Subgroups — #16018980

Order 3328.

562 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125495	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117108	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117088	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117081	C₂	1	1	yes	maximal, characteristic, complemented
5	#13117067	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116895	C₂	1	2	yes	maximal, complemented
7	#13116883	C₂	1	2	yes	maximal, complemented
8	#13116776	C₂	1	2	yes	maximal
9	#13116770	C₂	1	2	yes	maximal, complemented
10	#13116643	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#3156	C₂	1	1	yes	maximal, characteristic
12	C₂×C₁₃×#137	—	2	1	no	—
13	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
15	C₁₃×#536	—	2	1	no	—
16	#11719223	—	2	1	no	—
17	C₁₃×#497	—	2	1	no	—
18	C₁₃×#497	—	2	1	no	—
19	C₁₃×#495	C₂²	1	1	yes	characteristic
20	C₁₃×#495	C₂²	1	1	yes	characteristic
21	C₁₃×#494	C₂²	1	1	yes	characteristic
22	C₁₃×#494	C₂²	1	1	yes	characteristic
23	C₁₃×#405	C₂²	1	1	yes	characteristic
24	C₂×#88069	C₂²	1	2	yes	—
25	C₂×#88069	C₂²	1	2	yes	—
26	C₂×#88069	C₂²	1	2	yes	—
27	C₂×#88067	C₂²	1	2	yes	—
28	#11718798	—	2	1	no	—
29	#11718777	C₂²	1	2	yes	—
30	#11718377	—	2	1	no	—
31	#11718364	—	2	1	no	—
32	#11718360	C₂²	1	1	yes	characteristic
33	C₄×Dic₅₂	C₂²	1	2	yes	—
34	#11718344	C₂²	1	1	yes	characteristic
35	#11718156	C₂²	1	2	yes	—
36	#11718156	C₂²	1	2	yes	—
37	#11718156	C₂²	1	2	yes	—
38	#11718154	C₂²	1	2	yes	—
39	C₂²⋊#6090	C₂²	1	2	yes	—
40	C₄⋊#6090	C₂²	1	2	yes	—
41	#11718053	C₂²	1	1	yes	characteristic
42	C₄×#6090	C₂²	1	1	yes	characteristic
43	C₂×C₁₃×#41	—	2	1	no	—
44	C₂²×Q₈×C₁₃	—	2	1	no	—
45	C₁₃×#137	—	4	4	no	—
46	C₁₃×#135	—	4	2	no	—
47	C₂×C₁₃×Q₁₆	—	2	1	no	—
48	C₂×C₁₃×Q₁₆	—	2	1	no	—
49	C₂×C₁₃×Q₁₆	—	2	1	no	—
50	C₂×C₁₃×Q₁₆	—	4	2	no	—
51	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
52	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
54	C₂×C₁₃×SD₁₆	—	2	1	no	—
55	C₂×C₁₃×SD₁₆	—	2	1	no	—
56	C₂×C₁₃×SD₁₆	—	2	1	no	—
57	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
58	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
60	C₁₃×C₄⋊Q₈	—	2	1	no	—
61	C₁₃×C₄⋊Q₈	—	2	1	no	—
62	C₁₃×#124	—	2	1	no	—
63	C₁₃×#124	—	2	1	no	—
64	C₁₃×#123	—	4	2	no	—
65	C₁₃×C₂²⋊Q₈	—	4	2	no	—
66	C₁₃×#117	D₄	1	1	yes	characteristic
67	#88178	—	2	2	no	—
68	C₂×Dic₅₂	—	2	1	no	—
69	C₂×Dic₅₂	C₂³	1	2	yes	—
70	#88143	—	4	2	no	—
71	#88133	—	4	2	no	—
72	#88124	—	2	1	no	—
73	#88123	—	2	1	no	—
74	C₄×Dic₂₆	—	2	2	no	—
75	C₁₃×#97	D₄	1	1	yes	characteristic
76	C₁₃×#97	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	—	4	2	no	—
80	#88069	—	4	2	no	—
81	#88069	—	4	2	no	—
82	#88067	—	4	2	no	—
83	#88058	C₂³	1	2	yes	—
84	#88057	—	2	1	no	—
85	#88056	—	2	2	no	—
86	#88056	—	2	2	no	—
87	C₂×#6090	C₂³	1	2	yes	—
88	C₂×#6090	C₂³	1	2	yes	—
89	#33789	—	13	1	no	maximal, Sylow
90	C₁₃×#41	—	4	2	no	—
91	C₁₃×#41	—	4	2	no	—
92	C₂×Q₈×C₁₃	—	2	1	no	—
93	C₂×Q₈×C₁₃	—	2	1	no	—
94	C₂×Q₈×C₁₃	—	2	1	no	—
95	C₂×Q₈×C₁₃	—	4	2	no	—
96	C₂×Q₈×C₁₃	—	4	2	no	—
97	C₂×Q₈×C₁₃	—	4	1	no	—
98	C₂×D₄×C₁₃	—	2	1	no	—
99	C₂×D₄×C₁₃	—	4	1	no	—
100	C₂²×C₄×C₁₃	—	4	1	no	—
101	C₂²×C₄×C₁₃	—	4	1	no	—
102	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×Dic₂₆	—	2	2	no	—
104	C₁₃×Q₁₆	—	2	2	no	—
105	C₁₃×Q₁₆	—	2	2	no	—
106	C₁₃×Q₁₆	—	2	2	no	—
107	C₁₃×Q₁₆	—	4	2	no	—
108	C₁₃×Q₁₆	—	4	2	no	—
109	C₁₃×Q₁₆	—	4	2	no	—
110	C₁₃×SD₁₆	—	4	2	no	—
111	C₁₃×SD₁₆	—	4	2	no	—
112	C₁₃×SD₁₆	—	4	2	no	—
113	C₁₃×D₈	—	2	2	no	—
114	C₁₃×M₁₆	—	2	2	no	—
115	C₂×C₈×C₁₃	—	2	2	no	—
116	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
120	C₁₃×C₄⋊C₄	—	4	2	no	—
121	C₁₃×C₄⋊C₄	—	4	2	no	—
122	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
123	C₁₃×C₂²⋊C₄	—	4	2	no	—
124	C₁₃×C₂²⋊C₄	—	4	2	no	—
125	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
126	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
128	C₂²⋊Dic₁₃	—	4	2	no	—
129	C₄⋊Dic₁₃	—	2	2	no	—
130	C₄⋊Dic₁₃	—	4	2	no	—
131	#6101	—	4	2	no	—
132	#6101	—	4	2	no	—
133	#6101	—	4	2	no	—
134	C₄×Dic₁₃	—	4	2	no	—
135	Dic₅₂	—	2	2	no	—
136	Dic₅₂	—	2	2	no	—
137	Dic₅₂	—	4	2	no	—
138	#6090	—	2	2	no	—
139	#6090	—	2	2	no	—
140	#3177	—	13	1	no	—
141	#3156	D₁₃	1	1	yes	characteristic
142	#2251	—	13	1	no	—
143	#2247	—	13	1	no	—
144	#2245	—	13	1	no	—
145	#2242	—	13	1	no	—
146	#2211	—	13	2	no	—
147	#2209	—	13	2	no	—
148	#2189	—	13	2	no	—
149	#2188	—	13	2	no	—
150	#2164	—	13	1	no	—
151	C₂³×C₁₃	—	4	1	no	—
152	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
153	Q₈×C₁₃	—	4	1	no	—
154	Q₈×C₁₃	—	4	1	no	—
155	Q₈×C₁₃	—	4	1	no	—
156	Q₈×C₁₃	—	4	1	no	—
157	Q₈×C₁₃	—	4	1	no	—
158	Q₈×C₁₃	—	4	1	no	—
159	Q₈×C₁₃	—	8	2	no	—
160	D₄×C₁₃	—	4	1	no	—
161	D₄×C₁₃	—	4	1	no	—
162	D₄×C₁₃	—	8	2	no	—
163	C₂×C₄×C₁₃	—	2	1	no	—
164	C₂×C₄×C₁₃	—	2	1	no	—
165	C₂×C₄×C₁₃	—	4	1	no	—
166	C₂×C₄×C₁₃	—	4	1	no	—
167	C₂×C₄×C₁₃	—	4	1	no	—
168	C₂×C₄×C₁₃	—	8	2	no	—
169	C₂×C₄×C₁₃	—	8	2	no	—
170	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
171	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
172	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
173	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
174	C₂×Dic₁₃	—	4	2	no	—
175	C₂×Dic₁₃	—	4	2	no	—
176	Dic₂₆	—	4	2	no	—
177	Dic₂₆	—	4	2	no	—
178	C₈×C₁₃	—	2	1	no	—
179	C₈×C₁₃	—	2	1	no	—
180	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
181	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
182	C₂×#137	—	2	1	no	—
183	C₂×#135	C₂×D₁₃	1	1	yes	characteristic
184	C₂²×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
185	#541	—	26	1	no	—
186	#536	—	2	1	no	—
187	C₂×Q₃₂	—	13	2	no	—
188	C₂×Q₃₂	—	13	2	no	—
189	C₂×Q₃₂	—	13	2	no	—
190	C₂×SD₃₂	—	13	2	no	—
191	#499	—	13	1	no	—
192	#497	—	2	1	no	—
193	#497	—	2	1	no	—
194	#495	C₂×D₁₃	1	1	yes	characteristic
195	#495	C₂×D₁₃	1	1	yes	characteristic
196	#494	—	13	1	no	—
197	#494	C₂×D₁₃	1	1	yes	characteristic
198	#494	C₂×D₁₃	1	1	yes	characteristic
199	#491	—	26	1	no	—
200	#470	—	26	1	no	—
201	#467	—	13	2	no	—
202	#441	—	26	1	no	—
203	C₄×Q₁₆	—	13	2	no	—
204	#405	C₂×D₁₃	1	1	yes	characteristic
205	C₄⋊C₁₆	—	13	2	no	—
206	#358	—	13	2	no	—
207	#358	—	13	2	no	—
208	#358	—	13	2	no	—
209	#357	—	13	2	no	—
210	C₂²⋊C₁₆	—	13	2	no	—
211	#346	—	13	1	no	—
212	C₄×C₁₆	—	13	1	no	—
213	C₂²×C₁₃	—	2	2	no	—
214	C₂²×C₁₃	—	2	1	no	—
215	C₂²×C₁₃	—	8	2	no	—
216	C₂²×C₁₃	—	8	1	no	—
217	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
218	C₄×C₁₃	—	2	2	no	—
219	C₄×C₁₃	—	2	2	no	—
220	C₄×C₁₃	—	2	1	no	—
221	C₄×C₁₃	—	8	1	no	—
222	C₄×C₁₃	—	8	1	no	—
223	C₄×C₁₃	—	8	1	no	—
224	C₄×C₁₃	#535	1	1	yes	characteristic
225	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
226	Dic₁₃	—	8	2	no	—
227	Dic₁₃	—	8	2	no	—
228	C₂×#41	—	2	1	no	—
229	C₂²×Q₈	—	2	1	no	—
230	#137	—	4	4	no	—
231	#135	—	4	2	no	—
232	C₂×Q₁₆	—	2	1	no	—
233	C₂×Q₁₆	—	2	1	no	—
234	C₂×Q₁₆	—	2	1	no	—
235	C₂×Q₁₆	—	4	2	no	—
236	C₂×Q₁₆	—	13	2	no	—
237	C₂×Q₁₆	—	26	1	no	—
238	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
239	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
240	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
241	C₂×SD₁₆	—	2	1	no	—
242	C₂×SD₁₆	—	2	1	no	—
243	C₂×SD₁₆	—	2	1	no	—
244	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
245	C₂×M₁₆	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	C₄⋊Q₈	—	26	1	no	—
250	#126	—	52	2	no	—
251	#125	—	26	1	no	—
252	#125	—	52	2	no	—
253	#124	—	2	1	no	—
254	#124	—	2	1	no	—
255	#123	—	4	2	no	—
256	C₂²⋊Q₈	—	4	2	no	—
257	C₂²⋊Q₈	—	26	2	no	—
258	C₄×Q₈	—	26	2	no	—
259	#117	C₂²⋊D₁₃	1	1	yes	characteristic
260	Q₃₂	—	52	2	no	—
261	Q₃₂	—	52	2	no	—
262	Q₃₂	—	52	2	no	—
263	SD₃₂	—	52	2	no	—
264	C₂×C₁₆	—	13	2	no	—
265	C₂×C₁₆	—	13	2	no	—
266	#107	—	13	2	no	—
267	#106	—	26	1	no	—
268	#103	—	26	2	no	—
269	#103	—	26	2	no	—
270	#97	C₂²⋊D₁₃	1	1	yes	characteristic
271	#97	C₂²⋊D₁₃	1	1	yes	characteristic
272	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
274	C₂×C₁₃	—	4	1	no	—
275	C₂×C₁₃	—	16	1	no	—
276	C₂×C₁₃	#3205	1	1	yes	characteristic
277	C₂×C₁₃	#3417	1	2	yes	—
278	#41	—	4	2	no	—
279	#41	—	4	2	no	—
280	C₂×Q₈	—	2	1	no	—
281	C₂×Q₈	—	2	1	no	—
282	C₂×Q₈	—	2	1	no	—
283	C₂×Q₈	—	4	2	no	—
284	C₂×Q₈	—	4	2	no	—
285	C₂×Q₈	—	4	1	no	—
286	C₂×Q₈	—	26	2	no	—
287	C₂×D₄	—	2	1	no	—
288	C₂×D₄	—	4	1	no	—
289	C₂²×C₄	—	4	1	no	—
290	C₂²×C₄	—	4	1	no	—
291	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
292	Q₁₆	—	2	2	no	—
293	Q₁₆	—	2	2	no	—
294	Q₁₆	—	2	2	no	—
295	Q₁₆	—	4	2	no	—
296	Q₁₆	—	4	2	no	—
297	Q₁₆	—	4	2	no	—
298	Q₁₆	—	26	2	no	—
299	Q₁₆	—	26	2	no	—
300	Q₁₆	—	52	2	no	—
301	SD₁₆	—	4	2	no	—
302	SD₁₆	—	4	2	no	—
303	SD₁₆	—	4	2	no	—
304	D₈	—	2	2	no	—
305	M₁₆	—	2	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
308	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
309	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
310	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
311	C₄⋊C₄	—	4	2	no	—
312	C₄⋊C₄	—	4	2	no	—
313	C₄⋊C₄	—	26	2	no	—
314	C₄⋊C₄	—	52	2	no	—
315	C₄⋊C₄	—	52	2	no	—
316	C₄⋊C₄	—	52	2	no	—
317	C₄⋊C₄	—	52	2	no	—
318	C₄⋊C₄	#6104	1	2	yes	—
319	C₂²⋊C₄	—	4	2	no	—
320	C₂²⋊C₄	—	4	2	no	—
321	C₂²⋊C₄	—	52	2	no	—
322	C₂²⋊C₄	#6104	1	2	yes	—
323	C₄²	—	52	2	no	—
324	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
325	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
326	C₁₆	—	26	2	no	—
327	C₁₆	—	26	2	no	—
328	C₁₃	#33789	1	1	yes	characteristic, complemented, Sylow
329	C₂³	—	4	1	no	—
330	C₂³	C₂×#6104	1	1	yes	characteristic
331	Q₈	—	4	1	no	—
332	Q₈	—	4	1	no	—
333	Q₈	—	4	1	no	—
334	Q₈	—	4	1	no	—
335	Q₈	—	4	1	no	—
336	Q₈	—	4	1	no	—
337	Q₈	—	8	2	no	—
338	Q₈	—	52	2	no	—
339	Q₈	—	52	2	no	—
340	D₄	—	4	1	no	—
341	D₄	—	4	1	no	—
342	D₄	—	8	2	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	8	2	no	—
349	C₂×C₄	—	8	2	no	—
350	C₂×C₄	—	52	2	no	—
351	C₂×C₄	—	52	2	no	—
352	C₂×C₄	C₂×#6104	1	2	yes	—
353	C₂×C₄	C₂×#6104	1	2	yes	—
354	C₂×C₄	C₂×#6104	1	1	yes	characteristic
355	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	#88186	1	2	yes	—
359	C₈	#88251	1	2	yes	—
360	C₂²	—	2	2	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	8	2	no	—
363	C₂²	—	8	1	no	—
364	C₂²	C₂²×#6104	1	1	yes	characteristic
365	C₄	—	2	2	no	—
366	C₄	—	2	2	no	—
367	C₄	—	2	1	no	—
368	C₄	—	8	1	no	—
369	C₄	—	8	1	no	—
370	C₄	—	8	1	no	—
371	C₄	—	104	2	no	—
372	C₄	—	104	2	no	—
373	C₄	C₂×#88186	1	1	yes	characteristic
374	C₄	#11719510	1	1	yes	characteristic
375	C₂	—	4	1	no	—
376	C₂	—	16	1	no	—
377	C₂	#13125754	1	1	yes	characteristic
378	C₂	#13127987	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3156	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×C₁₃×#135	C₂²
hypercenter	#3156	D₁₃
hypocenter	C₁₃	#33789
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3156

Complements

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

N	H ≅ G/N	type
1. #13125495	C₂	semidirect
2. #13117108	C₂	semidirect
3. #13117088	C₂	semidirect
4. #13117081	C₂	semidirect
5. #13117067	C₂	semidirect
6. #13116895	C₂	semidirect
7. #13116883	C₂	semidirect
9. #13116770	C₂	semidirect
10. #13116643	C₂	semidirect
328. C₁₃	#33789	semidirect

Maximal subgroups

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

subgroup	index
#13125495	2
#13117108	2
#13117088	2
#13117081	2
#13117067	2
#13116895	2
#13116883	2
#13116776	2
#13116770	2
#13116643	2
C₁₃×#3156	2
#33789	13