Subgroups — #16795920

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	#13131782	C₂	1	2	yes	maximal, complemented
2	#13130485	C₂	1	1	yes	maximal, characteristic, complemented
3	#13130482	C₂	1	1	yes	maximal, characteristic, complemented
4	#13130471	C₂	1	2	yes	maximal, complemented
5	#13130434	C₂	1	1	yes	maximal, characteristic, complemented
6	#13130431	C₂	1	1	yes	maximal, characteristic, complemented
7	#13130421	C₂	1	2	yes	maximal, complemented
8	#13130391	C₂	1	1	yes	maximal, characteristic, complemented
9	#13130194	C₂	1	2	yes	maximal, complemented
10	#13130040	C₂	1	1	yes	maximal, characteristic, complemented
11	#13125408	C₂	1	1	yes	maximal, characteristic
12	#11719396	C₂²	1	2	yes	—
13	#11719080	C₂²	1	2	yes	—
14	#11719079	—	2	2	no	—
15	#11719077	—	2	2	no	—
16	#11719076	—	2	2	no	—
17	#11719076	C₂²	1	2	yes	—
18	#11719075	—	2	2	no	—
19	#11719074	C₂²	1	2	yes	—
20	#11719073	C₂²	1	2	yes	—
21	#11719052	C₂²	1	2	yes	—
22	#11719049	C₂²	1	1	yes	characteristic
23	#11719049	C₂²	1	1	yes	characteristic
24	C₁₃×#499	C₄	1	1	yes	characteristic
25	#11718729	C₄	1	2	yes	—
26	#11718548	C₄	1	1	yes	characteristic
27	#11718544	C₄	1	1	yes	characteristic
28	#11718534	C₂²	1	1	yes	characteristic
29	#11718533	C₂²	1	1	yes	characteristic
30	#11718525	C₄	1	2	yes	—
31	#11718513	C₂²	1	2	yes	—
32	#11718357	C₄	1	1	yes	characteristic
33	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
34	#11718253	C₂²	1	2	yes	—
35	#11718253	C₂²	1	1	yes	characteristic
36	#11718253	C₂²	1	1	yes	characteristic
37	#11718251	C₂²	1	1	yes	characteristic
38	#11718251	C₂²	1	1	yes	characteristic
39	C₈×#1057	C₂²	1	1	yes	characteristic
40	C₈×#1057	C₂²	1	1	yes	characteristic
41	#11718244	C₂²	1	2	yes	—
42	#11718244	C₂²	1	2	yes	—
43	#11718243	C₂²	1	2	yes	—
44	#11718243	C₂²	1	2	yes	—
45	C₂×#6118	—	2	2	no	—
46	C₂×#6118	—	2	2	no	—
47	C₂×D₁₃⋊C₈	—	2	2	no	—
48	C₁₃×#125	C₂×C₄	1	2	yes	—
49	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
50	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
51	C₂×#6095	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
54	#88152	—	2	2	no	—
55	#88151	C₂×C₄	1	2	yes	—
56	#88149	C₂×C₄	1	2	yes	—
57	#88148	—	2	2	no	—
58	#88147	—	2	2	no	—
59	#88147	C₂³	1	2	yes	—
60	#88146	—	2	2	no	—
61	#88141	C₂³	1	2	yes	—
62	#88129	C₂×C₄	1	2	yes	—
63	C₄²×D₁₃	C₂³	1	1	yes	characteristic
64	#88112	—	2	2	no	—
65	#88112	—	2	1	no	—
66	#88112	D₄	1	2	yes	—
67	#88111	—	2	1	no	—
68	#88111	—	2	2	no	—
69	#88111	—	2	2	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	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
82	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
83	C₁₃×#106	C₂×C₄	1	2	yes	—
84	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
85	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
86	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
87	#88050	C₂×C₄	1	2	yes	—
88	#88050	C₂×C₄	1	2	yes	—
89	#88049	C₂×C₄	1	2	yes	—
90	#88049	C₂×C₄	1	2	yes	—
91	C₄×#1055	C₂×C₄	1	1	yes	characteristic
92	#17974	—	13	1	no	maximal, Sylow
93	C₂×D₅₂	—	2	2	no	—
94	C₂×D₅₂	C₂²×C₄	1	2	yes	—
95	C₂×C₄×D₁₃	—	2	2	no	—
96	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
99	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
100	C₂×#1057	—	2	2	no	—
101	C₂×#1057	—	2	2	no	—
102	C₂×#1057	—	2	1	no	—
103	C₂×#1057	—	2	1	no	—
104	C₂×#1057	C₂×D₄	1	2	yes	—
105	C₂×#1057	C₂×D₄	1	2	yes	—
106	#6118	—	8	2	no	—
107	#6118	—	8	2	no	—
108	D₁₃⋊C₈	—	8	2	no	—
109	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
111	C₁₃×C₄⋊C₄	—	2	2	no	—
112	C₁₃×C₄⋊C₄	—	2	2	no	—
113	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
114	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
115	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
116	#6103	—	2	2	no	—
117	#6103	—	2	2	no	—
118	#6103	—	2	2	no	—
119	#6103	—	2	2	no	—
120	#6103	—	2	2	no	—
121	C₄⋊Dic₁₃	—	2	2	no	—
122	#6101	—	2	2	no	—
123	C₄×Dic₁₃	—	2	2	no	—
124	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
125	C₄×Dic₁₃	#41	1	1	yes	characteristic
126	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
127	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
128	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
129	Dic₅₂	—	2	2	no	—
130	D₁₀₄	—	2	2	no	—
131	#6095	—	2	4	no	—
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	C₂²×D₁₃	—	2	2	no	—
146	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃	—	2	2	no	—
148	C₂×C₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
152	C₂×Dic₁₃	—	2	2	no	—
153	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
154	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
156	D₅₂	—	2	2	no	—
157	D₅₂	—	2	2	no	—
158	D₅₂	—	8	2	no	—
159	C₄×D₁₃	—	2	1	no	—
160	C₄×D₁₃	—	2	1	no	—
161	C₄×D₁₃	—	2	2	no	—
162	C₄×D₁₃	—	4	1	no	—
163	C₄×D₁₃	—	4	2	no	—
164	C₄×D₁₃	—	4	2	no	—
165	Dic₂₆	—	2	2	no	—
166	Dic₂₆	—	2	2	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	#261	1	1	yes	characteristic
179	#495	—	13	1	no	—
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	C₂×C₄×D₄	1	1	yes	characteristic
211	C₂×D₁₃	—	2	1	no	—
212	C₂×D₁₃	—	2	1	no	—
213	C₂×D₁₃	—	2	1	no	—
214	C₂×D₁₃	—	4	2	no	—
215	C₂×D₁₃	—	4	2	no	—
216	C₂×D₁₃	—	4	2	no	—
217	C₄×C₁₃	—	2	1	no	—
218	C₄×C₁₃	—	2	1	no	—
219	C₄×C₁₃	—	4	2	no	—
220	C₄×C₁₃	—	4	2	no	—
221	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
222	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
223	Dic₁₃	—	2	1	no	—
224	Dic₁₃	—	2	1	no	—
225	Dic₁₃	—	4	2	no	—
226	Dic₁₃	#577	1	2	yes	—
227	C₂×Q₁₆	—	13	1	no	—
228	C₂×SD₁₆	—	13	2	no	—
229	C₂×D₈	—	13	1	no	—
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	C₂×#261	1	2	yes	—
237	#124	—	13	2	no	—
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	C₂×#261	1	1	yes	characteristic
247	#107	C₂×#261	1	1	yes	characteristic
248	#106	C₂×#261	1	2	yes	—
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	#2931	1	1	yes	characteristic
275	C₂×C₁₃	#2952	1	1	yes	characteristic
276	C₂×C₁₃	#2957	1	1	yes	characteristic
277	D₁₃	—	4	1	no	—
278	D₁₃	—	8	2	no	—
279	C₂×Q₈	—	13	2	no	—
280	C₂×D₄	—	13	2	no	—
281	C₂×D₄	—	26	2	no	—
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	—	26	2	no	—
287	SD₁₆	—	26	4	no	—
288	D₈	—	26	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	—	2	2	no	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	26	2	no	—
307	C₄⋊C₄	—	26	2	no	—
308	C₄⋊C₄	C₂²×#261	1	2	yes	—
309	C₄⋊C₄	C₂²×#261	1	2	yes	—
310	C₂²⋊C₄	—	26	2	no	—
311	C₂²⋊C₄	—	26	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	—	13	1	no	—
322	C₂³	—	26	2	no	—
323	Q₈	—	26	2	no	—
324	Q₈	—	26	2	no	—
325	D₄	—	26	2	no	—
326	D₄	—	26	2	no	—
327	D₄	—	104	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	13	1	no	—
331	C₂×C₄	—	13	1	no	—
332	C₂×C₄	—	13	1	no	—
333	C₂×C₄	—	26	2	no	—
334	C₂×C₄	—	26	1	no	—
335	C₂×C₄	—	26	1	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	#88240	1	2	yes	—
352	C₈	#88240	1	2	yes	—
353	C₂²	—	26	1	no	—
354	C₂²	—	26	1	no	—
355	C₂²	—	26	1	no	—
356	C₂²	—	52	2	no	—
357	C₂²	—	52	2	no	—
358	C₂²	—	52	2	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	—	4	2	no	—
364	C₄	—	13	2	no	—
365	C₄	—	26	1	no	—
366	C₄	—	26	1	no	—
367	C₄	—	52	2	no	—
368	C₄	C₂×#88240	1	1	yes	characteristic
369	C₄	C₂×#88240	1	1	yes	characteristic
370	C₂	—	52	1	no	—
371	C₂	—	104	2	no	—
372	C₂	#13131561	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×#499	C₄
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#11718544	C₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	#499	#261
hypocenter	C₁₃	#17974
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

Complements

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

N	H ≅ G/N	type
1. #13131782	C₂	semidirect
2. #13130485	C₂	semidirect
3. #13130482	C₂	semidirect
4. #13130471	C₂	semidirect
5. #13130434	C₂	semidirect
6. #13130431	C₂	semidirect
7. #13130421	C₂	semidirect
8. #13130391	C₂	semidirect
9. #13130194	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
#13131782	2
#13130485	2
#13130482	2
#13130471	2
#13130434	2
#13130431	2
#13130421	2
#13130391	2
#13130194	2
#13130040	2
#13125408	2
#17974	13