Subgroups — #16795916

Order 3328.

592 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131779	C₂	1	2	yes	maximal, complemented
2	#13130486	C₂	1	1	yes	maximal, characteristic, complemented
3	#13130483	C₂	1	1	yes	maximal, characteristic, complemented
4	#13130472	C₂	1	2	yes	maximal, complemented
5	#13130432	C₂	1	1	yes	maximal, characteristic, complemented
6	#13130429	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₁₆×#1057	C₂	1	2	yes	maximal, complemented
8	#13130391	C₂	1	1	yes	maximal, characteristic, complemented
9	#13130192	C₂	1	2	yes	maximal, complemented
10	#13130040	C₂	1	1	yes	maximal, characteristic, complemented
11	#13125403	C₂	1	1	yes	maximal, characteristic
12	#11719385	C₂²	1	2	yes	—
13	#11719158	C₂²	1	2	yes	—
14	Q₈×#1057	C₂²	1	2	yes	—
15	#11719145	C₂²	1	2	yes	—
16	D₄×#1057	C₂²	1	2	yes	—
17	#11719068	—	2	2	no	—
18	#11719067	—	2	2	no	—
19	#11719066	—	2	2	no	—
20	#11719064	—	2	2	no	—
21	#11719052	C₂²	1	2	yes	—
22	#11719049	C₂²	1	1	yes	characteristic
23	#11719049	C₂²	1	1	yes	characteristic
24	C₁₃×#495	C₄	1	1	yes	characteristic
25	#11718849	C₄	1	1	yes	characteristic
26	Q₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
27	#11718832	C₄	1	2	yes	—
28	SD₁₆×Dic₁₃	C₂²	1	2	yes	—
29	#11718818	C₄	1	1	yes	characteristic
30	D₈×Dic₁₃	C₂²	1	1	yes	characteristic
31	#11718717	C₄	1	2	yes	—
32	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
33	#11718344	C₄	1	1	yes	characteristic
34	Q₈⋊#1057	C₂²	1	2	yes	—
35	Q₈⋊#1057	C₂²	1	2	yes	—
36	D₄⋊#1057	C₂²	1	2	yes	—
37	D₄⋊#1057	C₂²	1	2	yes	—
38	#11718253	C₂²	1	2	yes	—
39	#11718253	C₂²	1	1	yes	characteristic
40	#11718253	C₂²	1	1	yes	characteristic
41	#11718251	C₂²	1	1	yes	characteristic
42	#11718251	C₂²	1	1	yes	characteristic
43	C₈×#1057	C₂²	1	1	yes	characteristic
44	C₈×#1057	C₂²	1	1	yes	characteristic
45	C₂×#6122	—	2	2	no	—
46	C₂²×#1057	—	2	2	no	—
47	C₂×#6118	—	2	2	no	—
48	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
49	C₂×C₁₃×SD₁₆	C₂×C₄	1	2	yes	—
50	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
51	C₁₃×#124	C₂×C₄	1	2	yes	—
52	#88200	C₂×C₄	1	2	yes	—
53	Q₈×Dic₁₃	C₂³	1	2	yes	—
54	#88192	C₂×C₄	1	2	yes	—
55	D₄×Dic₁₃	C₂³	1	2	yes	—
56	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
58	#88137	—	2	2	no	—
59	#88135	—	2	2	no	—
60	#88133	—	2	2	no	—
61	#88131	—	2	2	no	—
62	C₄²×D₁₃	C₂³	1	1	yes	characteristic
63	#88124	C₂×C₄	1	2	yes	—
64	C₂²⋊#1057	—	2	2	no	—
65	C₂²⋊#1057	—	2	2	no	—
66	#88112	—	2	2	no	—
67	#88112	—	2	1	no	—
68	#88112	D₄	1	2	yes	—
69	#88111	—	2	1	no	—
70	#88111	D₄	1	2	yes	—
71	#88110	—	2	2	no	—
72	C₄⋊#1057	—	2	2	no	—
73	C₄⋊#1057	—	2	2	no	—
74	C₄⋊#1057	C₂³	1	2	yes	—
75	C₄⋊#1057	C₂³	1	2	yes	—
76	C₄×#1057	—	2	2	no	—
77	C₄×#1057	C₂³	1	1	yes	characteristic
78	C₄×#1057	C₂³	1	1	yes	characteristic
79	C₄×#1057	C₂³	1	1	yes	characteristic
80	C₄×#1057	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
82	#88075	C₂×C₄	1	2	yes	—
83	#88075	C₂×C₄	1	2	yes	—
84	#88072	C₂×C₄	1	2	yes	—
85	#88072	C₂×C₄	1	2	yes	—
86	#88058	C₂×C₄	1	1	yes	characteristic
87	#88058	C₂×C₄	1	1	yes	characteristic
88	#88057	C₂×C₄	1	2	yes	—
89	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
90	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
91	C₄×#1055	C₂×C₄	1	1	yes	characteristic
92	#17974	—	13	1	no	maximal, Sylow
93	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
94	C₂×D₄×C₁₃	C₂²×C₄	1	2	yes	—
95	C₂×C₂²⋊D₁₃	—	2	2	no	—
96	C₂²×Dic₁₃	—	2	2	no	—
97	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
100	#6122	—	8	2	no	—
101	C₂×#1057	—	2	2	no	—
102	C₂×#1057	—	2	2	no	—
103	C₂×#1057	—	2	1	no	—
104	C₂×#1057	—	2	1	no	—
105	C₂×#1057	—	8	2	no	—
106	C₂×#1057	C₂×D₄	1	2	yes	—
107	C₂×#1057	C₂×D₄	1	2	yes	—
108	#6118	—	8	2	no	—
109	C₁₃×Q₁₆	—	2	2	no	—
110	C₁₃×SD₁₆	—	2	4	no	—
111	C₁₃×D₈	—	2	2	no	—
112	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
114	C₁₃×C₂²⋊C₄	—	2	2	no	—
115	C₁₃×C₂²⋊C₄	—	2	2	no	—
116	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
117	C₂²⋊Dic₁₃	—	2	2	no	—
118	C₂²⋊Dic₁₃	—	2	2	no	—
119	#6103	—	2	2	no	—
120	C₄⋊Dic₁₃	—	2	2	no	—
121	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
122	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
123	#6101	—	2	2	no	—
124	#6101	—	2	2	no	—
125	#6101	—	2	2	no	—
126	C₄×Dic₁₃	—	2	2	no	—
127	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
128	C₄×Dic₁₃	#41	1	1	yes	characteristic
129	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
130	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
131	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
132	C₈×D₁₃	—	2	2	no	—
133	C₈×D₁₃	—	2	2	no	—
134	#3168	—	13	1	no	—
135	#2993	—	13	2	no	—
136	#1594	—	13	1	no	—
137	#1593	—	13	1	no	—
138	#1590	—	13	2	no	—
139	#1580	—	13	1	no	—
140	#1579	—	13	1	no	—
141	C₈×SD₁₆	—	13	2	no	—
142	#1566	—	13	1	no	—
143	#1495	—	13	2	no	—
144	#1450	—	13	1	no	—
145	C₂³×C₁₃	—	2	2	no	—
146	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
147	Q₈×C₁₃	—	2	2	no	—
148	Q₈×C₁₃	—	2	2	no	—
149	D₄×C₁₃	—	2	2	no	—
150	D₄×C₁₃	—	2	2	no	—
151	C₂×C₄×C₁₃	—	2	2	no	—
152	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
155	C₂²⋊D₁₃	—	8	2	no	—
156	C₂×Dic₁₃	—	2	2	no	—
157	C₂×Dic₁₃	—	2	2	no	—
158	C₂×Dic₁₃	—	4	2	no	—
159	C₂×Dic₁₃	—	4	2	no	—
160	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
161	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
162	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
163	C₄×D₁₃	—	2	1	no	—
164	C₄×D₁₃	—	2	1	no	—
165	C₄×D₁₃	—	2	2	no	—
166	C₄×D₁₃	—	4	1	no	—
167	#1057	—	2	2	no	—
168	#1057	—	2	2	no	—
169	#1057	—	4	1	no	—
170	#1057	—	4	1	no	—
171	#1057	—	4	2	no	—
172	#1057	—	4	2	no	—
173	C₈×C₁₃	#131	1	2	yes	—
174	C₈×C₁₃	#131	1	2	yes	—
175	#1055	—	2	1	no	—
176	#1055	—	2	1	no	—
177	#529	—	13	2	no	—
178	#499	—	13	1	no	—
179	#495	#261	1	1	yes	characteristic
180	#488	—	13	2	no	—
181	#467	—	13	1	no	—
182	#466	—	13	1	no	—
183	#465	—	13	2	no	—
184	#446	—	13	2	no	—
185	C₈×Q₈	—	13	2	no	—
186	C₄×Q₁₆	—	13	1	no	—
187	C₄×SD₁₆	—	13	2	no	—
188	C₄×D₈	—	13	1	no	—
189	#436	—	13	2	no	—
190	#436	—	26	2	no	—
191	#435	—	26	2	no	—
192	C₈×D₄	—	13	2	no	—
193	#433	—	26	2	no	—
194	#431	—	13	1	no	—
195	#431	—	13	1	no	—
196	#424	—	26	2	no	—
197	#423	—	13	2	no	—
198	C₂×C₄×C₈	—	13	1	no	—
199	#334	—	13	2	no	—
200	#334	—	13	1	no	—
201	#334	—	13	1	no	—
202	Q₈⋊C₈	—	13	2	no	—
203	Q₈⋊C₈	—	13	2	no	—
204	D₄⋊C₈	—	13	2	no	—
205	D₄⋊C₈	—	13	2	no	—
206	#322	—	13	1	no	—
207	#322	—	13	1	no	—
208	C₈²	—	13	1	no	—
209	C₈²	—	13	1	no	—
210	C₂²×C₁₃	—	4	2	no	—
211	C₂²×C₁₃	—	4	2	no	—
212	C₂²×C₁₃	—	4	2	no	—
213	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
214	C₂×D₁₃	—	2	1	no	—
215	C₂×D₁₃	—	2	1	no	—
216	C₂×D₁₃	—	2	1	no	—
217	C₄×C₁₃	—	2	1	no	—
218	C₄×C₁₃	—	2	1	no	—
219	C₄×C₁₃	—	4	2	no	—
220	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
221	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
222	Dic₁₃	—	2	1	no	—
223	Dic₁₃	—	2	1	no	—
224	Dic₁₃	—	4	2	no	—
225	Dic₁₃	—	4	2	no	—
226	Dic₁₃	#577	1	2	yes	—
227	C₂×Q₁₆	C₂×#261	1	1	yes	characteristic
228	C₂×SD₁₆	C₂×#261	1	2	yes	—
229	C₂×D₈	C₂×#261	1	1	yes	characteristic
230	C₂×M₁₆	—	26	2	no	—
231	C₂×M₁₆	—	26	2	no	—
232	C₂²×C₈	—	13	1	no	—
233	C₂²×C₈	—	13	1	no	—
234	C₂²×C₈	—	26	2	no	—
235	#126	—	26	2	no	—
236	#125	—	13	2	no	—
237	#124	C₂×#261	1	2	yes	—
238	#123	—	26	2	no	—
239	C₂²⋊Q₈	—	13	2	no	—
240	#121	—	13	2	no	—
241	C₄×Q₈	—	13	2	no	—
242	C₄×D₄	—	13	2	no	—
243	C₄×D₄	—	26	2	no	—
244	#117	—	26	2	no	—
245	C₂×C₄²	—	13	1	no	—
246	#107	—	13	1	no	—
247	#107	—	13	1	no	—
248	#106	—	13	2	no	—
249	C₄⋊C₈	—	13	2	no	—
250	C₄⋊C₈	—	13	2	no	—
251	C₄⋊C₈	—	13	2	no	—
252	C₄⋊C₈	—	26	2	no	—
253	C₄⋊C₈	—	26	1	no	—
254	C₄⋊C₈	—	26	2	no	—
255	C₄⋊C₈	—	26	2	no	—
256	#103	—	13	2	no	—
257	#103	—	13	2	no	—
258	#102	—	13	2	no	—
259	#102	—	13	2	no	—
260	C₂²⋊C₈	—	13	2	no	—
261	C₂²⋊C₈	—	26	2	no	—
262	C₂²⋊C₈	—	26	2	no	—
263	C₂²⋊C₈	—	26	1	no	—
264	#97	—	26	2	no	—
265	C₄×C₈	—	13	1	no	—
266	C₄×C₈	—	13	1	no	—
267	C₄×C₈	—	13	1	no	—
268	C₄×C₈	—	13	1	no	—
269	C₄×C₈	—	13	1	no	—
270	C₄×C₈	—	13	1	no	—
271	C₄×C₈	—	13	1	no	—
272	C₄×C₈	—	26	2	no	—
273	C₄×C₈	C₂×#261	1	1	yes	characteristic
274	C₂×C₁₃	—	8	2	no	—
275	C₂×C₁₃	#2931	1	1	yes	characteristic
276	C₂×C₁₃	#2952	1	1	yes	characteristic
277	C₂×C₁₃	#2957	1	1	yes	characteristic
278	D₁₃	—	4	1	no	—
279	C₂×Q₈	C₂²×#261	1	2	yes	—
280	C₂×D₄	—	26	2	no	—
281	C₂×D₄	C₂²×#261	1	2	yes	—
282	C₂²×C₄	—	13	1	no	—
283	C₂²×C₄	—	13	1	no	—
284	C₂²×C₄	—	13	1	no	—
285	C₂²×C₄	—	26	2	no	—
286	Q₁₆	—	2	2	no	—
287	SD₁₆	—	2	4	no	—
288	D₈	—	2	2	no	—
289	M₁₆	—	104	2	no	—
290	M₁₆	—	104	2	no	—
291	C₂×C₈	—	13	2	no	—
292	C₂×C₈	—	13	2	no	—
293	C₂×C₈	—	13	1	no	—
294	C₂×C₈	—	13	1	no	—
295	C₂×C₈	—	26	2	no	—
296	C₂×C₈	—	26	2	no	—
297	C₂×C₈	—	26	1	no	—
298	C₂×C₈	—	26	1	no	—
299	C₂×C₈	—	26	2	no	—
300	C₂×C₈	—	26	2	no	—
301	C₂×C₈	—	104	2	no	—
302	C₂×C₈	C₂²×#261	1	1	yes	characteristic
303	C₂×C₈	C₂²×#261	1	1	yes	characteristic
304	C₄⋊C₄	—	13	2	no	—
305	C₄⋊C₄	—	13	2	no	—
306	C₄⋊C₄	—	26	2	no	—
307	C₄⋊C₄	—	26	2	no	—
308	C₄⋊C₄	—	26	2	no	—
309	C₄⋊C₄	—	26	2	no	—
310	C₂²⋊C₄	—	2	2	no	—
311	C₂²⋊C₄	—	2	2	no	—
312	C₂²⋊C₄	—	26	2	no	—
313	C₂²⋊C₄	—	26	2	no	—
314	C₂²⋊C₄	—	26	2	no	—
315	C₄²	—	13	1	no	—
316	C₄²	—	13	1	no	—
317	C₄²	—	13	1	no	—
318	C₄²	—	26	2	no	—
319	C₄²	C₂²×#261	1	1	yes	characteristic
320	C₁₃	#17974	1	1	yes	characteristic, complemented, Sylow
321	C₂³	—	2	2	no	—
322	C₂³	—	13	1	no	—
323	Q₈	—	2	2	no	—
324	Q₈	—	2	2	no	—
325	D₄	—	2	2	no	—
326	D₄	—	2	2	no	—
327	D₄	—	104	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	13	1	no	—
330	C₂×C₄	—	13	1	no	—
331	C₂×C₄	—	13	1	no	—
332	C₂×C₄	—	26	2	no	—
333	C₂×C₄	—	26	1	no	—
334	C₂×C₄	—	26	1	no	—
335	C₂×C₄	—	26	2	no	—
336	C₂×C₄	—	26	2	no	—
337	C₂×C₄	—	52	1	no	—
338	C₂×C₄	—	52	2	no	—
339	C₂×C₄	—	52	2	no	—
340	C₂×C₄	D₄×#261	1	1	yes	characteristic
341	C₂×C₄	D₄×#261	1	1	yes	characteristic
342	C₂×C₄	C₂³×#261	1	1	yes	characteristic
343	C₈	—	26	2	no	—
344	C₈	—	26	2	no	—
345	C₈	—	26	1	no	—
346	C₈	—	26	1	no	—
347	C₈	—	52	2	no	—
348	C₈	—	52	2	no	—
349	C₈	—	52	1	no	—
350	C₈	—	52	1	no	—
351	C₈	#88238	1	2	yes	—
352	C₈	#88238	1	2	yes	—
353	C₂²	—	4	2	no	—
354	C₂²	—	4	2	no	—
355	C₂²	—	4	2	no	—
356	C₂²	—	26	1	no	—
357	C₂²	—	26	1	no	—
358	C₂²	—	26	1	no	—
359	C₂²	C₂×D₄×#261	1	1	yes	characteristic
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	4	2	no	—
363	C₄	—	13	2	no	—
364	C₄	—	26	1	no	—
365	C₄	—	26	1	no	—
366	C₄	—	52	2	no	—
367	C₄	—	52	2	no	—
368	C₄	C₂×#88238	1	1	yes	characteristic
369	C₄	C₂×#88238	1	1	yes	characteristic
370	C₂	—	8	2	no	—
371	C₂	—	52	1	no	—
372	C₂	#13131558	1	1	yes	characteristic
373	C₂	#13131599	1	1	yes	characteristic
374	C₂	#13131615	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₁₃×#495	C₄
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#11718818	C₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	#495	#261
hypocenter	C₁₃	#17974
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #495

Complements

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

N	H ≅ G/N	type
1. #13131779	C₂	semidirect
2. #13130486	C₂	semidirect
3. #13130483	C₂	semidirect
4. #13130472	C₂	semidirect
5. #13130432	C₂	semidirect
6. #13130429	C₂	semidirect
7. SD₁₆×#1057	C₂	semidirect
8. #13130391	C₂	semidirect
9. #13130192	C₂	semidirect
10. #13130040	C₂	semidirect
320. C₁₃	#17974	semidirect

Maximal subgroups

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

subgroup	index
#13131779	2
#13130486	2
#13130483	2
#13130472	2
#13130432	2
#13130429	2
SD₁₆×#1057	2
#13130391	2
#13130192	2
#13130040	2
#13125403	2
#17974	13