Subgroups — #16785049

Order 3328.

566 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13130769	C₂	1	1	yes	maximal, characteristic
2	C₄²×#1057	C₂	1	1	yes	maximal, characteristic
3	#13129672	C₂	1	2	yes	maximal
4	#13129672	C₂	1	2	yes	maximal
5	#13117407	C₂	1	1	yes	maximal, characteristic
6	C₂×#88142	C₄	1	1	yes	characteristic
7	C₂×#88141	—	2	1	no	—
8	C₂×#88112	—	2	1	no	—
9	C₂×#88112	—	2	1	no	—
10	C₂×C₄⋊#1057	C₂²	1	2	yes	—
11	C₂×C₄×#1057	—	2	1	no	—
12	C₂×C₄×#1057	—	2	1	no	—
13	C₂×C₄×#1057	C₂²	1	2	yes	—
14	C₁₃×#389	C₄	1	1	yes	characteristic
15	#11718637	C₄	1	1	yes	characteristic
16	C₄⋊C₄×Dic₁₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×Dic₁₃	C₂²	1	1	yes	characteristic
18	#11718633	—	2	1	no	—
19	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
20	#11718589	C₄	1	1	yes	characteristic
21	#11718310	—	2	1	no	—
22	#11718307	—	2	1	no	—
23	C₂²×Dic₂₆	—	2	1	no	—
24	C₂²×#1057	—	2	2	no	—
25	C₂²×#1057	—	2	2	no	—
26	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
27	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×#6101	—	2	1	no	—
33	C₂×#6101	—	2	1	no	—
34	C₂×#6101	—	2	1	no	—
35	C₂×#6101	—	2	1	no	—
36	C₂×C₄×Dic₁₃	—	2	1	no	—
37	C₂×C₄×Dic₁₃	—	2	1	no	—
38	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
40	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
41	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
42	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
43	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
44	#88142	—	2	2	no	—
45	#88142	C₈	1	4	yes	complemented
46	#88141	—	4	2	no	—
47	#88141	—	4	2	no	—
48	#88112	—	4	2	no	—
49	#88112	—	4	2	no	—
50	C₄⋊#1057	—	2	4	no	—
51	C₄×#1057	—	2	4	no	—
52	C₄×#1057	—	2	4	no	—
53	C₄×#1057	—	2	2	no	—
54	C₄×#1057	Q₈	1	4	yes	complemented
55	C₁₃×#95	—	2	1	no	—
56	#88071	—	2	1	no	—
57	#88071	—	2	1	no	—
58	#88071	—	2	1	no	—
59	#9870	—	13	1	no	maximal, Sylow
60	C₂²×C₄×C₁₃	—	2	1	no	—
61	C₂²×C₄×C₁₃	—	2	1	no	—
62	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×Dic₁₃	—	2	1	no	—
66	C₂²×Dic₁₃	—	2	1	no	—
67	C₂²×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂×Dic₂₆	—	2	4	no	—
72	C₂×Dic₂₆	—	4	2	no	—
73	C₂×Dic₂₆	—	4	2	no	—
74	C₂×#1057	—	2	4	no	—
75	C₂×#1057	—	2	4	no	—
76	C₂×#1057	—	2	4	no	—
77	C₂×#1057	—	4	2	no	—
78	C₂×#1057	—	4	2	no	—
79	C₂×#1057	—	4	2	no	—
80	C₁₃×C₄⋊C₄	—	2	1	no	—
81	C₁₃×C₄⋊C₄	—	2	2	no	—
82	C₁₃×C₄⋊C₄	—	4	2	no	—
83	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
84	C₄²×C₁₃	—	2	2	no	—
85	C₄⋊Dic₁₃	—	2	1	no	—
86	C₄⋊Dic₁₃	—	2	2	no	—
87	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
88	#6101	—	2	4	no	—
89	#6101	—	4	2	no	—
90	#6101	—	4	2	no	—
91	#6101	—	4	2	no	—
92	C₄×Dic₁₃	—	2	2	no	—
93	C₄×Dic₁₃	—	2	2	no	—
94	C₄×Dic₁₃	—	4	1	no	—
95	C₄×Dic₁₃	—	4	1	no	—
96	C₄×Dic₁₃	—	4	1	no	—
97	C₄×Dic₁₃	—	4	1	no	—
98	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
99	C₄×Dic₁₃	M₁₆	1	2	yes	—
100	C₄×Dic₁₃	D₈	1	2	yes	—
101	C₄×Dic₁₃	SD₁₆	1	2	yes	—
102	C₄×Dic₁₃	SD₁₆	1	2	yes	—
103	C₄×Dic₁₃	Q₁₆	1	2	yes	—
104	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
105	C₄×Dic₁₃	#41	1	2	yes	—
106	C₄×C₄⋊Q₈	—	13	1	no	—
107	#1842	—	13	1	no	—
108	C₄²×C₈	—	13	1	no	—
109	#1279	—	13	2	no	—
110	#1279	—	13	2	no	—
111	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	4	1	no	—
118	C₂×C₄×C₁₃	—	4	1	no	—
119	C₂×C₄×C₁₃	—	4	1	no	—
120	C₂×C₄×C₁₃	—	4	1	no	—
121	C₂×C₄×C₁₃	—	4	1	no	—
122	C₂×C₄×C₁₃	#102	1	2	yes	—
123	C₂×C₄×C₁₃	#103	1	2	yes	—
124	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
125	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₁₃	#131	1	2	yes	—
128	C₂×Dic₁₃	—	2	1	no	—
129	C₂×Dic₁₃	—	2	1	no	—
130	C₂×Dic₁₃	—	2	2	no	—
131	C₂×Dic₁₃	—	4	1	no	—
132	C₂×Dic₁₃	—	4	1	no	—
133	C₂×Dic₁₃	—	4	1	no	—
134	C₂×Dic₁₃	—	4	1	no	—
135	C₂×Dic₁₃	—	4	1	no	—
136	C₂×Dic₁₃	C₂²⋊C₈	1	2	yes	—
137	C₂×Dic₁₃	C₂²⋊C₈	1	2	yes	—
138	C₂×Dic₁₃	#102	1	2	yes	—
139	C₂×Dic₁₃	#103	1	2	yes	—
140	C₂×Dic₁₃	#104	1	2	yes	—
141	C₂×Dic₁₃	#104	1	2	yes	—
142	C₂×Dic₁₃	C₂²⋊Q₈	1	2	yes	—
143	C₂×Dic₁₃	#124	1	1	yes	characteristic
144	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
145	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
146	C₂×Dic₁₃	C₂×SD₁₆	1	1	yes	characteristic
147	C₂×Dic₁₃	C₂×SD₁₆	1	1	yes	characteristic
148	C₂×Dic₁₃	C₂×Q₁₆	1	1	yes	characteristic
149	Dic₂₆	—	4	8	no	—
150	#1057	—	2	8	no	—
151	#1057	—	8	2	no	—
152	C₂×C₄⋊Q₈	—	13	1	no	—
153	C₂×C₄×Q₈	—	26	1	no	—
154	C₂×C₄⋊C₈	—	13	2	no	—
155	C₂×C₄⋊C₈	—	26	1	no	—
156	C₂×C₄⋊C₈	—	26	1	no	—
157	C₂×C₄×C₈	—	13	2	no	—
158	C₂×C₄×C₈	—	26	1	no	—
159	C₂×C₄×C₈	—	26	1	no	—
160	#391	—	26	1	no	—
161	#389	—	13	1	no	—
162	#389	—	26	1	no	—
163	#389	#261	1	1	yes	characteristic
164	#384	—	13	1	no	—
165	C₄×C₄⋊C₄	—	13	1	no	—
166	C₄×C₄⋊C₄	—	13	1	no	—
167	C₄×C₄⋊C₄	—	26	1	no	—
168	C₄³	—	13	1	no	—
169	C₂²×C₁₃	#391	1	1	yes	characteristic
170	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
171	C₂²×C₁₃	#408	1	1	yes	characteristic
172	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
173	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
174	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
175	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
176	C₄×C₁₃	—	2	2	no	—
177	C₄×C₁₃	—	2	2	no	—
178	C₄×C₁₃	—	4	2	no	—
179	C₄×C₁₃	—	8	1	no	—
180	C₄×C₁₃	C₈×Q₈	1	2	yes	—
181	C₄×C₁₃	#446	1	2	yes	—
182	Dic₁₃	—	2	2	no	—
183	Dic₁₃	—	4	2	no	—
184	Dic₁₃	—	8	1	no	—
185	Dic₁₃	D₄⋊C₈	1	4	yes	—
186	Dic₁₃	Q₈⋊C₈	1	4	yes	—
187	Dic₁₃	#486	1	1	yes	characteristic
188	Dic₁₃	#487	1	1	yes	characteristic
189	Dic₁₃	#498	1	1	yes	characteristic
190	Dic₁₃	#500	1	1	yes	characteristic
191	C₂²×Q₈	—	26	1	no	—
192	C₂²×C₈	—	26	2	no	—
193	C₂²×C₈	—	26	2	no	—
194	C₄⋊Q₈	—	13	4	no	—
195	C₄⋊Q₈	—	26	2	no	—
196	C₄×Q₈	—	52	2	no	—
197	C₄×Q₈	—	52	2	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	13	1	no	—
200	C₂×C₄⋊C₄	—	13	1	no	—
201	C₂×C₄⋊C₄	—	26	1	no	—
202	C₂×C₄⋊C₄	—	26	1	no	—
203	C₂×C₄⋊C₄	—	26	1	no	—
204	C₂×C₄⋊C₄	—	26	1	no	—
205	C₂×C₄⋊C₄	C₂×#261	1	1	yes	characteristic
206	C₂×C₄⋊C₄	C₂×#261	1	1	yes	characteristic
207	C₂×C₄²	—	13	1	no	—
208	C₂×C₄²	—	13	1	no	—
209	C₂×C₄²	—	13	1	no	—
210	C₂×C₄²	—	13	1	no	—
211	C₂×C₄²	—	13	1	no	—
212	C₂×C₄²	—	13	1	no	—
213	C₂×C₄²	—	26	1	no	—
214	C₂×C₄²	—	26	1	no	—
215	C₂×C₄²	C₂×#261	1	1	yes	characteristic
216	C₄⋊C₈	—	26	4	no	—
217	C₄⋊C₈	—	52	2	no	—
218	C₄⋊C₈	—	52	2	no	—
219	C₄×C₈	—	13	4	no	—
220	C₄×C₈	—	26	2	no	—
221	C₄×C₈	—	26	4	no	—
222	C₄×C₈	—	26	4	no	—
223	#95	—	2	1	no	—
224	#95	—	26	1	no	—
225	#95	—	26	1	no	—
226	#95	—	26	1	no	—
227	C₂×C₁₃	C₂×D₄⋊C₈	1	1	yes	characteristic
228	C₂×C₁₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
229	C₂×C₁₃	#1492	1	1	yes	characteristic
230	C₂×C₁₃	#1493	1	1	yes	characteristic
231	C₂×C₁₃	#1984	1	1	yes	characteristic
232	C₂×C₁₃	#1990	1	1	yes	characteristic
233	C₂×C₁₃	#1994	1	1	yes	characteristic
234	C₂×Q₈	—	26	4	no	—
235	C₂×Q₈	—	52	2	no	—
236	C₂×Q₈	—	52	2	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	13	1	no	—
240	C₂²×C₄	—	13	1	no	—
241	C₂²×C₄	—	13	1	no	—
242	C₂²×C₄	—	13	1	no	—
243	C₂²×C₄	—	26	1	no	—
244	C₂²×C₄	—	26	1	no	—
245	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
246	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
247	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
248	C₂×C₈	—	26	4	no	—
249	C₂×C₈	—	26	4	no	—
250	C₂×C₈	—	26	4	no	—
251	C₂×C₈	—	52	2	no	—
252	C₂×C₈	—	52	2	no	—
253	C₂×C₈	—	52	2	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	13	2	no	—
258	C₄⋊C₄	—	26	4	no	—
259	C₄⋊C₄	—	26	1	no	—
260	C₄⋊C₄	—	26	2	no	—
261	C₄⋊C₄	—	52	2	no	—
262	C₄⋊C₄	—	52	2	no	—
263	C₄⋊C₄	—	52	2	no	—
264	C₄⋊C₄	D₁₃⋊C₈	1	2	yes	—
265	C₄²	—	2	2	no	—
266	C₄²	—	13	2	no	—
267	C₄²	—	13	2	no	—
268	C₄²	—	13	2	no	—
269	C₄²	—	13	2	no	—
270	C₄²	—	13	2	no	—
271	C₄²	—	13	2	no	—
272	C₄²	—	13	2	no	—
273	C₄²	—	13	2	no	—
274	C₄²	—	26	2	no	—
275	C₄²	—	26	2	no	—
276	C₄²	—	52	1	no	—
277	C₄²	—	52	1	no	—
278	C₄²	—	52	1	no	—
279	C₄²	—	52	1	no	—
280	C₁₃	#9870	1	1	yes	characteristic, complemented, Sylow
281	C₂³	C₂×C₂²⋊#261	1	1	yes	characteristic
282	Q₈	—	52	8	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	13	2	no	—
294	C₂×C₄	—	13	2	no	—
295	C₂×C₄	—	13	1	no	—
296	C₂×C₄	—	13	1	no	—
297	C₂×C₄	—	13	2	no	—
298	C₂×C₄	—	13	1	no	—
299	C₂×C₄	—	13	2	no	—
300	C₂×C₄	—	13	1	no	—
301	C₂×C₄	—	13	1	no	—
302	C₂×C₄	—	13	1	no	—
303	C₂×C₄	—	13	2	no	—
304	C₂×C₄	—	13	2	no	—
305	C₂×C₄	—	13	2	no	—
306	C₂×C₄	—	26	1	no	—
307	C₂×C₄	—	26	1	no	—
308	C₂×C₄	—	26	2	no	—
309	C₂×C₄	—	52	1	no	—
310	C₂×C₄	—	52	1	no	—
311	C₂×C₄	—	52	1	no	—
312	C₂×C₄	—	52	1	no	—
313	C₂×C₄	—	52	1	no	—
314	C₂×C₄	#88115	1	2	yes	—
315	C₂×C₄	#88117	1	2	yes	—
316	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
317	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
318	C₂×C₄	#88238	1	2	yes	—
319	C₂×C₄	Q₈×#261	1	2	yes	—
320	C₈	—	26	8	no	—
321	C₈	—	104	2	no	—
322	C₂²	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
323	C₂²	C₂×#88115	1	1	yes	characteristic
324	C₂²	#11719146	1	1	yes	characteristic
325	C₂²	C₂×#88117	1	1	yes	characteristic
326	C₂²	(C₂×#261)⋊Q₈	1	1	yes	characteristic
327	C₂²	D₁₃⋊#104	1	1	yes	characteristic
328	C₂²	D₁₃⋊#104	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	4	2	no	—
332	C₄	—	8	1	no	—
333	C₄	—	13	1	no	—
334	C₄	—	13	1	no	—
335	C₄	—	13	1	no	—
336	C₄	—	13	1	no	—
337	C₄	—	13	4	no	—
338	C₄	—	13	4	no	—
339	C₄	—	26	2	no	—
340	C₄	—	52	2	no	—
341	C₄	—	104	1	no	—
342	C₄	#11719074	1	2	yes	—
343	C₄	#11719078	1	2	yes	—
344	C₂	D₁₃⋊(D₄⋊C₈)	1	1	yes	characteristic
345	C₂	D₁₃⋊(Q₈⋊C₈)	1	1	yes	characteristic
346	C₂	#13130181	1	1	yes	characteristic
347	C₂	#13130184	1	1	yes	characteristic
348	C₂	#13131007	1	1	yes	characteristic
349	C₂	#13131017	1	1	yes	characteristic
350	C₂	#13131022	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₁₃×#389	C₄
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	#389	#261
hypocenter	C₁₃	#9870
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

Complements

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

N	H ≅ G/N	type
45. #88142	C₈	semidirect
54. C₄×#1057	Q₈	semidirect
280. C₁₃	#9870	semidirect

Maximal subgroups

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

subgroup	index
#13130769	2
C₄²×#1057	2
#13129672	2
#13117407	2
#9870	13