Subgroups — #16018005

Order 3328.

548 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125422	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117044	C₂	1	2	yes	maximal, complemented
3	#13116988	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116983	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116886	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116881	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116776	C₂	1	2	yes	maximal
8	#13116705	C₂	1	2	yes	maximal, complemented
9	#13116700	C₂	1	2	yes	maximal, complemented
10	#13116639	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₁₃×#135	C₂	1	1	yes	maximal, characteristic
12	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
13	C₄×C₁₃×#41	—	2	1	no	—
14	#11719175	—	2	1	no	—
15	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
16	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
17	C₄×C₁₃×SD₁₆	—	2	1	no	—
18	C₄×C₁₃×SD₁₆	—	2	1	no	—
19	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
20	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
21	C₁₃×#427	C₂²	1	1	yes	characteristic
22	C₁₃×#416	—	2	1	no	—
23	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₂×#88069	C₂²	1	2	yes	—
25	C₂×#88067	C₂²	1	2	yes	—
26	#11718777	—	2	1	no	—
27	#11718777	C₂²	1	2	yes	—
28	#11718360	C₂²	1	1	yes	characteristic
29	C₄×Dic₅₂	—	2	1	no	—
30	C₄×Dic₅₂	C₂²	1	2	yes	—
31	#11718343	C₂²	1	1	yes	characteristic
32	#11718342	—	2	1	no	—
33	#11718156	C₂²	1	2	yes	—
34	#11718154	C₂²	1	2	yes	—
35	C₂²⋊#6090	C₂²	1	2	yes	—
36	#11718084	C₂²	1	2	yes	—
37	#11718084	C₂²	1	2	yes	—
38	#11718082	C₂²	1	2	yes	—
39	#11718081	C₂²	1	2	yes	—
40	C₄⋊#6090	C₂²	1	1	yes	characteristic
41	C₄⋊#6090	C₂²	1	1	yes	characteristic
42	#11718053	C₂²	1	2	yes	—
43	C₂×C₁₃×#41	—	2	1	no	—
44	C₁₃×#135	—	2	2	no	—
45	C₁₃×#135	—	2	2	no	—
46	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
47	C₂×C₁₃×SD₁₆	—	2	1	no	—
48	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₄×Q₈×C₁₃	—	2	1	no	—
52	C₄×Q₈×C₁₃	—	2	1	no	—
53	C₄×D₄×C₁₃	—	2	1	no	—
54	C₄×D₄×C₁₃	—	2	1	no	—
55	C₄×D₄×C₁₃	—	4	1	no	—
56	C₄×D₄×C₁₃	—	4	1	no	—
57	C₁₃×#117	—	2	1	no	—
58	C₁₃×#117	—	4	1	no	—
59	C₂×C₄²×C₁₃	—	4	1	no	—
60	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
61	#88180	—	4	1	no	—
62	#88178	—	2	2	no	—
63	#88178	—	4	1	no	—
64	C₂×Dic₅₂	—	2	1	no	—
65	C₂×Dic₅₂	C₂³	1	2	yes	—
66	#88124	—	4	1	no	—
67	#88123	—	2	1	no	—
68	#88123	—	2	1	no	—
69	C₄×Dic₂₆	—	2	2	no	—
70	C₄×Dic₂₆	—	4	1	no	—
71	C₁₃×#107	C₂³	1	1	yes	characteristic
72	C₁₃×#107	C₂³	1	1	yes	characteristic
73	C₁₃×#106	—	2	1	no	—
74	C₁₃×#103	—	2	1	no	—
75	C₁₃×#103	—	2	1	no	—
76	C₁₃×#102	—	2	1	no	—
77	C₁₃×#102	—	2	1	no	—
78	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
79	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
80	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
82	#88069	—	2	4	no	—
83	#88067	—	2	4	no	—
84	#88058	—	2	1	no	—
85	#88058	C₂³	1	2	yes	—
86	#88056	—	2	2	no	—
87	#88056	—	2	2	no	—
88	C₂×#6090	C₂³	1	2	yes	—
89	C₂×#6090	C₂³	1	2	yes	—
90	#33676	—	13	1	no	maximal, Sylow
91	C₁₃×#41	—	2	4	no	—
92	C₁₃×#41	—	4	2	no	—
93	C₂×Q₈×C₁₃	—	2	1	no	—
94	C₂×D₄×C₁₃	—	2	1	no	—
95	C₂×D₄×C₁₃	—	4	1	no	—
96	C₂²×C₄×C₁₃	—	4	1	no	—
97	C₂²×C₄×C₁₃	—	4	1	no	—
98	C₂²×C₄×C₁₃	—	4	1	no	—
99	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
100	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
101	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×Dic₂₆	—	2	2	no	—
103	C₂×Dic₂₆	—	4	1	no	—
104	C₁₃×Q₁₆	—	2	1	no	—
105	C₁₃×Q₁₆	#41	1	2	yes	—
106	C₁₃×SD₁₆	—	2	4	no	—
107	C₁₃×D₈	—	2	1	no	—
108	C₁₃×D₈	#41	1	2	yes	—
109	C₂×C₈×C₁₃	—	2	1	no	—
110	C₂×C₈×C₁₃	—	2	1	no	—
111	C₂×C₈×C₁₃	—	4	1	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	C₂⁴	1	1	yes	characteristic
116	C₁₃×C₄⋊C₄	—	2	1	no	—
117	C₁₃×C₄⋊C₄	—	2	1	no	—
118	C₁₃×C₄⋊C₄	—	4	1	no	—
119	C₁₃×C₄⋊C₄	—	4	1	no	—
120	C₁₃×C₂²⋊C₄	—	4	1	no	—
121	C₁₃×C₂²⋊C₄	—	4	1	no	—
122	C₁₃×C₂²⋊C₄	—	4	1	no	—
123	C₄²×C₁₃	—	4	1	no	—
124	C₄²×C₁₃	—	4	1	no	—
125	C₄²×C₁₃	—	4	1	no	—
126	C₄²×C₁₃	D₈	1	1	yes	characteristic
127	C₄²×C₁₃	D₈	1	1	yes	characteristic
128	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
130	C₂²⋊Dic₁₃	—	4	2	no	—
131	C₄⋊Dic₁₃	—	2	2	no	—
132	C₄⋊Dic₁₃	—	4	1	no	—
133	C₄⋊Dic₁₃	—	4	1	no	—
134	C₄⋊Dic₁₃	—	4	1	no	—
135	#6101	—	4	2	no	—
136	#6101	—	4	2	no	—
137	C₄×Dic₁₃	—	4	2	no	—
138	Dic₅₂	—	2	2	no	—
139	Dic₅₂	—	2	2	no	—
140	Dic₅₂	—	4	2	no	—
141	#6090	—	2	2	no	—
142	#6090	—	2	2	no	—
143	#3170	—	13	1	no	—
144	C₄×#135	D₁₃	1	1	yes	characteristic
145	#2237	—	13	2	no	—
146	#2227	—	13	1	no	—
147	#2226	—	13	1	no	—
148	#2209	—	13	1	no	—
149	#2208	—	13	1	no	—
150	#2189	—	13	2	no	—
151	#2176	—	13	2	no	—
152	#2175	—	13	2	no	—
153	#2163	—	13	1	no	—
154	C₂³×C₁₃	—	4	1	no	—
155	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
156	Q₈×C₁₃	—	2	2	no	—
157	Q₈×C₁₃	—	4	1	no	—
158	D₄×C₁₃	—	2	2	no	—
159	D₄×C₁₃	—	4	4	no	—
160	D₄×C₁₃	—	4	1	no	—
161	C₂×C₄×C₁₃	—	2	1	no	—
162	C₂×C₄×C₁₃	—	2	1	no	—
163	C₂×C₄×C₁₃	—	2	2	no	—
164	C₂×C₄×C₁₃	—	4	1	no	—
165	C₂×C₄×C₁₃	—	4	4	no	—
166	C₂×C₄×C₁₃	—	4	1	no	—
167	C₂×C₄×C₁₃	—	4	1	no	—
168	C₂×C₄×C₁₃	—	4	1	no	—
169	C₂×C₄×C₁₃	—	8	2	no	—
170	C₂×C₄×C₁₃	—	8	2	no	—
171	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
172	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
173	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
174	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
175	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
176	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
177	C₂×Dic₁₃	—	4	2	no	—
178	C₂×Dic₁₃	—	4	2	no	—
179	Dic₂₆	—	4	2	no	—
180	Dic₂₆	—	4	2	no	—
181	Dic₂₆	—	8	2	no	—
182	C₈×C₁₃	—	2	1	no	—
183	C₈×C₁₃	—	2	1	no	—
184	C₈×C₁₃	—	2	1	no	—
185	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
186	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
187	C₂×#135	C₂×D₁₃	1	1	yes	characteristic
188	#533	—	26	1	no	—
189	C₄×#41	—	2	1	no	—
190	C₂×Q₃₂	—	13	2	no	—
191	C₂×SD₃₂	—	13	2	no	—
192	#500	—	13	1	no	—
193	#494	—	13	1	no	—
194	#487	—	26	1	no	—
195	#467	—	13	2	no	—
196	#467	—	26	1	no	—
197	C₄×Q₁₆	—	13	2	no	—
198	C₄×Q₁₆	—	26	1	no	—
199	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
200	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
201	C₄×SD₁₆	—	2	1	no	—
202	C₄×SD₁₆	—	2	1	no	—
203	C₄×D₈	C₂×D₁₃	1	1	yes	characteristic
204	C₄×D₈	C₂×D₁₃	1	1	yes	characteristic
205	#427	C₂×D₁₃	1	1	yes	characteristic
206	#416	—	2	1	no	—
207	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
208	#367	—	13	2	no	—
209	#366	—	13	2	no	—
210	C₄⋊C₁₆	—	13	1	no	—
211	C₄⋊C₁₆	—	13	1	no	—
212	#358	—	13	2	no	—
213	#358	—	13	2	no	—
214	#358	—	13	2	no	—
215	#357	—	13	2	no	—
216	C₂²⋊C₁₆	—	13	2	no	—
217	#346	—	13	2	no	—
218	C₂²×C₁₃	—	2	1	no	—
219	C₂²×C₁₃	—	2	1	no	—
220	C₂²×C₁₃	—	2	1	no	—
221	C₂²×C₁₃	—	4	2	no	—
222	C₂²×C₁₃	—	8	1	no	—
223	C₂²×C₁₃	—	8	1	no	—
224	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
225	C₄×C₁₃	—	2	1	no	—
226	C₄×C₁₃	—	2	1	no	—
227	C₄×C₁₃	—	2	1	no	—
228	C₄×C₁₃	—	2	1	no	—
229	C₄×C₁₃	—	4	2	no	—
230	C₄×C₁₃	—	8	1	no	—
231	C₄×C₁₃	—	8	1	no	—
232	C₄×C₁₃	#510	1	2	yes	—
233	C₄×C₁₃	#525	1	1	yes	characteristic
234	C₄×C₁₃	#575	1	1	yes	characteristic
235	Dic₁₃	—	8	2	no	—
236	Dic₁₃	—	8	2	no	—
237	C₂×#41	—	2	1	no	—
238	#135	—	2	2	no	—
239	#135	—	2	2	no	—
240	C₂×Q₁₆	—	13	2	no	—
241	C₂×Q₁₆	—	26	1	no	—
242	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
243	C₂×SD₁₆	—	2	1	no	—
244	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
245	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
246	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
247	C₄⋊Q₈	—	26	1	no	—
248	C₄⋊Q₈	—	26	1	no	—
249	#125	—	52	1	no	—
250	C₂²⋊Q₈	—	26	2	no	—
251	C₂²⋊Q₈	—	52	1	no	—
252	C₄×Q₈	—	2	1	no	—
253	C₄×Q₈	—	2	1	no	—
254	C₄×Q₈	—	26	2	no	—
255	C₄×Q₈	—	52	1	no	—
256	C₄×D₄	—	2	1	no	—
257	C₄×D₄	—	2	1	no	—
258	C₄×D₄	—	4	1	no	—
259	C₄×D₄	—	4	1	no	—
260	#117	—	2	1	no	—
261	#117	—	4	1	no	—
262	#117	—	52	1	no	—
263	C₂×C₄²	—	4	1	no	—
264	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
265	Q₃₂	—	26	4	no	—
266	SD₃₂	—	26	4	no	—
267	C₂×C₁₆	—	13	2	no	—
268	C₂×C₁₆	—	13	2	no	—
269	#107	—	13	2	no	—
270	#107	—	26	1	no	—
271	#107	C₂²×D₁₃	1	1	yes	characteristic
272	#107	C₂²×D₁₃	1	1	yes	characteristic
273	#106	—	2	1	no	—
274	#103	—	2	1	no	—
275	#103	—	2	1	no	—
276	#103	—	26	2	no	—
277	#103	—	26	2	no	—
278	#102	—	2	1	no	—
279	#102	—	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	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
284	C₂×C₁₃	—	4	1	no	—
285	C₂×C₁₃	—	8	2	no	—
286	C₂×C₁₃	#3100	1	1	yes	characteristic
287	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
288	C₂×C₁₃	#3414	1	1	yes	characteristic
289	#41	—	2	4	no	—
290	#41	—	4	2	no	—
291	C₂×Q₈	—	2	1	no	—
292	C₂×Q₈	—	26	2	no	—
293	C₂×Q₈	—	52	1	no	—
294	C₂×D₄	—	2	1	no	—
295	C₂×D₄	—	4	1	no	—
296	C₂²×C₄	—	4	1	no	—
297	C₂²×C₄	—	4	1	no	—
298	C₂²×C₄	—	4	1	no	—
299	C₂²×C₄	#6104	1	1	yes	characteristic
300	C₂²×C₄	#6104	1	1	yes	characteristic
301	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
302	Q₁₆	—	2	1	no	—
303	Q₁₆	—	26	2	no	—
304	Q₁₆	—	26	2	no	—
305	Q₁₆	—	52	2	no	—
306	Q₁₆	#6127	1	2	yes	—
307	SD₁₆	—	2	4	no	—
308	D₈	—	2	1	no	—
309	D₈	#6127	1	2	yes	—
310	C₂×C₈	—	2	1	no	—
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	4	1	no	—
313	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
314	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
315	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
316	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
317	C₄⋊C₄	—	2	1	no	—
318	C₄⋊C₄	—	2	1	no	—
319	C₄⋊C₄	—	4	1	no	—
320	C₄⋊C₄	—	4	1	no	—
321	C₄⋊C₄	—	26	2	no	—
322	C₄⋊C₄	—	52	1	no	—
323	C₄⋊C₄	—	52	1	no	—
324	C₄⋊C₄	—	52	2	no	—
325	C₄⋊C₄	—	52	2	no	—
326	C₄⋊C₄	—	52	1	no	—
327	C₂²⋊C₄	—	4	1	no	—
328	C₂²⋊C₄	—	4	1	no	—
329	C₂²⋊C₄	—	4	1	no	—
330	C₂²⋊C₄	—	52	2	no	—
331	C₄²	—	4	1	no	—
332	C₄²	—	4	1	no	—
333	C₄²	—	4	1	no	—
334	C₄²	—	52	2	no	—
335	C₄²	#6104	1	1	yes	characteristic
336	C₄²	#6104	1	1	yes	characteristic
337	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
338	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
339	C₁₆	—	26	2	no	—
340	C₁₆	—	26	2	no	—
341	C₁₃	#33676	1	1	yes	characteristic, complemented, Sylow
342	C₂³	—	4	1	no	—
343	C₂³	C₂×#6104	1	1	yes	characteristic
344	Q₈	—	2	2	no	—
345	Q₈	—	4	1	no	—
346	Q₈	—	52	2	no	—
347	Q₈	—	52	2	no	—
348	Q₈	—	104	2	no	—
349	D₄	—	2	2	no	—
350	D₄	—	4	4	no	—
351	D₄	—	4	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	4	1	no	—
356	C₂×C₄	—	4	4	no	—
357	C₂×C₄	—	4	1	no	—
358	C₂×C₄	—	4	1	no	—
359	C₂×C₄	—	4	1	no	—
360	C₂×C₄	—	8	2	no	—
361	C₂×C₄	—	8	2	no	—
362	C₂×C₄	—	52	2	no	—
363	C₂×C₄	—	52	2	no	—
364	C₂×C₄	C₂×#6104	1	1	yes	characteristic
365	C₂×C₄	C₂×#6104	1	1	yes	characteristic
366	C₂×C₄	C₂×#6104	1	1	yes	characteristic
367	C₂×C₄	C₂×#6104	1	1	yes	characteristic
368	C₂×C₄	C₂×#6104	1	1	yes	characteristic
369	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	C₂×#6127	1	1	yes	characteristic
374	C₈	C₂×#6127	1	1	yes	characteristic
375	C₂²	—	2	1	no	—
376	C₂²	—	2	1	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	4	2	no	—
379	C₂²	—	8	1	no	—
380	C₂²	—	8	1	no	—
381	C₂²	C₂²×#6104	1	1	yes	characteristic
382	C₄	—	2	1	no	—
383	C₄	—	2	1	no	—
384	C₄	—	2	1	no	—
385	C₄	—	2	1	no	—
386	C₄	—	4	2	no	—
387	C₄	—	8	1	no	—
388	C₄	—	8	1	no	—
389	C₄	—	104	2	no	—
390	C₄	—	104	2	no	—
391	C₄	#11718856	1	2	yes	—
392	C₄	#11719496	1	1	yes	characteristic
393	C₄	#11719527	1	1	yes	characteristic
394	C₂	—	4	1	no	—
395	C₂	—	8	2	no	—
396	C₂	#13124836	1	1	yes	characteristic
397	C₂	C₂×#11718856	1	1	yes	characteristic
398	C₂	#13127970	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₄×C₁₃×#135	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×C₁₃×#135	C₂²
hypercenter	C₄×#135	D₁₃
hypocenter	C₁₃	#33676
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #13125422	C₂	semidirect
2. #13117044	C₂	semidirect
3. #13116988	C₂	semidirect
4. #13116983	C₂	semidirect
5. #13116886	C₂	semidirect
6. #13116881	C₂	semidirect
8. #13116705	C₂	semidirect
9. #13116700	C₂	semidirect
10. #13116639	C₂	semidirect
341. C₁₃	#33676	semidirect

Maximal subgroups

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

subgroup	index
#13125422	2
#13117044	2
#13116988	2
#13116983	2
#13116886	2
#13116881	2
#13116776	2
#13116705	2
#13116700	2
#13116639	2
C₄×C₁₃×#135	2
#33676	13