Subgroups — #15846016

Order 3328.

662 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125964	C₂	1	1	yes	maximal, characteristic
2	#13125761	C₂	1	1	yes	maximal, characteristic
3	#13123549	C₂	1	1	yes	maximal, characteristic
4	#13114332	C₂	1	4	yes	maximal, complemented
5	#13114208	C₂	1	4	yes	maximal, complemented
6	#13113584	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113521	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113517	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×#88155	—	2	1	no	—
13	#11719182	C₂²	1	1	yes	characteristic
14	C₁₃×#498	—	2	1	no	—
15	C₁₃×#498	—	2	1	no	—
16	C₁₃×#498	C₂²	1	2	yes	—
17	C₁₃×#498	C₂²	1	2	yes	—
18	C₂×C₁₃×#106	—	2	1	no	—
19	C₂×C₁₃×#106	—	2	1	no	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	#11718888	—	2	2	no	—
22	#11718771	—	2	1	no	—
23	#11718749	—	2	1	no	—
24	#11718746	—	2	1	no	—
25	#11718745	C₂²	1	4	yes	—
26	#11718744	C₂²	1	4	yes	—
27	#11718742	—	2	1	no	—
28	#11718740	—	2	1	no	—
29	#11718629	—	2	2	no	—
30	#11718586	C₂²	1	1	yes	characteristic
31	#11718586	C₂²	1	1	yes	characteristic
32	#11718585	C₂²	1	1	yes	characteristic
33	#11718349	—	2	1	no	—
34	#11718349	C₂²	1	2	yes	—
35	C₈×D₅₂	—	2	1	no	—
36	C₈×D₅₂	C₂²	1	2	yes	—
37	#11718139	C₂²	1	4	yes	—
38	#11718138	C₂²	1	4	yes	—
39	#11718050	C₂²	1	2	yes	—
40	#11718049	C₂²	1	2	yes	—
41	C₂²×Q₈×C₁₃	—	2	1	no	—
42	C₂²×Q₈×C₁₃	—	2	1	no	—
43	C₂×#6127	—	2	1	no	—
44	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
46	C₁₃×C₄⋊Q₈	—	2	1	no	—
47	C₁₃×C₄⋊Q₈	—	2	1	no	—
48	C₁₃×C₄⋊Q₈	—	2	1	no	—
49	C₁₃×C₄⋊Q₈	—	2	1	no	—
50	C₁₃×C₄⋊Q₈	—	2	1	no	—
51	C₁₃×C₄⋊Q₈	—	2	1	no	—
52	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
53	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
54	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
55	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
56	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
57	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
58	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
59	C₂×#6106	—	2	4	no	—
60	#88184	—	2	1	no	—
61	#88184	D₄	1	2	yes	—
62	C₂×#6099	—	2	1	no	—
63	#88155	—	4	4	no	—
64	C₂×#6094	—	2	2	no	—
65	C₂×C₈×D₁₃	—	2	2	no	—
66	#88129	—	2	1	no	—
67	#88128	D₄	1	1	yes	characteristic
68	C₄×D₅₂	—	2	1	no	—
69	C₄×D₅₂	C₂³	1	2	yes	—
70	#88123	D₄	1	1	yes	characteristic
71	C₁₃×#106	—	2	2	no	—
72	C₁₃×#106	—	2	2	no	—
73	C₁₃×#106	—	2	2	no	—
74	C₁₃×#106	—	2	2	no	—
75	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
76	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
79	#88075	—	2	4	no	—
80	C₂²⋊#1055	—	2	1	no	—
81	C₂²⋊#1055	—	2	1	no	—
82	#88060	—	2	2	no	—
83	#88060	—	2	2	no	—
84	#88049	—	2	4	no	—
85	#88048	—	2	4	no	—
86	C₄⋊#1055	—	2	1	no	—
87	C₄⋊#1055	C₂³	1	2	yes	—
88	C₄⋊#1055	C₂³	1	2	yes	—
89	C₄⋊#1055	C₂³	1	2	yes	—
90	C₄×#1055	—	2	1	no	—
91	C₄×#1055	—	2	1	no	—
92	#19191	—	13	1	no	maximal, Sylow
93	C₂×Q₈×C₁₃	—	2	2	no	—
94	C₂×Q₈×C₁₃	—	2	2	no	—
95	C₂×Q₈×C₁₃	—	4	1	no	—
96	C₂×Q₈×C₁₃	—	4	1	no	—
97	C₂×Q₈×C₁₃	—	8	2	no	—
98	C₂×Q₈×C₁₃	—	8	2	no	—
99	C₂²×C₄×C₁₃	—	4	1	no	—
100	C₂²×C₄×C₁₃	—	4	1	no	—
101	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
104	C₂×C₂²⋊D₁₃	—	2	2	no	—
105	#6127	—	4	2	no	—
106	#6127	—	4	2	no	—
107	C₂×D₅₂	—	2	1	no	—
108	C₂×D₅₂	—	2	1	no	—
109	C₂×D₅₂	C₂×D₄	1	2	yes	—
110	C₂×C₄×D₁₃	—	2	2	no	—
111	C₂×Dic₂₆	—	2	1	no	—
112	C₂×C₈×C₁₃	—	2	2	no	—
113	C₂×C₈×C₁₃	SD₁₆	1	4	yes	complemented
114	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
118	C₁₃×C₄⋊C₄	—	2	2	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	—	4	1	no	—
123	C₁₃×C₄⋊C₄	—	4	1	no	—
124	C₁₃×C₄⋊C₄	—	4	1	no	—
125	C₁₃×C₄⋊C₄	—	4	1	no	—
126	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₁₃	#41	1	1	yes	characteristic
129	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
130	#6106	—	8	4	no	—
131	#6103	—	2	2	no	—
132	#6103	—	2	2	no	—
133	C₄⋊Dic₁₃	—	2	1	no	—
134	C₄⋊Dic₁₃	C₂×D₄	1	2	yes	—
135	#6099	—	4	2	no	—
136	C₂×#1055	—	2	2	no	—
137	C₂×#1055	—	2	2	no	—
138	C₂×#1055	—	2	2	no	—
139	#6094	—	4	4	no	—
140	C₈×D₁₃	—	4	4	no	—
141	#3227	—	13	1	no	—
142	#3206	—	13	1	no	—
143	C₂×#498	D₁₃	1	1	yes	characteristic, complemented
144	#2931	—	13	1	no	—
145	#1691	—	13	4	no	—
146	#1671	—	13	4	no	—
147	#1564	—	13	1	no	—
148	#1549	—	13	1	no	—
149	#1548	—	13	1	no	—
150	C₂³×C₁₃	#121	1	1	yes	characteristic
151	C₂²×D₁₃	—	2	2	no	—
152	Q₈×C₁₃	—	4	4	no	—
153	Q₈×C₁₃	—	8	2	no	—
154	Q₈×C₁₃	—	8	2	no	—
155	Q₈×C₁₃	—	8	2	no	—
156	C₂×C₄×C₁₃	—	2	2	no	—
157	C₂×C₄×C₁₃	—	2	2	no	—
158	C₂×C₄×C₁₃	—	4	2	no	—
159	C₂×C₄×C₁₃	—	4	2	no	—
160	C₂×C₄×C₁₃	—	8	2	no	—
161	C₂×C₄×C₁₃	—	8	2	no	—
162	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
163	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
164	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
165	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
166	C₂×C₄×C₁₃	#137	1	2	yes	—
167	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
170	C₂²⋊D₁₃	—	4	4	no	—
171	C₂×Dic₁₃	—	2	2	no	—
172	D₅₂	—	2	4	no	—
173	D₅₂	—	4	2	no	—
174	D₅₂	—	4	2	no	—
175	C₄×D₁₃	—	4	2	no	—
176	C₄×D₁₃	—	4	2	no	—
177	Dic₂₆	—	4	2	no	—
178	C₈×C₁₃	—	2	1	no	—
179	C₈×C₁₃	—	2	1	no	—
180	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
181	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
182	#1055	—	4	2	no	—
183	#1055	—	4	2	no	—
184	#1055	—	4	2	no	—
185	C₂×#131	—	26	1	no	—
186	#532	—	13	1	no	—
187	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
188	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
189	#498	—	2	1	no	—
190	#498	—	2	1	no	—
191	#498	—	26	1	no	—
192	#498	C₂×D₁₃	1	2	yes	—
193	#498	C₂×D₁₃	1	2	yes	—
194	#492	—	26	1	no	—
195	#487	—	26	1	no	—
196	#486	—	26	1	no	—
197	#481	—	26	2	no	—
198	#476	—	13	4	no	—
199	#461	—	13	4	no	—
200	#448	—	26	2	no	—
201	#436	—	13	2	no	—
202	#436	—	26	1	no	—
203	C₈×D₄	—	13	2	no	—
204	C₈×D₄	—	26	1	no	—
205	#431	—	13	1	no	—
206	#431	—	13	1	no	—
207	C₂×#106	—	2	1	no	—
208	C₂×#106	—	2	1	no	—
209	#423	—	13	1	no	—
210	#408	—	26	1	no	—
211	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
212	#335	—	13	2	no	—
213	#334	—	13	2	no	—
214	#333	—	13	4	no	—
215	#332	—	13	4	no	—
216	C₂²×C₁₃	—	2	1	no	—
217	C₂²×C₁₃	—	2	1	no	—
218	C₂²×C₁₃	#461	1	2	yes	—
219	C₂²×C₁₃	C₂×#121	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	1	no	—
224	C₄×C₁₃	—	2	1	no	—
225	C₄×C₁₃	—	8	2	no	—
226	C₄×C₁₃	—	8	2	no	—
227	C₄×C₁₃	#461	1	2	yes	—
228	C₄×C₁₃	#471	1	2	yes	—
229	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
230	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
231	C₄×C₁₃	#577	1	2	yes	—
232	Dic₁₃	—	4	2	no	—
233	C₂×#41	—	26	1	no	—
234	C₂²×Q₈	—	2	1	no	—
235	C₂²×Q₈	—	2	1	no	—
236	C₂×SD₁₆	—	26	4	no	—
237	#131	—	52	4	no	—
238	C₂×M₁₆	—	26	1	no	—
239	C₂×M₁₆	—	26	2	no	—
240	C₂²×C₈	—	26	2	no	—
241	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
242	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
243	C₄⋊Q₈	—	2	1	no	—
244	C₄⋊Q₈	—	2	1	no	—
245	C₄⋊Q₈	—	2	1	no	—
246	C₄⋊Q₈	—	2	1	no	—
247	C₄⋊Q₈	—	2	1	no	—
248	C₄⋊Q₈	—	2	1	no	—
249	C₄⋊Q₈	—	13	1	no	—
250	C₄⋊Q₈	C₂²×D₁₃	1	2	yes	—
251	C₄⋊Q₈	C₂²×D₁₃	1	2	yes	—
252	#127	—	13	1	no	—
253	#124	—	26	1	no	—
254	#121	—	13	2	no	—
255	#121	—	26	1	no	—
256	C₄×D₄	—	13	2	no	—
257	C₄×D₄	—	26	1	no	—
258	C₂×C₄⋊C₄	—	2	1	no	—
259	C₂×C₄⋊C₄	—	2	1	no	—
260	C₂×C₄⋊C₄	—	2	1	no	—
261	C₂×C₄⋊C₄	—	2	1	no	—
262	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
263	#106	—	2	2	no	—
264	#106	—	2	2	no	—
265	#106	—	2	2	no	—
266	#106	—	2	2	no	—
267	#106	—	26	4	no	—
268	C₄⋊C₈	—	13	2	no	—
269	C₄⋊C₈	—	13	2	no	—
270	C₄⋊C₈	—	13	2	no	—
271	C₄⋊C₈	—	26	1	no	—
272	#103	—	26	4	no	—
273	#102	—	26	4	no	—
274	C₂²⋊C₈	—	26	1	no	—
275	C₂²⋊C₈	—	26	1	no	—
276	C₂²⋊C₈	—	26	2	no	—
277	C₂²⋊C₈	—	26	2	no	—
278	C₄×C₈	—	26	1	no	—
279	C₄×C₈	—	26	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	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
284	C₂×C₁₃	—	2	2	no	—
285	C₂×C₁₃	C₂×#461	1	1	yes	characteristic
286	C₂×C₁₃	#3057	1	1	yes	characteristic
287	C₂×C₁₃	C₂×#471	1	1	yes	characteristic
288	D₁₃	—	8	2	no	—
289	#41	—	52	2	no	—
290	#41	—	52	2	no	—
291	C₂×Q₈	—	2	2	no	—
292	C₂×Q₈	—	2	2	no	—
293	C₂×Q₈	—	4	1	no	—
294	C₂×Q₈	—	4	1	no	—
295	C₂×Q₈	—	8	2	no	—
296	C₂×Q₈	—	8	2	no	—
297	C₂×Q₈	—	26	1	no	—
298	C₂×D₄	—	13	2	no	—
299	C₂×D₄	—	26	2	no	—
300	C₂×D₄	—	26	1	no	—
301	C₂×D₄	—	26	1	no	—
302	C₂²×C₄	—	4	1	no	—
303	C₂²×C₄	—	4	1	no	—
304	C₂²×C₄	—	26	2	no	—
305	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
306	C₂²×C₄	#6129	1	1	yes	characteristic
307	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
308	SD₁₆	—	104	4	no	—
309	M₁₆	—	52	2	no	—
310	M₁₆	—	52	4	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	26	2	no	—
313	C₂×C₈	—	26	2	no	—
314	C₂×C₈	—	26	2	no	—
315	C₂×C₈	—	52	4	no	—
316	C₂×C₈	#6106	1	4	yes	complemented
317	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
318	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
319	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
320	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	4	1	no	—
326	C₄⋊C₄	—	4	1	no	—
327	C₄⋊C₄	—	4	1	no	—
328	C₄⋊C₄	—	4	1	no	—
329	C₄⋊C₄	—	13	2	no	—
330	C₄⋊C₄	—	26	1	no	—
331	C₂²⋊C₄	—	26	2	no	—
332	C₂²⋊C₄	—	26	2	no	—
333	C₄²	D₄×D₁₃	1	1	yes	characteristic
334	C₄²	#6129	1	1	yes	characteristic
335	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
336	C₄²	C₂³×D₁₃	1	1	yes	characteristic
337	C₁₃	#19191	1	1	yes	characteristic, complemented, Sylow
338	C₂³	—	26	2	no	—
339	C₂³	#88192	1	1	yes	characteristic
340	Q₈	—	4	4	no	—
341	Q₈	—	8	2	no	—
342	Q₈	—	8	2	no	—
343	Q₈	—	8	2	no	—
344	Q₈	—	52	2	no	—
345	D₄	—	26	4	no	—
346	D₄	—	52	4	no	—
347	D₄	—	52	2	no	—
348	D₄	—	52	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	8	2	no	—
354	C₂×C₄	—	8	2	no	—
355	C₂×C₄	—	26	2	no	—
356	C₂×C₄	—	52	2	no	—
357	C₂×C₄	—	52	2	no	—
358	C₂×C₄	#88192	1	1	yes	characteristic
359	C₂×C₄	#88192	1	1	yes	characteristic
360	C₂×C₄	#88192	1	1	yes	characteristic
361	C₂×C₄	C₂×#6106	1	2	yes	—
362	C₂×C₄	#88196	1	2	yes	—
363	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
364	C₂×C₄	C₂×#6129	1	1	yes	characteristic
365	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
366	C₈	—	2	1	no	—
367	C₈	—	2	1	no	—
368	C₈	—	52	2	no	—
369	C₈	—	52	2	no	—
370	C₈	—	52	2	no	—
371	C₈	C₂×#6106	1	2	yes	—
372	C₈	C₂×#6106	1	2	yes	—
373	C₂²	—	2	1	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	52	2	no	—
376	C₂²	—	52	2	no	—
377	C₂²	—	52	2	no	—
378	C₂²	#11718744	1	2	yes	—
379	C₂²	C₂×#88192	1	1	yes	characteristic
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	—	8	2	no	—
383	C₄	—	8	2	no	—
384	C₄	—	52	2	no	—
385	C₄	#11718744	1	2	yes	—
386	C₄	#11718838	1	2	yes	—
387	C₄	#11719467	1	2	yes	—
388	C₄	C₂²×#6106	1	1	yes	characteristic
389	C₄	C₂×#88196	1	1	yes	characteristic
390	C₂	—	2	2	no	—
391	C₂	—	104	2	no	—
392	C₂	C₂×#11718744	1	1	yes	characteristic
393	C₂	#13124471	1	1	yes	characteristic
394	C₂	C₂×#11718838	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88192
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#498	C₂
socle	C₂²×C₁₃	C₂×#121
prime radical	#11719182	C₂²
hypercenter	C₂×#498	D₁₃
hypocenter	C₁₃	#19191
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#498

Complements

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

N	H ≅ G/N	type
4. #13114332	C₂	semidirect
5. #13114208	C₂	semidirect
6. #13113584	C₂	semidirect
7. #13113521	C₂	semidirect
8. #13113517	C₂	semidirect
9. C₂×C₁₃×#498	C₂	semidirect
113. C₂×C₈×C₁₃	SD₁₆	semidirect
143. C₂×#498	D₁₃	semidirect
316. C₂×C₈	#6106	semidirect
337. C₁₃	#19191	semidirect

Maximal subgroups

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

subgroup	index
#13125964	2
#13125761	2
#13123549	2
#13114332	2
#13114208	2
#13113584	2
#13113521	2
#13113517	2
C₂×C₁₃×#498	2
#19191	13