Subgroups — #16794923

Order 3328.

734 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131753	C₂	1	2	yes	maximal
2	#13130389	C₂	1	2	yes	maximal, complemented
3	#13130388	C₂	1	4	yes	maximal, complemented
4	#13130386	C₂	1	2	yes	maximal, complemented
5	#13130383	C₂	1	1	yes	maximal, characteristic, complemented
6	D₁₃⋊#335	C₂	1	2	yes	maximal, complemented
7	C₂×#11718254	C₂	1	1	yes	maximal, characteristic, complemented
8	#13123258	C₂	1	1	yes	maximal, characteristic
9	C₂×#88161	C₄	1	1	yes	characteristic
10	C₂×#88155	C₄	1	1	yes	characteristic
11	#11719227	C₂²	1	1	yes	characteristic
12	#11719119	C₂²	1	2	yes	—
13	C₂×C₄⋊#1057	C₂²	1	2	yes	—
14	#11719074	—	2	2	no	—
15	#11719073	—	2	2	no	—
16	#11719063	—	2	4	no	—
17	#11719051	C₂²	1	4	yes	—
18	D₁₃⋊(C₄⋊C₈)	C₂²	1	4	yes	—
19	C₁₃×#405	C₄	1	1	yes	characteristic
20	M₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
22	#11718658	C₄	1	1	yes	characteristic
23	#11718369	C₄	1	2	yes	—
24	#11718368	C₂²	1	2	yes	—
25	#11718350	C₄	1	2	yes	—
26	C₄×C₈×D₁₃	C₂²	1	2	yes	—
27	#11718256	C₂²	1	4	yes	—
28	#11718255	C₂²	1	4	yes	—
29	#11718254	C₂²	1	2	yes	—
30	#11718254	C₂²	1	2	yes	—
31	#11718254	C₂²	1	1	yes	characteristic
32	#11718254	C₂²	1	1	yes	characteristic
33	#11718253	C₂²	1	2	yes	—
34	C₂×#6127	C₂×C₄	1	1	yes	characteristic
35	C₂²×#1057	—	2	2	no	—
36	C₂×D₁₃⋊C₈	—	2	4	no	—
37	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
41	C₂×#6099	C₂×C₄	1	1	yes	characteristic
42	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
43	#88161	—	2	4	no	—
44	#88155	C₈	1	8	yes	complemented
45	C₂×#6094	C₂×C₄	1	2	yes	—
46	C₂×#6094	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
48	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
49	#88147	D₄	1	1	yes	characteristic
50	#88147	Q₈	1	1	yes	characteristic
51	#88141	D₄	1	1	yes	characteristic
52	#88141	Q₈	1	1	yes	characteristic
53	#88135	D₄	1	2	yes	—
54	#88135	Q₈	1	2	yes	—
55	#88126	C₂×C₄	1	2	yes	—
56	C₄²×D₁₃	C₂³	1	2	yes	—
57	C₂²⋊#1057	—	2	2	no	—
58	#88112	—	2	4	no	—
59	#88111	—	2	4	no	—
60	C₄⋊#1057	—	2	2	no	—
61	C₄⋊#1057	C₂³	1	2	yes	—
62	C₄⋊#1057	C₂³	1	2	yes	—
63	C₄⋊#1057	C₂³	1	4	yes	—
64	C₄×#1057	—	2	4	no	—
65	C₄×#1057	—	2	2	no	—
66	C₁₃×#97	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
68	#88055	C₂×C₄	1	2	yes	—
69	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
72	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
73	C₈×Dic₁₃	C₂³	1	2	yes	—
74	#88043	C₂×C₄	1	2	yes	—
75	C₄×#1055	C₂×C₄	1	2	yes	—
76	#17789	—	13	1	no	maximal, Sylow
77	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₂²⋊D₁₃	C₄⋊C₄	1	2	yes	—
79	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
81	#6127	C₂×C₈	1	4	yes	—
82	#6127	M₁₆	1	4	yes	—
83	C₂×D₅₂	C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
86	C₂×C₄×D₁₃	C₂×Q₈	1	2	yes	—
87	C₂×Dic₂₆	C₄⋊C₄	1	1	yes	characteristic
88	C₂×#1057	—	2	2	no	—
89	C₂×#1057	—	2	4	no	—
90	C₂×#1057	—	2	2	no	—
91	C₂×#1057	—	8	2	no	—
92	D₁₃⋊C₈	—	8	4	no	—
93	C₁₃×M₁₆	—	2	2	no	—
94	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	complemented
95	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
99	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
100	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
101	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
102	#6103	C₄⋊C₄	1	2	yes	—
103	#6103	C₄⋊C₄	1	2	yes	—
104	#6101	C₄⋊C₄	1	2	yes	—
105	#6101	C₄⋊C₄	1	2	yes	—
106	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
108	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
109	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
110	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
111	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
112	#6099	C₂×C₈	1	4	yes	complemented
113	C₂×#1055	—	2	2	no	—
114	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
115	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
116	C₂×#1055	C₂²×C₄	1	2	yes	—
117	#6094	—	2	4	no	—
118	#6094	C₂×C₈	1	8	yes	complemented
119	C₈×D₁₃	—	2	4	no	—
120	C₈×D₁₃	C₂×C₈	1	8	yes	complemented
121	#2988	—	13	2	no	—
122	C₄×#131	—	13	1	no	—
123	#1565	—	13	4	no	—
124	#1565	—	13	2	no	—
125	#1564	—	13	1	no	—
126	#1564	—	13	2	no	—
127	C₂×#335	—	13	1	no	—
128	C₂×#335	—	13	2	no	—
129	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
136	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
137	C₂²⋊D₁₃	C₄⋊C₈	1	8	yes	—
138	C₂×Dic₁₃	—	2	1	no	—
139	C₂×Dic₁₃	—	2	1	no	—
140	C₂×Dic₁₃	—	4	1	no	—
141	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
144	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
146	D₅₂	C₄⋊C₈	1	4	yes	—
147	C₄×D₁₃	—	2	2	no	—
148	C₄×D₁₃	—	2	2	no	—
149	C₄×D₁₃	—	4	2	no	—
150	C₄×D₁₃	C₂²×C₈	1	4	yes	—
151	C₄×D₁₃	C₂×M₁₆	1	4	yes	—
152	Dic₂₆	C₄⋊C₈	1	4	yes	—
153	#1057	—	4	2	no	—
154	#1057	—	4	4	no	—
155	#1057	—	4	2	no	—
156	C₈×C₁₃	—	2	2	no	—
157	C₈×C₁₃	C₂²×C₈	1	2	yes	—
158	C₈×C₁₃	C₂²×C₈	1	2	yes	—
159	#1055	—	2	1	no	—
160	#1055	—	2	1	no	—
161	#1055	C₂²×C₈	1	4	yes	—
162	C₂×#131	—	13	1	no	—
163	C₂×#131	—	13	1	no	—
164	C₄×#41	—	13	1	no	—
165	C₈×Q₈	—	26	2	no	—
166	C₈×D₄	—	26	4	no	—
167	C₈×D₄	—	26	2	no	—
168	#431	—	13	4	no	—
169	#431	—	13	2	no	—
170	C₂×C₄⋊C₈	—	13	4	no	—
171	C₂×C₄⋊C₈	—	13	2	no	—
172	#405	—	13	1	no	—
173	#405	—	13	2	no	—
174	#405	—	13	2	no	—
175	#405	#261	1	1	yes	characteristic
176	C₄×M₁₆	—	13	1	no	—
177	C₄×M₁₆	—	13	2	no	—
178	C₂×C₄×C₈	—	13	1	no	—
179	C₂×C₄×C₈	—	13	2	no	—
180	#335	—	13	2	no	—
181	#335	—	13	1	no	—
182	#335	—	13	2	no	—
183	#335	—	13	4	no	—
184	#335	—	13	1	no	—
185	#334	—	13	2	no	—
186	#334	—	13	4	no	—
187	C₂²×C₁₃	—	2	1	no	—
188	C₂²×C₁₃	—	2	1	no	—
189	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
190	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
191	C₂×D₁₃	—	2	2	no	—
192	C₂×D₁₃	—	2	2	no	—
193	C₂×D₁₃	C₂×C₄⋊C₈	1	4	yes	—
194	C₄×C₁₃	—	2	2	no	—
195	C₄×C₁₃	—	2	2	no	—
196	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
197	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
198	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
199	Dic₁₃	—	2	1	no	—
200	Dic₁₃	—	2	1	no	—
201	Dic₁₃	—	2	1	no	—
202	Dic₁₃	C₂×C₄⋊C₈	1	4	yes	—
203	Dic₁₃	#576	1	1	yes	characteristic
204	Dic₁₃	#578	1	1	yes	characteristic
205	C₂×#41	—	13	1	no	—
206	#131	—	13	8	no	—
207	#131	—	26	4	no	—
208	C₂×M₁₆	—	13	1	no	—
209	C₂×M₁₆	—	13	2	no	—
210	C₂×M₁₆	—	13	2	no	—
211	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
212	C₂²×C₈	—	13	2	no	—
213	C₂²×C₈	—	13	1	no	—
214	C₂²×C₈	—	13	2	no	—
215	C₂²×C₈	—	26	4	no	—
216	C₂²×C₈	—	26	2	no	—
217	C₂²×C₈	C₂×#261	1	1	yes	characteristic
218	C₄×Q₈	—	13	1	no	—
219	C₄×Q₈	—	13	1	no	—
220	C₄×D₄	—	13	1	no	—
221	C₄×D₄	—	13	1	no	—
222	C₄×D₄	—	13	2	no	—
223	C₄×D₄	—	13	2	no	—
224	#117	—	13	2	no	—
225	#117	C₂×#261	1	1	yes	characteristic
226	C₂×C₄²	—	13	1	no	—
227	C₂×C₄²	—	13	2	no	—
228	C₄⋊C₈	—	13	2	no	—
229	C₄⋊C₈	—	13	2	no	—
230	C₄⋊C₈	—	13	4	no	—
231	C₄⋊C₈	—	26	2	no	—
232	C₄⋊C₈	—	26	4	no	—
233	C₂²⋊C₈	—	26	4	no	—
234	C₂²⋊C₈	—	26	2	no	—
235	#97	—	13	2	no	—
236	#97	—	13	2	no	—
237	#97	C₂×#261	1	2	yes	—
238	C₄×C₈	—	13	2	no	—
239	C₄×C₈	—	13	1	no	—
240	C₄×C₈	—	13	1	no	—
241	C₄×C₈	—	13	1	no	—
242	C₄×C₈	—	13	1	no	—
243	C₄×C₈	—	13	2	no	—
244	C₄×C₈	—	26	4	no	—
245	C₄×C₈	—	26	2	no	—
246	C₄×C₈	C₂×#261	1	2	yes	—
247	C₂×C₁₃	—	2	2	no	—
248	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
249	C₂×C₁₃	#2913	1	1	yes	characteristic
250	C₂×C₁₃	#2916	1	1	yes	characteristic
251	D₁₃	—	2	4	no	—
252	#41	—	13	4	no	—
253	#41	—	13	4	no	—
254	C₂×Q₈	—	13	1	no	—
255	C₂×D₄	—	13	2	no	—
256	C₂×D₄	—	13	1	no	—
257	C₂²×C₄	—	13	2	no	—
258	C₂²×C₄	—	13	1	no	—
259	C₂²×C₄	—	13	1	no	—
260	C₂²×C₄	—	13	2	no	—
261	C₂²×C₄	—	13	2	no	—
262	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
263	M₁₆	—	2	2	no	—
264	M₁₆	—	13	4	no	—
265	M₁₆	—	13	8	no	—
266	M₁₆	—	26	4	no	—
267	C₂×C₈	—	13	2	no	—
268	C₂×C₈	—	13	8	no	—
269	C₂×C₈	—	13	1	no	—
270	C₂×C₈	—	13	1	no	—
271	C₂×C₈	—	26	4	no	—
272	C₂×C₈	—	26	2	no	—
273	C₂×C₈	—	26	2	no	—
274	C₂×C₈	—	26	4	no	—
275	C₂×C₈	—	26	2	no	—
276	C₂×C₈	—	104	4	no	—
277	C₂×C₈	—	104	2	no	—
278	C₂×C₈	D₁₃⋊C₈	1	4	yes	complemented
279	C₂×C₈	C₂²×#261	1	2	yes	—
280	C₂×C₈	C₂²×#261	1	1	yes	characteristic
281	C₂×C₈	C₂²×#261	1	1	yes	characteristic
282	C₄⋊C₄	—	13	2	no	—
283	C₄⋊C₄	—	13	2	no	—
284	C₄⋊C₄	C₄⋊#261	1	1	yes	characteristic
285	C₄⋊C₄	C₄⋊#261	1	1	yes	characteristic
286	C₂²⋊C₄	—	13	2	no	—
287	C₂²⋊C₄	—	13	2	no	—
288	C₂²⋊C₄	C₄⋊#261	1	2	yes	—
289	C₄²	—	13	2	no	—
290	C₄²	—	13	1	no	—
291	C₄²	—	13	2	no	—
292	C₄²	—	13	1	no	—
293	C₄²	—	13	1	no	—
294	C₄²	—	13	1	no	—
295	C₄²	C₂²×#261	1	2	yes	—
296	C₁₃	#17789	1	1	yes	characteristic, complemented, Sylow
297	C₂³	—	13	2	no	—
298	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
299	Q₈	—	13	4	no	—
300	D₄	—	13	8	no	—
301	D₄	—	13	4	no	—
302	C₂×C₄	—	13	1	no	—
303	C₂×C₄	—	13	2	no	—
304	C₂×C₄	—	13	4	no	—
305	C₂×C₄	—	13	4	no	—
306	C₂×C₄	—	13	1	no	—
307	C₂×C₄	—	13	1	no	—
308	C₂×C₄	—	13	1	no	—
309	C₂×C₄	—	26	2	no	—
310	C₂×C₄	—	26	2	no	—
311	C₂×C₄	—	26	1	no	—
312	C₂×C₄	—	26	1	no	—
313	C₂×C₄	—	52	2	no	—
314	C₂×C₄	—	52	1	no	—
315	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
316	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
317	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
318	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
319	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
320	C₂×C₄	C₂³×#261	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	—	13	4	no	—
323	C₈	—	26	1	no	—
324	C₈	—	26	1	no	—
325	C₈	—	52	2	no	—
326	C₈	—	52	4	no	—
327	C₈	—	52	2	no	—
328	C₈	C₂×D₁₃⋊C₈	1	2	yes	—
329	C₈	C₂×D₁₃⋊C₈	1	2	yes	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	13	4	no	—
333	C₂²	—	26	2	no	—
334	C₂²	—	26	2	no	—
335	C₂²	D₁₃⋊(C₄⋊C₈)	1	2	yes	—
336	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
337	C₄	—	2	2	no	—
338	C₄	—	2	2	no	—
339	C₄	—	13	4	no	—
340	C₄	—	13	1	no	—
341	C₄	—	13	1	no	—
342	C₄	—	26	1	no	—
343	C₄	—	26	1	no	—
344	C₄	—	26	1	no	—
345	C₄	D₁₃⋊(C₄⋊C₈)	1	2	yes	—
346	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
347	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
348	C₂	—	2	2	no	—
349	C₂	—	26	4	no	—
350	C₂	C₂×D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
351	C₂	#13131491	1	1	yes	characteristic
352	C₂	#13131498	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#261
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₁₃×#405	C₄
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂×#6127	C₂×C₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	#405	#261
hypocenter	C₁₃	#17789
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
2. #13130389	C₂	semidirect
3. #13130388	C₂	semidirect
4. #13130386	C₂	semidirect
5. #13130383	C₂	semidirect
6. D₁₃⋊#335	C₂	semidirect
7. C₂×#11718254	C₂	semidirect
44. #88155	C₈	semidirect
94. C₂×C₈×C₁₃	C₂×C₈	semidirect
112. #6099	C₂×C₈	semidirect
118. #6094	C₂×C₈	semidirect
120. C₈×D₁₃	C₂×C₈	semidirect
278. C₂×C₈	D₁₃⋊C₈	semidirect
296. C₁₃	#17789	semidirect

Maximal subgroups

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

subgroup	index
#13131753	2
#13130389	2
#13130388	2
#13130386	2
#13130383	2
D₁₃⋊#335	2
C₂×#11718254	2
#13123258	2
#17789	13