	—
247	C₄⋊C₈	—	2	1	no	—
248	C₄⋊C₈	—	13	2	no	—
249	C₄⋊C₈	—	13	2	no	—
250	C₄⋊C₈	—	26	4	no	—
251	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
252	#103	—	13	2	no	—
253	#103	—	13	2	no	—
254	#102	—	13	2	no	—
255	#102	—	13	2	no	—
256	C₂²⋊C₈	—	2	1	no	—
257	C₂²⋊C₈	—	2	1	no	—
258	C₂²⋊C₈	—	13	2	no	—
259	C₂²⋊C₈	—	26	4	no	—
260	C₂²⋊C₈	—	26	1	no	—
261	#97	—	26	4	no	—
262	C₄×C₈	—	2	1	no	—
263	C₄×C₈	—	13	1	no	—
264	C₄×C₈	—	13	1	no	—
265	C₄×C₈	—	13	1	no	—
266	C₄×C₈	—	13	2	no	—
267	C₄×C₈	—	13	1	no	—
268	C₄×C₈	—	26	1	no	—
269	C₄×C₈	C₂²×D₁₃	1	2	yes	—
270	C₂×C₁₃	—	2	2	no	—
271	C₂×C₁₃	C₂×#436	1	1	yes	characteristic
272	C₂×C₁₃	#2949	1	1	yes	characteristic
273	C₂×C₁₃	#2957	1	1	yes	characteristic
274	D₁₃	—	8	2	no	—
275	C₂⁴	—	26	1	no	—
276	C₂×Q₈	—	2	1	no	—
277	C₂×Q₈	—	2	2	no	—
278	C₂×Q₈	—	2	2	no	—
279	C₂×Q₈	C₂×C₄×D₁₃	1	2	yes	—
280	C₂×D₄	—	13	2	no	—
281	C₂×D₄	—	26	1	no	—
282	C₂×D₄	—	26	2	no	—
283	C₂×D₄	—	26	2	no	—
284	C₂²×C₄	—	2	1	no	—
285	C₂²×C₄	—	2	1	no	—
286	C₂²×C₄	—	26	1	no	—
287	C₂²×C₄	—	26	2	no	—
288	C₂²×C₄	—	26	2	no	—
289	C₂²×C₄	—	104	2	no	—
290	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
291	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
292	C₂²×C₄	#6129	1	1	yes	characteristic
293	SD₁₆	—	26	4	no	—
294	SD₁₆	—	26	4	no	—
295	M₁₆	—	104	4	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	13	4	no	—
299	C₂×C₈	—	13	1	no	—
300	C₂×C₈	—	13	1	no	—
301	C₂×C₈	—	13	1	no	—
302	C₂×C₈	—	13	1	no	—
303	C₂×C₈	—	13	4	no	—
304	C₂×C₈	—	26	2	no	—
305	C₂×C₈	—	26	2	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	2	1	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	13	2	no	—
310	C₄⋊C₄	—	26	1	no	—
311	C₄⋊C₄	C₂×C₄×D₁₃	1	2	yes	—
312	C₂²⋊C₄	—	26	2	no	—
313	C₂²⋊C₄	—	26	2	no	—
314	C₄²	—	2	2	no	—
315	C₄²	—	2	2	no	—
316	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
317	C₄²	D₄×D₁₃	1	1	yes	characteristic
318	C₄²	#6129	1	1	yes	characteristic
319	C₄²	C₂³×D₁₃	1	1	yes	characteristic
320	C₁₃	#17896	1	1	yes	characteristic, complemented, Sylow
321	C₂³	—	26	2	no	—
322	C₂³	—	52	2	no	—
323	C₂³	—	104	1	no	—
324	C₂³	—	104	1	no	—
325	C₂³	#88135	1	1	yes	characteristic
326	Q₈	—	2	2	no	—
327	Q₈	—	2	2	no	—
328	Q₈	—	4	2	no	—
329	D₄	—	26	2	no	—
330	D₄	—	26	2	no	—
331	D₄	—	52	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	8	1	no	—
338	C₂×C₄	—	26	2	no	—
339	C₂×C₄	—	52	4	no	—
340	C₂×C₄	—	104	2	no	—
341	C₂×C₄	—	104	1	no	—
342	C₂×C₄	#88135	1	1	yes	characteristic
343	C₂×C₄	#88135	1	1	yes	characteristic
344	C₂×C₄	#88135	1	1	yes	characteristic
345	C₂×C₄	#88155	1	2	yes	—
346	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
347	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
348	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
349	C₂×C₄	C₂×#6129	1	1	yes	characteristic
350	C₈	—	4	2	no	—
351	C₈	—	4	2	no	—
352	C₈	—	13	2	no	—
353	C₈	—	13	2	no	—
354	C₈	—	26	1	no	—
355	C₈	—	26	1	no	—
356	C₈	—	26	4	no	—
357	C₈	—	52	2	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	52	2	no	—
361	C₂²	—	52	2	no	—
362	C₂²	—	52	2	no	—
363	C₂²	—	104	4	no	—
364	C₂²	#11718390	1	2	yes	—
365	C₂²	C₂×#88135	1	1	yes	characteristic
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	4	2	no	—
370	C₄	—	4	2	no	—
371	C₄	—	52	2	no	—
372	C₄	#11718390	1	2	yes	—
373	C₄	C₂×#88155	1	1	yes	characteristic
374	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
375	C₄	#11719483	1	2	yes	—
376	C₂	—	2	2	no	—
377	C₂	—	104	2	no	—
378	C₂	C₂×#11718390	1	1	yes	characteristic
379	C₂	#13123691	1	1	yes	characteristic
380	C₂	#13123749	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×#1479	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂²×D₅₂	C₂×C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1479	D₁₃
hypocenter	C₁₃	#17896
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1479

Complements

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

N	H ≅ G/N	type
1. #13123921	C₂	semidirect
4. #13113741	C₂	semidirect
5. #13113723	C₂	semidirect
6. #13113582	C₂	semidirect
8. #13113120	C₂	semidirect
9. C₁₃×#1479	C₂	semidirect
21. #11718685	C₂²	semidirect
31. C₁₃×Q₈⋊C₈	C₂²	semidirect
32. #11718072	C₂²	semidirect
33. #11718071	C₂²	semidirect
35. #11718046	C₂²	semidirect
36. C₈×#1055	C₂²	semidirect
124. C₂×#1055	M₁₆	semidirect
137. #1479	D₁₃	semidirect
200. Q₈⋊C₈	C₂×D₁₃	semidirect
320. C₁₃	#17896	semidirect

Maximal subgroups

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

subgroup	index
#13123921	2
#13123876	2
C₂×C₄×#6106	2
#13113741	2
#13113723	2
#13113582	2
#13113228	2
#13113120	2
C₁₃×#1479	2
#17896	13