Subgroups — #16017999

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	C₄×#88158	C₂	1	1	yes	maximal, characteristic
2	#13117041	C₂	1	2	yes	maximal, complemented
3	#13116985	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116980	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116883	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116878	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116774	C₂	1	2	yes	maximal
8	#13116705	C₂	1	2	yes	maximal, complemented
9	#13116698	C₂	1	2	yes	maximal, complemented
10	#13116639	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#533	—	2	1	no	—
13	C₂×#88158	C₂²	1	1	yes	characteristic
14	C₄×#6127	—	2	1	no	—
15	C₁₃×#500	C₂²	1	1	yes	characteristic
16	C₁₃×#494	C₂²	1	1	yes	characteristic
17	C₁₃×#487	—	2	1	no	—
18	C₁₃×#467	—	2	1	no	—
19	C₁₃×#467	C₂²	1	2	yes	—
20	C₄×C₁₃×Q₁₆	—	2	1	no	—
21	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
23	C₂×#88069	C₂²	1	2	yes	—
24	C₂×#88068	C₂²	1	2	yes	—
25	#11718774	—	2	1	no	—
26	#11718767	C₂²	1	1	yes	characteristic
27	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
28	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
29	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
30	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
31	C₄×#6095	—	2	1	no	—
32	C₄×#6095	—	2	1	no	—
33	#11718156	C₂²	1	2	yes	—
34	#11718156	C₂²	1	2	yes	—
35	C₂²⋊#6090	C₂²	1	2	yes	—
36	#11718084	C₂²	1	2	yes	—
37	#11718083	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×#6127	—	2	1	no	—
44	C₂×C₁₃×Q₁₆	—	2	1	no	—
45	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
46	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
48	C₁₃×C₄⋊Q₈	—	2	1	no	—
49	C₁₃×C₄⋊Q₈	—	2	1	no	—
50	C₁₃×#125	—	4	1	no	—
51	C₁₃×C₂²⋊Q₈	—	2	2	no	—
52	C₁₃×C₂²⋊Q₈	—	4	1	no	—
53	C₄×Q₈×C₁₃	—	2	2	no	—
54	C₄×Q₈×C₁₃	—	4	1	no	—
55	C₁₃×#117	—	4	1	no	—
56	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
57	C₄×C₂²⋊D₁₃	—	4	1	no	—
58	C₄×C₂²⋊D₁₃	—	4	1	no	—
59	#88180	—	2	1	no	—
60	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
61	#88158	—	2	2	no	—
62	#88158	—	2	2	no	—
63	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
64	C₂×#6095	—	2	1	no	—
65	C₄×D₅₂	—	2	1	no	—
66	C₄×D₅₂	—	2	1	no	—
67	#88126	—	4	1	no	—
68	C₄²×D₁₃	—	4	1	no	—
69	C₄×Dic₂₆	—	2	1	no	—
70	C₄×Dic₂₆	—	2	1	no	—
71	C₁₃×#107	—	2	1	no	—
72	C₁₃×#107	C₂³	1	2	yes	—
73	C₁₃×#103	—	2	2	no	—
74	C₁₃×#103	—	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	#88069	—	2	4	no	—
80	#88068	—	2	4	no	—
81	#88061	—	2	1	no	—
82	#88061	—	2	1	no	—
83	#88058	C₂³	1	1	yes	characteristic
84	#88058	C₂³	1	1	yes	characteristic
85	#88057	—	2	1	no	—
86	#88056	—	2	1	no	—
87	#88056	—	2	1	no	—
88	C₂×#6090	C₂³	1	2	yes	—
89	C₂×#6090	C₂³	1	2	yes	—
90	#33676	—	13	1	no	maximal, Sylow
91	C₂×Q₈×C₁₃	—	2	2	no	—
92	C₂×Q₈×C₁₃	—	4	1	no	—
93	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
94	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
95	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₂²⋊D₁₃	—	4	1	no	—
97	#6127	—	2	4	no	—
98	#6127	—	4	2	no	—
99	C₂×D₅₂	—	2	1	no	—
100	C₂×C₄×D₁₃	—	4	1	no	—
101	C₂×C₄×D₁₃	—	4	1	no	—
102	C₂×C₄×D₁₃	—	4	1	no	—
103	C₂×Dic₂₆	—	2	1	no	—
104	C₁₃×Q₁₆	—	2	2	no	—
105	C₁₃×Q₁₆	—	2	2	no	—
106	C₁₃×Q₁₆	—	4	2	no	—
107	C₂×C₈×C₁₃	—	2	1	no	—
108	C₂×C₈×C₁₃	—	2	1	no	—
109	C₂×C₈×C₁₃	—	4	1	no	—
110	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
114	C₁₃×C₄⋊C₄	—	2	2	no	—
115	C₁₃×C₄⋊C₄	—	4	1	no	—
116	C₁₃×C₄⋊C₄	—	4	1	no	—
117	C₁₃×C₄⋊C₄	—	4	2	no	—
118	C₁₃×C₄⋊C₄	—	4	2	no	—
119	C₁₃×C₄⋊C₄	—	4	1	no	—
120	C₁₃×C₂²⋊C₄	—	4	2	no	—
121	C₄²×C₁₃	—	4	2	no	—
122	C₄²×C₁₃	D₈	1	1	yes	characteristic
123	C₄²×C₁₃	D₈	1	1	yes	characteristic
124	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
126	C₂²⋊Dic₁₃	—	4	1	no	—
127	#6103	—	4	1	no	—
128	#6103	—	4	1	no	—
129	C₄⋊Dic₁₃	—	2	1	no	—
130	C₄⋊Dic₁₃	—	2	1	no	—
131	#6101	—	4	1	no	—
132	#6101	—	4	1	no	—
133	C₄×Dic₁₃	—	4	1	no	—
134	C₄×Dic₁₃	—	4	1	no	—
135	C₄×Dic₁₃	—	4	1	no	—
136	Dic₅₂	—	2	1	no	—
137	Dic₅₂	#41	1	2	yes	—
138	D₁₀₄	—	2	1	no	—
139	D₁₀₄	#41	1	2	yes	—
140	#6095	—	2	4	no	—
141	#6090	—	2	2	no	—
142	#6090	—	2	2	no	—
143	#3170	D₁₃	1	1	yes	characteristic, complemented
144	C₄×#135	—	13	1	no	—
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	C₂×D₈	1	1	yes	characteristic
155	C₂²×D₁₃	—	4	1	no	—
156	Q₈×C₁₃	—	4	2	no	—
157	Q₈×C₁₃	—	4	2	no	—
158	Q₈×C₁₃	—	8	2	no	—
159	C₂×C₄×C₁₃	—	2	1	no	—
160	C₂×C₄×C₁₃	—	2	1	no	—
161	C₂×C₄×C₁₃	—	2	2	no	—
162	C₂×C₄×C₁₃	—	4	1	no	—
163	C₂×C₄×C₁₃	—	4	2	no	—
164	C₂×C₄×C₁₃	—	4	2	no	—
165	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
166	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
167	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
168	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
169	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
170	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
171	C₂²⋊D₁₃	—	4	4	no	—
172	C₂×Dic₁₃	—	4	1	no	—
173	C₂×Dic₁₃	—	4	1	no	—
174	C₂×Dic₁₃	—	4	1	no	—
175	D₅₂	—	2	2	no	—
176	D₅₂	—	4	1	no	—
177	C₄×D₁₃	—	4	4	no	—
178	C₄×D₁₃	—	8	2	no	—
179	C₄×D₁₃	—	8	2	no	—
180	Dic₂₆	—	2	2	no	—
181	Dic₂₆	—	4	1	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	—	13	1	no	—
188	#533	—	2	1	no	—
189	C₄×#41	—	26	1	no	—
190	C₂×Q₃₂	—	13	2	no	—
191	C₂×SD₃₂	—	13	2	no	—
192	#500	C₂×D₁₃	1	1	yes	characteristic
193	#494	C₂×D₁₃	1	1	yes	characteristic
194	#487	—	2	1	no	—
195	#467	—	2	1	no	—
196	#467	C₂×D₁₃	1	2	yes	—
197	C₄×Q₁₆	—	2	1	no	—
198	C₄×Q₁₆	—	13	1	no	—
199	C₄×Q₁₆	—	13	1	no	—
200	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
201	C₄×SD₁₆	—	26	1	no	—
202	C₄×SD₁₆	—	26	1	no	—
203	C₄×D₈	—	13	1	no	—
204	C₄×D₈	—	13	1	no	—
205	#427	—	13	1	no	—
206	#416	—	26	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	C₂²×D₈	1	1	yes	characteristic
222	C₂×D₁₃	—	4	2	no	—
223	C₂×D₁₃	—	8	1	no	—
224	C₂×D₁₃	—	8	1	no	—
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	—	8	2	no	—
230	C₄×C₁₃	—	8	2	no	—
231	C₄×C₁₃	#510	1	2	yes	—
232	C₄×C₁₃	#525	1	1	yes	characteristic
233	C₄×C₁₃	#575	1	1	yes	characteristic
234	Dic₁₃	—	4	2	no	—
235	Dic₁₃	—	8	1	no	—
236	Dic₁₃	—	8	1	no	—
237	C₂×#41	—	26	1	no	—
238	#135	—	26	2	no	—
239	#135	—	26	2	no	—
240	C₂×Q₁₆	—	2	1	no	—
241	C₂×Q₁₆	—	13	1	no	—
242	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
243	C₂×SD₁₆	—	26	1	no	—
244	C₂×D₈	—	13	1	no	—
245	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
246	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
247	C₄⋊Q₈	—	2	1	no	—
248	C₄⋊Q₈	—	2	1	no	—
249	#125	—	4	1	no	—
250	C₂²⋊Q₈	—	2	2	no	—
251	C₂²⋊Q₈	—	4	1	no	—
252	C₄×Q₈	—	2	2	no	—
253	C₄×Q₈	—	4	1	no	—
254	C₄×Q₈	—	26	1	no	—
255	C₄×Q₈	—	26	1	no	—
256	C₄×D₄	—	26	1	no	—
257	C₄×D₄	—	26	1	no	—
258	C₄×D₄	—	52	1	no	—
259	C₄×D₄	—	52	1	no	—
260	#117	—	4	1	no	—
261	#117	—	26	1	no	—
262	#117	—	52	1	no	—
263	C₂×C₄²	—	52	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	—	2	1	no	—
270	#107	—	13	1	no	—
271	#107	—	13	1	no	—
272	#107	C₂²×D₁₃	1	2	yes	—
273	#106	—	26	1	no	—
274	#103	—	2	2	no	—
275	#103	—	2	2	no	—
276	#103	—	26	1	no	—
277	#103	—	26	1	no	—
278	#102	—	26	1	no	—
279	#102	—	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	—	4	1	no	—
285	C₂×C₁₃	#3100	1	1	yes	characteristic
286	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
287	C₂×C₁₃	#3414	1	1	yes	characteristic
288	D₁₃	—	8	2	no	—
289	#41	—	26	4	no	—
290	#41	—	52	2	no	—
291	C₂×Q₈	—	2	2	no	—
292	C₂×Q₈	—	4	1	no	—
293	C₂×Q₈	—	26	1	no	—
294	C₂×D₄	—	26	1	no	—
295	C₂×D₄	—	52	1	no	—
296	C₂²×C₄	—	52	1	no	—
297	C₂²×C₄	—	52	1	no	—
298	C₂²×C₄	—	52	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	2	no	—
303	Q₁₆	—	2	2	no	—
304	Q₁₆	—	4	2	no	—
305	Q₁₆	—	13	2	no	—
306	Q₁₆	—	26	1	no	—
307	SD₁₆	—	26	4	no	—
308	D₈	—	13	2	no	—
309	D₈	—	26	1	no	—
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	2	no	—
318	C₄⋊C₄	—	4	1	no	—
319	C₄⋊C₄	—	4	1	no	—
320	C₄⋊C₄	—	4	2	no	—
321	C₄⋊C₄	—	4	2	no	—
322	C₄⋊C₄	—	4	1	no	—
323	C₄⋊C₄	—	26	1	no	—
324	C₄⋊C₄	—	26	1	no	—
325	C₄⋊C₄	—	52	1	no	—
326	C₄⋊C₄	—	52	1	no	—
327	C₂²⋊C₄	—	4	2	no	—
328	C₂²⋊C₄	—	52	1	no	—
329	C₂²⋊C₄	—	52	1	no	—
330	C₂²⋊C₄	—	52	1	no	—
331	C₄²	—	4	2	no	—
332	C₄²	—	52	1	no	—
333	C₄²	—	52	1	no	—
334	C₄²	—	52	1	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	—	52	1	no	—
343	C₂³	C₂×#6104	1	1	yes	characteristic
344	Q₈	—	4	2	no	—
345	Q₈	—	4	2	no	—
346	Q₈	—	8	2	no	—
347	Q₈	—	26	2	no	—
348	Q₈	—	52	1	no	—
349	D₄	—	26	2	no	—
350	D₄	—	52	4	no	—
351	D₄	—	52	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	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	52	4	no	—
359	C₂×C₄	—	52	1	no	—
360	C₂×C₄	—	52	1	no	—
361	C₂×C₄	—	52	1	no	—
362	C₂×C₄	—	104	2	no	—
363	C₂×C₄	—	104	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	#41×D₁₃	1	1	yes	characteristic
374	C₈	#41×D₁₃	1	1	yes	characteristic
375	C₂²	—	2	1	no	—
376	C₂²	—	2	1	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	52	2	no	—
379	C₂²	—	104	1	no	—
380	C₂²	—	104	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	—	8	2	no	—
387	C₄	—	8	2	no	—
388	C₄	—	52	2	no	—
389	C₄	—	104	1	no	—
390	C₄	—	104	1	no	—
391	C₄	#11718840	1	2	yes	—
392	C₄	#11719527	1	1	yes	characteristic
393	C₄	#11719533	1	1	yes	characteristic
394	C₂	—	4	1	no	—
395	C₂	—	104	2	no	—
396	C₂	#13124832	1	1	yes	characteristic
397	C₂	C₂×#11718840	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	C₁₃×#3170	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×#88158	C₂²
hypercenter	#3170	D₁₃
hypocenter	C₁₃	#33676
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #13117041	C₂	semidirect
3. #13116985	C₂	semidirect
4. #13116980	C₂	semidirect
5. #13116883	C₂	semidirect
6. #13116878	C₂	semidirect
8. #13116705	C₂	semidirect
9. #13116698	C₂	semidirect
10. #13116639	C₂	semidirect
11. C₁₃×#3170	C₂	semidirect
143. #3170	D₁₃	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
C₄×#88158	2
#13117041	2
#13116985	2
#13116980	2
#13116883	2
#13116878	2
#13116774	2
#13116705	2
#13116698	2
#13116639	2
C₁₃×#3170	2
#33676	13