Subgroups — #16018251

Order 3328.

602 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125375	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117077	C₂	1	2	yes	maximal, complemented
3	#13117072	C₂	1	2	yes	maximal, complemented
4	#13117067	C₂	1	2	yes	maximal, complemented
5	#13117059	C₂	1	2	yes	maximal, complemented
6	#13116552	C₂	1	1	yes	maximal, characteristic, complemented
7	#13116548	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116543	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11718154	C₂	1	1	yes	maximal, characteristic, complemented
10	#13116433	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#3155	C₂	1	1	yes	maximal, characteristic
12	C₂×C₁₃×AGL(1,ℤ/8)	—	2	1	no	—
13	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
15	C₁₃×#535	—	2	1	no	—
16	#11719224	—	2	1	no	—
17	C₁₃×#496	—	2	2	no	—
18	C₁₃×#495	C₂²	1	2	yes	—
19	C₁₃×#493	C₂²	1	2	yes	—
20	C₁₃×#405	C₂²	1	1	yes	characteristic
21	#11718782	—	2	1	no	—
22	#11718767	C₂²	1	1	yes	characteristic
23	C₂×#88058	C₂²	1	1	yes	characteristic
24	C₂×#88052	C₂²	1	1	yes	characteristic
25	C₂²×#6090	C₂²	1	1	yes	characteristic
26	#11718363	—	2	2	no	—
27	#11718344	C₂²	1	2	yes	complemented
28	#11718343	C₂²	1	2	yes	complemented
29	#11718156	C₂²	1	1	yes	characteristic, complemented
30	#11718156	C₂²	1	1	yes	characteristic, complemented
31	#11718156	C₂²	1	2	yes	complemented
32	#11718154	C₂²	1	2	yes	—
33	#11718154	C₂²	1	2	yes	—
34	#11718154	C₂²	1	2	yes	—
35	#11718154	C₂²	1	2	yes	—
36	#11718154	C₂²	1	1	yes	characteristic, complemented
37	#11718154	C₂²	1	1	yes	characteristic, complemented
38	#11718154	C₂²	1	2	yes	complemented
39	#11718053	C₂²	1	2	yes	complemented
40	C₄×#6090	C₂²	1	2	yes	complemented
41	C₂×C₁₃×#41	—	2	1	no	—
42	C₂²×D₄×C₁₃	—	2	1	no	—
43	C₁₃×AGL(1,ℤ/8)	—	4	4	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	2	no	—
48	C₂×C₁₃×SD₁₆	—	2	1	no	—
49	C₂×C₁₃×D₈	—	2	2	no	—
50	C₂×C₁₃×D₈	—	2	1	no	—
51	C₂×C₁₃×D₈	—	2	2	no	—
52	C₂×C₁₃×D₈	—	2	2	no	—
53	C₂×C₁₃×D₈	C₂³	1	2	yes	—
54	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
55	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
57	C₁₃×#127	—	2	2	no	—
58	C₁₃×#124	—	2	2	no	—
59	C₁₃×#121	—	4	1	no	—
60	C₁₃×#121	—	4	1	no	—
61	C₁₃×#120	—	4	2	no	—
62	C₁₃×#117	D₄	1	1	yes	characteristic
63	#88180	—	2	1	no	—
64	C₂×C₄⋊Dic₁₃	—	2	1	no	—
65	#88144	—	4	1	no	—
66	#88142	—	4	1	no	—
67	#88132	—	4	2	no	—
68	#88124	—	2	2	no	—
69	#88123	—	2	2	no	—
70	C₁₃×#97	D₄	1	2	yes	—
71	C₄×C₈×C₁₃	C₂³	1	2	yes	—
72	#88058	—	2	1	no	—
73	#88058	—	2	2	no	—
74	#88058	C₂³	1	2	yes	—
75	#88058	C₂³	1	1	yes	characteristic
76	#88058	C₂³	1	1	yes	characteristic
77	#88057	—	2	1	no	—
78	#88057	—	2	2	no	—
79	#88052	—	2	2	no	—
80	C₂×#6090	—	2	2	no	—
81	C₂×#6090	C₂³	1	1	yes	characteristic
82	C₂×#6090	C₂³	1	1	yes	characteristic
83	C₂×#6090	C₂³	1	2	yes	—
84	#33702	—	13	1	no	maximal, Sylow
85	C₂⁴×C₁₃	—	4	1	no	—
86	C₁₃×#41	—	4	2	no	—
87	C₁₃×#41	—	4	2	no	—
88	C₂×Q₈×C₁₃	—	2	1	no	—
89	C₂×D₄×C₁₃	—	2	2	no	—
90	C₂×D₄×C₁₃	—	2	1	no	—
91	C₂×D₄×C₁₃	—	4	1	no	—
92	C₂×D₄×C₁₃	—	4	2	no	—
93	C₂×D₄×C₁₃	—	4	1	no	—
94	C₂×D₄×C₁₃	—	4	2	no	—
95	C₂×D₄×C₁₃	—	4	2	no	—
96	C₂×D₄×C₁₃	—	4	2	no	—
97	C₂²×C₄×C₁₃	—	4	1	no	—
98	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₂²×Dic₁₃	—	4	1	no	—
100	C₂×Dic₂₆	—	4	2	no	—
101	C₁₃×Q₁₆	—	2	1	no	—
102	C₁₃×Q₁₆	—	2	1	no	—
103	C₁₃×SD₁₆	—	4	2	no	—
104	C₁₃×SD₁₆	—	4	4	no	—
105	C₁₃×D₈	—	2	2	no	—
106	C₁₃×D₈	—	2	2	no	—
107	C₁₃×D₈	—	2	1	no	—
108	C₁₃×D₈	—	2	1	no	—
109	C₁₃×D₈	—	4	2	no	—
110	C₁₃×D₈	—	4	4	no	—
111	C₁₃×M₁₆	SD₁₆	1	4	yes	—
112	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
114	C₂×C₈×C₁₃	#41	1	2	yes	—
115	C₂×C₈×C₁₃	#41	1	2	yes	—
116	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
117	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
118	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
119	C₁₃×C₂²⋊C₄	—	4	2	no	—
120	C₁₃×C₂²⋊C₄	—	4	2	no	—
121	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
122	C₄²×C₁₃	C₂×D₄	1	2	yes	—
123	C₂²⋊Dic₁₃	—	4	1	no	—
124	C₄⋊Dic₁₃	—	2	1	no	—
125	C₄⋊Dic₁₃	—	2	2	no	—
126	C₄⋊Dic₁₃	—	2	1	no	—
127	C₄⋊Dic₁₃	—	4	2	no	—
128	C₄⋊Dic₁₃	—	4	1	no	—
129	#6101	—	4	2	no	—
130	#6101	—	4	2	no	—
131	C₄×Dic₁₃	—	4	1	no	—
132	#6090	—	2	1	no	—
133	#6090	—	2	1	no	—
134	#6090	—	2	2	no	—
135	#3164	—	13	1	no	—
136	#3155	D₁₃	1	1	yes	characteristic
137	#2244	—	13	2	no	—
138	#2243	—	13	2	no	—
139	#2242	—	13	2	no	—
140	#2240	—	13	2	no	—
141	#2140	—	13	1	no	—
142	#2139	—	13	1	no	—
143	#2138	—	13	1	no	—
144	C₂×#357	—	13	1	no	—
145	#2108	—	13	1	no	—
146	C₂³×C₁₃	—	4	2	no	—
147	C₂³×C₁₃	—	4	1	no	—
148	C₂³×C₁₃	—	8	2	no	—
149	C₂³×C₁₃	—	16	1	no	—
150	C₂³×C₁₃	—	16	1	no	—
151	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
152	Q₈×C₁₃	—	4	1	no	—
153	Q₈×C₁₃	—	4	1	no	—
154	D₄×C₁₃	—	4	2	no	—
155	D₄×C₁₃	—	4	1	no	—
156	D₄×C₁₃	—	4	1	no	—
157	D₄×C₁₃	—	4	2	no	—
158	D₄×C₁₃	—	8	2	no	—
159	D₄×C₁₃	—	8	2	no	—
160	D₄×C₁₃	—	16	2	no	—
161	D₄×C₁₃	—	16	2	no	—
162	C₂×C₄×C₁₃	—	4	1	no	—
163	C₂×C₄×C₁₃	—	8	2	no	—
164	C₂×C₄×C₁₃	#124	1	2	yes	—
165	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
166	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
167	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
168	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
169	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
170	C₂×Dic₁₃	—	4	1	no	—
171	C₂×Dic₁₃	—	4	1	no	—
172	C₂×Dic₁₃	—	4	2	no	—
173	C₂×Dic₁₃	—	16	1	no	—
174	Dic₂₆	—	16	2	no	—
175	C₈×C₁₃	#124	1	2	yes	—
176	C₈×C₁₃	C₂×SD₁₆	1	4	yes	—
177	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
178	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
179	C₂×AGL(1,ℤ/8)	—	2	1	no	—
180	C₂×#135	C₂×D₁₃	1	1	yes	characteristic
181	C₂²×D₈	C₂×D₁₃	1	1	yes	characteristic
182	#544	—	26	1	no	—
183	#535	—	2	1	no	—
184	C₂×M₃₂	—	13	1	no	—
185	C₂²×C₁₆	—	13	1	no	—
186	#501	—	26	2	no	—
187	#500	—	13	2	no	—
188	#499	—	13	2	no	—
189	#496	—	2	2	no	—
190	#495	C₂×D₁₃	1	2	yes	—
191	#493	C₂×D₁₃	1	2	yes	—
192	#428	—	26	1	no	—
193	#427	—	13	1	no	—
194	C₂×#107	—	13	1	no	—
195	#405	C₂×D₁₃	1	1	yes	characteristic
196	#358	—	13	1	no	—
197	#358	—	13	1	no	—
198	#358	—	13	2	no	—
199	#357	—	13	2	no	—
200	#357	—	13	2	no	—
201	#357	—	13	2	no	—
202	#357	—	13	2	no	—
203	#357	—	13	1	no	—
204	#357	—	13	1	no	—
205	#357	—	13	2	no	—
206	#346	—	13	2	no	—
207	C₄×C₁₆	—	13	2	no	—
208	C₂²×C₁₃	—	2	1	no	—
209	C₂²×C₁₃	—	2	1	no	—
210	C₂²×C₁₃	—	8	1	no	—
211	C₂²×C₁₃	—	8	2	no	—
212	C₂²×C₁₃	—	8	1	no	—
213	C₂²×C₁₃	—	8	2	no	—
214	C₂²×C₁₃	—	8	1	no	—
215	C₂²×C₁₃	—	8	2	no	—
216	C₂²×C₁₃	—	16	4	no	—
217	C₂²×C₁₃	#486	1	2	yes	—
218	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
219	C₄×C₁₃	—	2	2	no	—
220	C₄×C₁₃	—	2	2	no	—
221	C₄×C₁₃	—	8	1	no	—
222	C₄×C₁₃	#486	1	2	yes	—
223	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
224	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
225	Dic₁₃	—	8	1	no	—
226	Dic₁₃	—	8	2	no	—
227	Dic₁₃	—	8	1	no	—
228	C₂×#41	—	2	1	no	—
229	C₂²×D₄	—	2	1	no	—
230	AGL(1,ℤ/8)	—	4	4	no	—
231	#135	—	2	2	no	—
232	#135	—	2	2	no	—
233	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
234	C₂×SD₁₆	—	2	1	no	—
235	C₂×SD₁₆	—	2	2	no	—
236	C₂×D₈	—	2	2	no	—
237	C₂×D₈	—	2	2	no	—
238	C₂×D₈	—	2	1	no	—
239	C₂×D₈	—	2	2	no	—
240	C₂×D₈	C₂²×D₁₃	1	2	yes	—
241	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
242	C₂×M₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
243	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
244	C₄⋊Q₈	—	26	2	no	—
245	C₄⋊Q₈	—	52	1	no	—
246	#127	—	2	2	no	—
247	#125	—	26	2	no	—
248	#125	—	52	1	no	—
249	#124	—	2	2	no	—
250	C₂²⋊Q₈	—	52	2	no	—
251	#121	—	4	1	no	—
252	#121	—	4	1	no	—
253	#120	—	4	2	no	—
254	#117	—	26	1	no	—
255	#117	C₂²⋊D₁₃	1	1	yes	characteristic
256	C₂×C₄⋊C₄	—	26	1	no	—
257	M₃₂	—	26	2	no	—
258	C₂×C₁₆	—	13	1	no	—
259	C₂×C₁₆	—	13	2	no	—
260	C₂×C₁₆	—	13	1	no	—
261	C₂×C₁₆	—	26	2	no	—
262	#107	—	13	2	no	—
263	#107	—	13	1	no	—
264	#107	—	13	1	no	—
265	#107	—	26	1	no	—
266	#107	—	26	2	no	—
267	#106	—	26	1	no	—
268	#106	—	26	2	no	—
269	#97	C₂²⋊D₁₃	1	2	yes	—
270	C₄×C₈	C₂²×D₁₃	1	2	yes	—
271	C₂×C₁₃	—	2	2	no	—
272	C₂×C₁₃	—	16	1	no	—
273	C₂×C₁₃	—	16	2	no	—
274	C₂×C₁₃	C₂×#486	1	1	yes	characteristic
275	C₂×C₁₃	#3415	1	1	yes	characteristic
276	C₂×C₁₃	#3416	1	1	yes	characteristic
277	C₂⁴	—	4	1	no	—
278	#41	—	4	2	no	—
279	#41	—	4	2	no	—
280	C₂×Q₈	—	2	1	no	—
281	C₂×Q₈	—	52	2	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	—	2	1	no	—
284	C₂×D₄	—	4	1	no	—
285	C₂×D₄	—	4	1	no	—
286	C₂×D₄	—	4	2	no	—
287	C₂×D₄	—	4	2	no	—
288	C₂×D₄	—	4	2	no	—
289	C₂×D₄	—	4	2	no	—
290	C₂²×C₄	—	4	1	no	—
291	C₂²×C₄	—	52	1	no	—
292	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
293	Q₁₆	—	2	1	no	—
294	Q₁₆	—	2	1	no	—
295	SD₁₆	—	4	4	no	—
296	SD₁₆	—	4	2	no	—
297	D₈	—	2	2	no	—
298	D₈	—	2	2	no	—
299	D₈	—	2	1	no	—
300	D₈	—	2	1	no	—
301	D₈	—	4	4	no	—
302	D₈	—	4	2	no	—
303	M₁₆	#6105	1	4	yes	—
304	C₂×C₈	#6129	1	2	yes	—
305	C₂×C₈	#6129	1	2	yes	—
306	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
307	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
308	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
309	C₄⋊C₄	—	26	1	no	—
310	C₄⋊C₄	—	26	2	no	—
311	C₄⋊C₄	—	26	1	no	—
312	C₄⋊C₄	—	52	1	no	—
313	C₄⋊C₄	—	52	2	no	—
314	C₄⋊C₄	—	52	2	no	—
315	C₄⋊C₄	—	52	2	no	—
316	C₄⋊C₄	#6104	1	1	yes	characteristic
317	C₄⋊C₄	#6104	1	1	yes	characteristic
318	C₂²⋊C₄	—	4	2	no	—
319	C₂²⋊C₄	—	4	2	no	—
320	C₂²⋊C₄	—	52	1	no	—
321	C₂²⋊C₄	#6104	1	2	yes	—
322	C₄²	—	52	1	no	—
323	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
324	C₁₆	—	26	1	no	—
325	C₁₆	—	26	1	no	—
326	C₁₆	—	26	2	no	—
327	C₁₃	#33702	1	1	yes	characteristic, complemented, Sylow
328	C₂³	—	4	2	no	—
329	C₂³	—	4	1	no	—
330	C₂³	—	8	2	no	—
331	C₂³	—	16	1	no	—
332	C₂³	—	16	1	no	—
333	C₂³	C₂×#6104	1	1	yes	characteristic
334	Q₈	—	4	1	no	—
335	Q₈	—	4	1	no	—
336	Q₈	—	208	2	no	—
337	D₄	—	4	1	no	—
338	D₄	—	4	2	no	—
339	D₄	—	4	2	no	—
340	D₄	—	4	1	no	—
341	D₄	—	8	2	no	—
342	D₄	—	8	2	no	—
343	D₄	—	16	2	no	—
344	D₄	—	16	2	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	8	2	no	—
347	C₂×C₄	—	52	1	no	—
348	C₂×C₄	—	52	1	no	—
349	C₂×C₄	—	52	2	no	—
350	C₂×C₄	—	208	1	no	—
351	C₂×C₄	C₂×#6104	1	2	yes	—
352	C₂×C₄	C₂×#6104	1	2	yes	—
353	C₂×C₄	C₂×#6104	1	1	yes	characteristic
354	C₂×C₄	C₂×#6105	1	2	yes	—
355	C₂×C₄	#88190	1	2	yes	—
356	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
357	C₈	C₂×#6105	1	4	yes	—
358	C₈	#88190	1	2	yes	—
359	C₈	C₂×#6129	1	1	yes	characteristic
360	C₈	C₂×#6129	1	1	yes	characteristic
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	8	1	no	—
364	C₂²	—	8	2	no	—
365	C₂²	—	8	1	no	—
366	C₂²	—	8	2	no	—
367	C₂²	—	8	1	no	—
368	C₂²	—	8	2	no	—
369	C₂²	—	16	4	no	—
370	C₂²	#11718731	1	2	yes	—
371	C₂²	C₂²×#6104	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	2	2	no	—
374	C₄	—	8	1	no	—
375	C₄	—	104	2	no	—
376	C₄	—	104	1	no	—
377	C₄	—	104	1	no	—
378	C₄	#11718731	1	2	yes	—
379	C₄	C₂²×#6105	1	1	yes	characteristic
380	C₄	C₂×#88190	1	1	yes	characteristic
381	C₂	—	2	2	no	—
382	C₂	—	16	2	no	—
383	C₂	—	16	1	no	—
384	C₂	C₂×#11718731	1	1	yes	characteristic
385	C₂	#13127975	1	1	yes	characteristic
386	C₂	#13127983	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting, prime radical	C₁₃×#3155	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	#3155	D₁₃
hypocenter	C₁₃	#33702
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3155

Complements

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

N	H ≅ G/N	type
1. #13125375	C₂	semidirect
2. #13117077	C₂	semidirect
3. #13117072	C₂	semidirect
4. #13117067	C₂	semidirect
5. #13117059	C₂	semidirect
6. #13116552	C₂	semidirect
7. #13116548	C₂	semidirect
8. #13116543	C₂	semidirect
9. C₂×#11718154	C₂	semidirect
10. #13116433	C₂	semidirect
27. #11718344	C₂²	semidirect
28. #11718343	C₂²	semidirect
29. #11718156	C₂²	semidirect
30. #11718156	C₂²	semidirect
31. #11718156	C₂²	semidirect
36. #11718154	C₂²	semidirect
37. #11718154	C₂²	semidirect
38. #11718154	C₂²	semidirect
39. #11718053	C₂²	semidirect
40. C₄×#6090	C₂²	semidirect
327. C₁₃	#33702	semidirect

Maximal subgroups

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

subgroup	index
#13125375	2
#13117077	2
#13117072	2
#13117067	2
#13117059	2
#13116552	2
#13116548	2
#13116543	2
C₂×#11718154	2
#13116433	2
C₁₃×#3155	2
#33702	13