Subgroups — #16016272

Order 3328.

660 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718840	C₂	1	1	yes	maximal, characteristic
2	#13123713	C₂	1	1	yes	maximal, characteristic
3	#13116705	C₂	1	2	yes	maximal, complemented
4	#13116698	C₂	1	2	yes	maximal, complemented
5	C₄×#88069	C₂	1	2	yes	maximal, complemented
6	C₄×#88068	C₂	1	2	yes	maximal, complemented
7	#13116625	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116541	C₂	1	1	yes	maximal, characteristic
9	C₂×#11718156	C₂	1	1	yes	maximal, characteristic, complemented
10	#13116433	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2950	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
13	C₁₃×#519	—	2	1	no	—
14	C₂×#88158	C₂²	1	1	yes	characteristic
15	#11719228	—	2	1	no	—
16	C₁₃×#441	—	2	2	no	—
17	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
18	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
19	C₁₃×#427	C₂²	1	1	yes	characteristic
20	C₁₃×#419	—	2	1	no	—
21	C₁₃×#405	C₂²	1	1	yes	characteristic
22	C₂×#88069	C₂²	1	2	yes	—
23	#11718840	C₄	1	8	yes	complemented
24	C₂×#88068	C₂²	1	2	yes	—
25	#11718794	—	2	1	no	—
26	C₂×#88058	C₂²	1	1	yes	characteristic
27	C₂×#88052	C₂²	1	1	yes	characteristic
28	C₂²×#6090	C₂²	1	1	yes	characteristic
29	#11718371	—	2	2	no	—
30	C₄×Dic₅₂	C₂²	1	2	yes	—
31	C₄×D₁₀₄	C₂²	1	2	yes	—
32	#11718156	C₂²	1	2	yes	—
33	#11718156	C₂²	1	2	yes	—
34	#11718084	C₂²	1	1	yes	characteristic
35	#11718084	C₂²	1	1	yes	characteristic
36	#11718083	C₂²	1	1	yes	characteristic
37	#11718083	C₂²	1	1	yes	characteristic
38	#11718082	C₂²	1	2	yes	—
39	#11718081	C₂²	1	2	yes	—
40	#11718053	C₂²	1	2	yes	—
41	C₄×#6090	C₂²	1	2	yes	—
42	C₂²×Q₈×C₁₃	—	2	1	no	—
43	C₂×#6127	—	2	1	no	—
44	C₂×C₁₃×Q₁₆	—	2	1	no	—
45	C₂×C₁₃×Q₁₆	—	2	2	no	—
46	C₂×C₁₃×Q₁₆	C₂×C₄	1	4	yes	complemented
47	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
48	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₄×Q₈×C₁₃	—	2	2	no	—
51	C₄×Q₈×C₁₃	—	2	2	no	—
52	C₄×Q₈×C₁₃	—	4	1	no	—
53	C₄×Q₈×C₁₃	—	4	1	no	—
54	C₁₃×#117	—	2	1	no	—
55	C₁₃×#117	—	4	2	no	—
56	C₁₃×#117	D₄	1	1	yes	characteristic
57	C₂×C₄⋊Dic₁₃	—	2	1	no	—
58	C₂×Dic₅₂	C₂³	1	1	yes	characteristic
59	#88158	—	2	2	no	—
60	#88158	C₂×C₄	1	4	yes	—
61	C₂×D₁₀₄	C₂³	1	1	yes	characteristic
62	C₂×#6095	—	2	1	no	—
63	#88146	—	4	1	no	—
64	C₄⋊C₄×D₁₃	—	4	1	no	—
65	#88135	—	4	2	no	—
66	C₄×D₅₂	—	2	2	no	—
67	C₄×Dic₂₆	—	2	2	no	—
68	C₁₃×#107	C₂³	1	1	yes	characteristic
69	C₁₃×#107	C₂³	1	1	yes	characteristic
70	C₁₃×#106	—	2	1	no	—
71	C₁₃×#103	—	2	2	no	—
72	C₁₃×#103	—	2	2	no	—
73	C₁₃×#97	D₄	1	2	yes	—
74	C₄×C₈×C₁₃	C₂³	1	2	yes	—
75	#88069	C₂×C₄	1	8	yes	complemented
76	#88068	C₂×C₄	1	8	yes	complemented
77	#88061	—	2	2	no	—
78	#88058	—	2	1	no	—
79	#88058	C₂³	1	2	yes	—
80	#88056	—	2	2	no	—
81	#88052	C₂×C₄	1	4	yes	complemented
82	C₂×#6090	—	2	2	no	—
83	C₂×#6090	C₂³	1	1	yes	characteristic
84	C₂×#6090	C₂³	1	1	yes	characteristic
85	C₂×#6090	C₂³	1	2	yes	—
86	#33455	—	13	1	no	maximal, Sylow
87	C₂×Q₈×C₁₃	—	2	4	no	—
88	C₂×Q₈×C₁₃	—	2	2	no	—
89	C₂×Q₈×C₁₃	—	4	1	no	—
90	C₂×Q₈×C₁₃	—	4	2	no	—
91	C₂²×C₄×C₁₃	—	4	1	no	—
92	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₂²⋊D₁₃	—	4	1	no	—
94	C₂²×Dic₁₃	—	4	1	no	—
95	#6127	—	2	4	no	—
96	#6127	—	4	2	no	—
97	C₂×D₅₂	—	2	1	no	—
98	C₂×C₄×D₁₃	—	4	1	no	—
99	C₂×C₄×D₁₃	—	4	2	no	—
100	C₂×Dic₂₆	—	2	1	no	—
101	C₁₃×Q₁₆	—	2	4	no	—
102	C₁₃×Q₁₆	—	2	2	no	—
103	C₁₃×Q₁₆	C₂²×C₄	1	4	yes	—
104	C₁₃×M₁₆	—	2	2	no	—
105	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	#41	1	2	yes	—
109	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
110	C₁₃×C₄⋊C₄	—	2	1	no	—
111	C₁₃×C₄⋊C₄	—	2	1	no	—
112	C₁₃×C₄⋊C₄	—	4	2	no	—
113	C₁₃×C₄⋊C₄	—	4	2	no	—
114	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
115	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
116	C₁₃×C₂²⋊C₄	—	4	1	no	—
117	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
118	C₄²×C₁₃	—	4	1	no	—
119	C₄²×C₁₃	—	4	2	no	—
120	C₄²×C₁₃	—	4	2	no	—
121	C₄²×C₁₃	C₂×D₄	1	2	yes	—
122	#6103	—	4	2	no	—
123	C₄⋊Dic₁₃	—	2	2	no	—
124	C₄⋊Dic₁₃	—	4	1	no	—
125	#6101	—	4	2	no	—
126	C₄×Dic₁₃	—	4	2	no	—
127	Dic₅₂	—	2	1	no	—
128	Dic₅₂	C₂²×C₄	1	2	yes	—
129	D₁₀₄	—	2	1	no	—
130	D₁₀₄	C₂²×C₄	1	2	yes	—
131	#6095	—	2	4	no	—
132	#6090	—	2	1	no	—
133	#6090	—	2	1	no	—
134	#6090	C₂²×C₄	1	4	yes	—
135	C₂×#510	—	13	1	no	—
136	#2951	—	13	1	no	—
137	#2950	D₁₃	1	1	yes	characteristic, complemented
138	#2176	—	13	2	no	—
139	#2175	—	13	2	no	—
140	C₄×Q₃₂	—	13	2	no	—
141	C₄×SD₃₂	—	13	2	no	—
142	#2159	—	13	1	no	—
143	#2138	—	13	1	no	—
144	C₂×#358	—	13	1	no	—
145	#2108	—	13	1	no	—
146	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
147	C₂²×D₁₃	—	4	1	no	—
148	Q₈×C₁₃	—	2	4	no	—
149	Q₈×C₁₃	—	4	2	no	—
150	Q₈×C₁₃	—	4	4	no	—
151	C₂×C₄×C₁₃	—	4	2	no	—
152	C₂×C₄×C₁₃	—	4	4	no	—
153	C₂×C₄×C₁₃	—	4	1	no	—
154	C₂×C₄×C₁₃	—	4	1	no	—
155	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
156	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
157	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
158	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₁₃	#135	1	2	yes	—
160	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
161	C₂²⋊D₁₃	—	4	4	no	—
162	C₂×Dic₁₃	—	4	1	no	—
163	C₂×Dic₁₃	—	4	2	no	—
164	C₂×Dic₁₃	—	16	1	no	—
165	D₅₂	—	2	2	no	—
166	D₅₂	—	4	1	no	—
167	C₄×D₁₃	—	4	4	no	—
168	C₄×D₁₃	—	16	2	no	—
169	Dic₂₆	—	2	2	no	—
170	Dic₂₆	—	4	1	no	—
171	C₈×C₁₃	—	2	2	no	—
172	C₈×C₁₃	C₄×D₄	1	2	yes	—
173	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
174	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
175	C₂×#135	—	13	1	no	—
176	C₂²×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
177	#520	—	26	1	no	—
178	#519	—	2	1	no	—
179	#510	—	13	8	no	—
180	C₂×Q₃₂	—	13	2	no	—
181	C₂×SD₃₂	—	13	2	no	—
182	C₂×M₃₂	—	13	1	no	—
183	C₂²×C₁₆	—	13	1	no	—
184	#441	—	2	2	no	—
185	#440	—	26	2	no	—
186	C₄×Q₁₆	—	13	2	no	—
187	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
188	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
189	C₄×D₈	—	13	2	no	—
190	#427	C₂×D₁₃	1	1	yes	characteristic
191	C₂×#107	—	13	1	no	—
192	#419	—	2	1	no	—
193	#417	—	26	1	no	—
194	#405	C₂×D₁₃	1	1	yes	characteristic
195	#367	—	13	2	no	—
196	#366	—	13	2	no	—
197	#358	—	13	2	no	—
198	#358	—	13	2	no	—
199	#358	—	13	1	no	—
200	#358	—	13	1	no	—
201	#357	—	13	1	no	—
202	#357	—	13	1	no	—
203	#346	—	13	2	no	—
204	C₄×C₁₆	—	13	2	no	—
205	C₂²×C₁₃	—	2	1	no	—
206	C₂²×C₁₃	—	2	1	no	—
207	C₂²×C₁₃	C₄×D₈	1	2	yes	—
208	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
209	C₂×D₁₃	—	4	2	no	—
210	C₂×D₁₃	—	8	1	no	—
211	C₂×D₁₃	—	8	1	no	—
212	C₄×C₁₃	—	2	2	no	—
213	C₄×C₁₃	—	2	2	no	—
214	C₄×C₁₃	—	4	4	no	—
215	C₄×C₁₃	—	8	1	no	—
216	C₄×C₁₃	—	8	1	no	—
217	C₄×C₁₃	C₄×D₈	1	2	yes	—
218	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
219	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
220	Dic₁₃	—	4	2	no	—
221	Dic₁₃	—	8	2	no	—
222	C₂×#41	—	26	1	no	—
223	C₂²×Q₈	—	2	1	no	—
224	#135	—	13	4	no	—
225	#135	—	26	2	no	—
226	C₂×Q₁₆	—	2	2	no	—
227	C₂×Q₁₆	—	2	1	no	—
228	C₂×Q₁₆	—	13	1	no	—
229	C₂×Q₁₆	C₄×D₁₃	1	4	yes	complemented
230	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
231	C₂×SD₁₆	—	26	1	no	—
232	C₂×D₈	—	13	1	no	—
233	C₂×M₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
234	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
235	C₄×Q₈	—	2	2	no	—
236	C₄×Q₈	—	2	2	no	—
237	C₄×Q₈	—	4	1	no	—
238	C₄×Q₈	—	4	1	no	—
239	C₄×Q₈	—	26	2	no	—
240	C₄×D₄	—	26	2	no	—
241	C₄×D₄	—	52	2	no	—
242	#117	—	2	1	no	—
243	#117	—	4	2	no	—
244	#117	—	52	1	no	—
245	#117	C₂²⋊D₁₃	1	1	yes	characteristic
246	C₂×C₄⋊C₄	—	26	1	no	—
247	C₂×C₄⋊C₄	—	52	1	no	—
248	Q₃₂	—	13	8	no	—
249	SD₃₂	—	13	8	no	—
250	M₃₂	—	13	4	no	—
251	C₂×C₁₆	—	13	2	no	—
252	C₂×C₁₆	—	13	1	no	—
253	C₂×C₁₆	—	13	1	no	—
254	C₂×C₁₆	—	26	2	no	—
255	#107	—	13	2	no	—
256	#107	—	26	1	no	—
257	#107	C₂²×D₁₃	1	1	yes	characteristic
258	#107	C₂²×D₁₃	1	1	yes	characteristic
259	#106	—	2	1	no	—
260	#103	—	2	2	no	—
261	#103	—	2	2	no	—
262	#103	—	26	2	no	—
263	#102	—	26	2	no	—
264	#97	C₂²⋊D₁₃	1	2	yes	—
265	C₄×C₈	C₂²×D₁₃	1	2	yes	—
266	C₂×C₁₃	—	2	2	no	—
267	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
268	C₂×C₁₃	#3416	1	1	yes	characteristic
269	C₂×C₁₃	#3417	1	1	yes	characteristic
270	D₁₃	—	8	2	no	—
271	#41	—	26	4	no	—
272	#41	—	52	2	no	—
273	C₂×Q₈	—	2	4	no	—
274	C₂×Q₈	—	2	2	no	—
275	C₂×Q₈	—	4	1	no	—
276	C₂×Q₈	—	4	2	no	—
277	C₂×Q₈	—	26	1	no	—
278	C₂×D₄	—	26	1	no	—
279	C₂×D₄	—	52	1	no	—
280	C₂²×C₄	—	4	1	no	—
281	C₂²×C₄	—	52	1	no	—
282	C₂²×C₄	—	52	1	no	—
283	C₂²×C₄	—	52	2	no	—
284	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
285	Q₁₆	—	2	4	no	—
286	Q₁₆	—	2	2	no	—
287	Q₁₆	—	13	2	no	—
288	Q₁₆	—	26	1	no	—
289	Q₁₆	C₂×C₄×D₁₃	1	4	yes	—
290	SD₁₆	—	26	4	no	—
291	D₈	—	13	2	no	—
292	D₈	—	26	1	no	—
293	M₁₆	—	2	2	no	—
294	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
295	C₂×C₈	#6127	1	2	yes	—
296	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
297	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
298	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	4	2	no	—
302	C₄⋊C₄	—	4	2	no	—
303	C₄⋊C₄	—	26	2	no	—
304	C₄⋊C₄	—	52	1	no	—
305	C₄⋊C₄	—	52	2	no	—
306	C₄⋊C₄	#6104	1	1	yes	characteristic
307	C₄⋊C₄	#6104	1	1	yes	characteristic
308	C₂²⋊C₄	—	4	1	no	—
309	C₂²⋊C₄	—	52	2	no	—
310	C₂²⋊C₄	#6104	1	2	yes	—
311	C₄²	—	4	1	no	—
312	C₄²	—	4	2	no	—
313	C₄²	—	4	2	no	—
314	C₄²	—	52	2	no	—
315	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
316	C₁₆	—	13	4	no	—
317	C₁₆	—	26	1	no	—
318	C₁₆	—	26	1	no	—
319	C₁₃	#33455	1	1	yes	characteristic, complemented, Sylow
320	C₂³	—	52	1	no	—
321	C₂³	C₂×#6104	1	1	yes	characteristic
322	Q₈	—	2	4	no	—
323	Q₈	—	4	4	no	—
324	Q₈	—	4	2	no	—
325	Q₈	—	26	2	no	—
326	Q₈	—	52	1	no	—
327	D₄	—	26	2	no	—
328	D₄	—	52	4	no	—
329	D₄	—	52	1	no	—
330	C₂×C₄	—	4	4	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	52	1	no	—
335	C₂×C₄	—	52	4	no	—
336	C₂×C₄	—	52	2	no	—
337	C₂×C₄	—	208	2	no	—
338	C₂×C₄	—	208	1	no	—
339	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
340	C₂×C₄	C₂×#6104	1	2	yes	—
341	C₂×C₄	C₂×#6104	1	2	yes	—
342	C₂×C₄	C₂×#6104	1	1	yes	characteristic
343	C₂×C₄	#88204	1	2	yes	—
344	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
345	C₈	—	2	2	no	—
346	C₈	C₄×C₂²⋊D₁₃	1	2	yes	—
347	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
348	C₈	C₂×#6127	1	1	yes	characteristic
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	52	2	no	—
352	C₂²	—	104	1	no	—
353	C₂²	—	104	1	no	—
354	C₂²	C₄×#6104	1	2	yes	—
355	C₂²	C₂²×#6104	1	1	yes	characteristic
356	C₄	—	2	2	no	—
357	C₄	—	2	2	no	—
358	C₄	—	4	4	no	—
359	C₄	—	8	1	no	—
360	C₄	—	8	1	no	—
361	C₄	—	52	2	no	—
362	C₄	—	104	2	no	—
363	C₄	C₄×#6104	1	2	yes	—
364	C₄	C₂×C₄×C₂²⋊D₁₃	1	1	yes	characteristic
365	C₄	C₂×#88204	1	1	yes	characteristic
366	C₂	—	2	2	no	—
367	C₂	—	104	2	no	—
368	C₂	C₂×C₄×#6104	1	1	yes	characteristic
369	C₂	#13127980	1	1	yes	characteristic
370	C₂	#13127987	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2950	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×#88158	C₂²
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2950	D₁₃
hypocenter	C₁₃	#33455
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2950

Complements

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

N	H ≅ G/N	type
3. #13116705	C₂	semidirect
4. #13116698	C₂	semidirect
5. C₄×#88069	C₂	semidirect
6. C₄×#88068	C₂	semidirect
7. #13116625	C₂	semidirect
9. C₂×#11718156	C₂	semidirect
10. #13116433	C₂	semidirect
11. C₁₃×#2950	C₂	semidirect
23. #11718840	C₄	semidirect
46. C₂×C₁₃×Q₁₆	C₂×C₄	semidirect
75. #88069	C₂×C₄	semidirect
76. #88068	C₂×C₄	semidirect
81. #88052	C₂×C₄	semidirect
137. #2950	D₁₃	semidirect
229. C₂×Q₁₆	C₄×D₁₃	semidirect
319. C₁₃	#33455	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718840	2
#13123713	2
#13116705	2
#13116698	2
C₄×#88069	2
C₄×#88068	2
#13116625	2
#13116541	2
C₂×#11718156	2
#13116433	2
C₁₃×#2950	2
#33455	13