Subgroups — #16798251

Order 3328.

608 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13130766	C₂	1	2	yes	maximal, complemented
2	#13130755	C₂	1	2	yes	maximal, complemented
3	#13130746	C₂	1	2	yes	maximal
4	#13130721	C₂	1	1	yes	maximal, characteristic
5	#13130719	C₂	1	1	yes	maximal, characteristic
6	#13130642	C₂	1	2	yes	maximal, complemented
7	#13130591	C₂	1	2	yes	maximal
8	#13130558	C₂	1	2	yes	maximal, complemented
9	#13123263	C₂	1	1	yes	maximal, characteristic
10	C₂×#88161	C₂²	1	1	yes	characteristic
11	C₂×#88155	C₂²	1	1	yes	characteristic
12	#11719227	C₂²	1	1	yes	characteristic
13	#11719132	C₂²	1	2	yes	—
14	#11719116	C₂²	1	2	yes	—
15	D₁₃⋊#108	C₂²	1	2	yes	—
16	#11719092	C₂²	1	2	yes	—
17	#11719089	C₂²	1	4	yes	—
18	C₁₃×#405	C₄	1	1	yes	characteristic, complemented
19	M₁₆×Dic₁₃	C₄	1	1	yes	characteristic, complemented
20	C₂×#88055	C₄	1	1	yes	characteristic, complemented
21	#11718658	C₄	1	1	yes	characteristic, complemented
22	#11718369	C₄	1	1	yes	characteristic, complemented
23	#11718369	C₂²	1	1	yes	characteristic
24	D₁₃×#97	C₄	1	1	yes	characteristic, complemented
25	D₁₃×#97	C₂²	1	1	yes	characteristic
26	#11718350	C₄	1	1	yes	characteristic, complemented
27	#11718350	C₂²	1	1	yes	characteristic
28	C₄×#6094	C₄	1	1	yes	characteristic, complemented
29	C₄×#6094	C₂²	1	1	yes	characteristic
30	#11718269	C₂²	1	2	yes	—
31	#11718267	C₂²	1	2	yes	—
32	#11718266	C₂²	1	4	yes	—
33	#11718235	C₂²	1	4	yes	—
34	#11718234	C₂²	1	4	yes	—
35	C₂×#6127	C₂³	1	1	yes	characteristic
36	#88237	C₂³	1	4	yes	—
37	D₁₃⋊M₁₆	C₂³	1	4	yes	—
38	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×#6099	C₂×C₄	1	1	yes	characteristic
43	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
44	#88161	—	2	2	no	—
45	#88161	—	2	2	no	—
46	#88155	—	2	4	no	—
47	C₂×#6094	C₂×C₄	1	1	yes	characteristic
48	C₂×#6094	C₂×C₄	1	1	yes	characteristic
49	C₂×#6094	C₂³	1	1	yes	characteristic
50	C₂×#6094	C₂³	1	1	yes	characteristic
51	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
55	#88147	—	2	1	no	—
56	#88141	—	2	1	no	—
57	#88135	—	2	1	no	—
58	#88135	—	2	1	no	—
59	#88126	C₂×C₄	1	1	yes	characteristic
60	#88126	C₂³	1	1	yes	characteristic
61	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
62	C₄²×D₁₃	C₂³	1	1	yes	characteristic
63	#88121	—	2	2	no	—
64	#88119	—	2	2	no	—
65	#88113	—	2	2	no	—
66	#88107	—	2	2	no	—
67	#88104	—	2	2	no	—
68	#88103	—	2	2	no	—
69	#88102	—	2	2	no	—
70	#88101	—	2	2	no	—
71	#88100	—	2	4	no	—
72	C₈×#261	—	2	4	no	—
73	C₁₃×#97	C₂×C₄	1	1	yes	characteristic
74	C₁₃×#97	C₂×C₄	1	1	yes	characteristic
75	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
77	#88055	C₂×C₄	1	1	yes	characteristic
78	#88055	C₂×C₄	1	1	yes	characteristic
79	#88055	C₂×C₄	1	1	yes	characteristic
80	#88055	C₂×C₄	1	1	yes	characteristic
81	#88055	C₂×C₄	1	1	yes	characteristic
82	#88055	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
85	#88043	C₂×C₄	1	1	yes	characteristic
86	#88043	C₂×C₄	1	1	yes	characteristic
87	C₄×#1055	C₂×C₄	1	1	yes	characteristic
88	C₄×#1055	C₂×C₄	1	1	yes	characteristic
89	#20788	—	13	1	no	maximal, Sylow
90	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₂²⋊D₁₃	#41	1	1	yes	characteristic
92	C₂×C₂²⋊D₁₃	#41	1	1	yes	characteristic
93	C₂²×Dic₁₃	—	2	1	no	—
94	#6127	—	2	2	no	—
95	#6127	—	2	2	no	—
96	C₂×D₅₂	#41	1	1	yes	characteristic
97	C₂×C₄×D₁₃	—	2	1	no	—
98	C₂×C₄×D₁₃	—	2	1	no	—
99	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×D₁₃	C₂⁴	1	1	yes	characteristic
101	C₂×Dic₂₆	#41	1	1	yes	characteristic
102	C₂²⋊#261	—	2	4	no	—
103	#6122	—	2	4	no	—
104	C₄⋊#261	—	2	4	no	—
105	C₄×#261	—	2	4	no	—
106	#6118	—	2	4	no	—
107	D₁₃⋊C₈	—	2	4	no	—
108	C₁₃×M₁₆	—	2	2	no	—
109	C₂×C₈×C₁₃	—	2	2	no	—
110	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
114	C₁₃×C₄⋊C₄	—	2	1	no	—
115	C₁₃×C₂²⋊C₄	—	2	1	no	—
116	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
118	#6103	—	2	1	no	—
119	#6103	—	2	1	no	—
120	#6101	—	2	1	no	—
121	#6101	—	2	1	no	—
122	C₄×Dic₁₃	—	2	1	no	—
123	C₄×Dic₁₃	—	2	1	no	—
124	C₄×Dic₁₃	—	2	1	no	—
125	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
126	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
127	#6099	—	2	2	no	—
128	C₂×#1055	—	2	1	no	—
129	C₂×#1055	—	2	1	no	—
130	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
131	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
132	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
133	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
134	#6094	—	2	2	no	—
135	#6094	—	2	2	no	—
136	#6094	—	2	2	no	—
137	#6094	—	2	2	no	—
138	C₈×D₁₃	—	2	2	no	—
139	C₈×D₁₃	—	2	1	no	—
140	C₈×D₁₃	—	2	1	no	—
141	C₈×D₁₃	#41	1	4	yes	—
142	C₈×D₁₃	#41	1	2	yes	—
143	C₈×D₁₃	#41	1	2	yes	—
144	#2875	—	13	1	no	—
145	#1841	—	13	2	no	—
146	#1833	—	13	2	no	—
147	#1827	—	13	2	no	—
148	#1813	—	13	1	no	—
149	#1812	—	13	1	no	—
150	#1773	—	13	2	no	—
151	#1757	—	13	2	no	—
152	#1747	—	13	2	no	—
153	C₂³×C₁₃	#117	1	1	yes	characteristic
154	C₂²×D₁₃	#117	1	1	yes	characteristic
155	C₂²×D₁₃	C₂×#41	1	1	yes	characteristic
156	C₂×#261	—	4	4	no	—
157	C₂×C₄×C₁₃	—	2	1	no	—
158	C₂×C₄×C₁₃	—	2	1	no	—
159	C₂×C₄×C₁₃	—	2	1	no	—
160	C₂×C₄×C₁₃	—	2	1	no	—
161	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
162	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
163	C₂²⋊D₁₃	—	2	2	no	—
164	C₂²⋊D₁₃	—	2	2	no	—
165	C₂×Dic₁₃	—	2	1	no	—
166	C₂×Dic₁₃	—	2	1	no	—
167	C₂×Dic₁₃	—	8	1	no	—
168	C₂×Dic₁₃	#117	1	1	yes	characteristic
169	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
170	D₅₂	—	2	2	no	—
171	C₄×D₁₃	—	2	1	no	—
172	C₄×D₁₃	—	2	1	no	—
173	C₄×D₁₃	—	8	1	no	—
174	C₄×D₁₃	—	8	1	no	—
175	C₄×D₁₃	C₂×#41	1	2	yes	—
176	C₄×D₁₃	C₂×#41	1	2	yes	—
177	Dic₂₆	—	2	2	no	—
178	#1057	—	4	4	no	—
179	C₈×C₁₃	—	2	1	no	—
180	C₈×C₁₃	—	2	1	no	—
181	C₈×C₁₃	#117	1	2	yes	—
182	C₈×C₁₃	#117	1	2	yes	—
183	#1055	—	2	1	no	—
184	#1055	—	2	1	no	—
185	#1055	#117	1	1	yes	characteristic
186	#1055	#117	1	1	yes	characteristic
187	#1055	#117	1	2	yes	—
188	C₂×#131	—	13	1	no	—
189	C₂×#131	—	13	1	no	—
190	C₄×#41	—	13	1	no	—
191	C₂×#108	—	13	2	no	—
192	#427	—	13	2	no	—
193	#413	—	13	2	no	—
194	#410	—	13	2	no	—
195	#405	—	13	1	no	—
196	#405	—	13	1	no	—
197	#405	—	13	1	no	—
198	#405	—	13	1	no	—
199	#405	—	13	1	no	—
200	#405	—	13	4	no	—
201	#405	#261	1	1	yes	characteristic, complemented
202	C₄×M₁₆	—	13	1	no	—
203	C₄×M₁₆	—	13	1	no	—
204	C₄×M₁₆	—	13	1	no	—
205	C₂×#97	—	13	1	no	—
206	C₂×#97	—	13	1	no	—
207	C₂×#97	—	13	1	no	—
208	#344	—	13	4	no	—
209	#339	—	13	4	no	—
210	#338	—	13	2	no	—
211	H(3,ℤ/4)	—	13	2	no	—
212	H(3,ℤ/4)	—	13	4	no	—
213	C₂²×C₁₃	—	2	1	no	—
214	C₂²×C₁₃	—	4	1	no	—
215	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
216	C₂×D₁₃	—	2	1	no	—
217	C₂×D₁₃	—	4	1	no	—
218	C₂×D₁₃	—	4	1	no	—
219	C₂×D₁₃	#529	1	2	yes	—
220	#261	—	8	4	no	—
221	C₄×C₁₃	—	2	1	no	—
222	C₄×C₁₃	—	4	1	no	—
223	C₄×C₁₃	—	4	1	no	—
224	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
225	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
226	Dic₁₃	—	2	1	no	—
227	Dic₁₃	—	4	1	no	—
228	Dic₁₃	—	4	1	no	—
229	Dic₁₃	#529	1	2	yes	—
230	C₂×#41	—	13	1	no	—
231	#131	—	26	4	no	—
232	#131	—	26	2	no	—
233	#131	—	26	2	no	—
234	C₂×M₁₆	—	13	1	no	—
235	C₂×M₁₆	—	13	1	no	—
236	C₂×M₁₆	—	13	1	no	—
237	C₂×M₁₆	—	13	1	no	—
238	C₂×M₁₆	—	13	1	no	—
239	C₂×M₁₆	—	13	4	no	—
240	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
241	C₂²×C₈	—	13	1	no	—
242	C₂²×C₈	—	13	1	no	—
243	C₂²×C₈	—	13	1	no	—
244	C₂²×C₈	—	13	1	no	—
245	C₂²×C₈	—	13	1	no	—
246	C₂²×C₈	C₂×#261	1	1	yes	characteristic
247	C₄×Q₈	—	26	1	no	—
248	C₄×D₄	—	26	1	no	—
249	C₄×D₄	—	26	1	no	—
250	C₄×D₄	—	26	1	no	—
251	#117	—	13	1	no	—
252	#117	—	13	1	no	—
253	#117	—	13	4	no	—
254	#117	C₂×#261	1	1	yes	characteristic
255	C₂×C₄²	—	13	1	no	—
256	C₂×C₄²	—	13	1	no	—
257	C₂×C₄²	—	13	1	no	—
258	#108	—	26	2	no	—
259	#108	—	26	2	no	—
260	#107	—	26	2	no	—
261	#106	—	26	2	no	—
262	#101	—	26	2	no	—
263	#100	—	26	2	no	—
264	#99	—	26	2	no	—
265	#99	—	26	2	no	—
266	#97	—	13	1	no	—
267	#97	—	13	1	no	—
268	#97	—	13	1	no	—
269	#97	—	13	1	no	—
270	#97	—	13	1	no	—
271	#97	—	13	1	no	—
272	#97	—	13	1	no	—
273	#97	—	13	1	no	—
274	#97	—	26	4	no	—
275	#97	C₂×#261	1	1	yes	characteristic
276	#97	C₂×#261	1	1	yes	characteristic
277	C₄×C₈	—	13	1	no	—
278	C₄×C₈	—	13	1	no	—
279	C₄×C₈	—	13	1	no	—
280	C₄×C₈	—	13	1	no	—
281	C₄×C₈	—	26	4	no	—
282	C₄×C₈	C₂×#261	1	1	yes	characteristic
283	C₄×C₈	C₂×#261	1	1	yes	characteristic
284	C₂×C₁₃	—	2	1	no	—
285	C₂×C₁₃	—	4	1	no	—
286	C₂×C₁₃	#2283	1	1	yes	characteristic
287	D₁₃	—	2	2	no	—
288	D₁₃	—	4	1	no	—
289	#41	—	26	2	no	—
290	#41	—	26	2	no	—
291	C₂×Q₈	—	13	1	no	—
292	C₂×D₄	—	13	1	no	—
293	C₂×D₄	—	13	1	no	—
294	C₂×D₄	—	13	1	no	—
295	C₂²×C₄	—	13	1	no	—
296	C₂²×C₄	—	13	1	no	—
297	C₂²×C₄	—	26	1	no	—
298	C₂²×C₄	—	26	1	no	—
299	C₂²×C₄	—	26	1	no	—
300	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
301	M₁₆	—	2	2	no	—
302	M₁₆	—	26	2	no	—
303	M₁₆	—	26	2	no	—
304	M₁₆	—	26	2	no	—
305	M₁₆	—	26	2	no	—
306	M₁₆	—	26	2	no	—
307	M₁₆	—	26	4	no	—
308	M₁₆	—	26	4	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	13	1	no	—
311	C₂×C₈	—	13	1	no	—
312	C₂×C₈	—	13	4	no	—
313	C₂×C₈	—	13	2	no	—
314	C₂×C₈	—	13	2	no	—
315	C₂×C₈	—	13	1	no	—
316	C₂×C₈	—	13	1	no	—
317	C₂×C₈	—	26	1	no	—
318	C₂×C₈	—	26	1	no	—
319	C₂×C₈	—	26	1	no	—
320	C₂×C₈	—	26	1	no	—
321	C₂×C₈	—	26	2	no	—
322	C₂×C₈	—	26	4	no	—
323	C₂×C₈	C₂²×#261	1	1	yes	characteristic
324	C₂×C₈	C₂²×#261	1	1	yes	characteristic
325	C₂×C₈	C₂²×#261	1	1	yes	characteristic
326	C₂×C₈	C₂²×#261	1	1	yes	characteristic
327	C₄⋊C₄	—	2	1	no	—
328	C₄⋊C₄	—	26	1	no	—
329	C₄⋊C₄	—	26	1	no	—
330	C₄⋊C₄	—	26	4	no	—
331	C₂²⋊C₄	—	2	1	no	—
332	C₂²⋊C₄	—	26	1	no	—
333	C₂²⋊C₄	—	26	1	no	—
334	C₂²⋊C₄	—	26	4	no	—
335	C₄²	—	13	1	no	—
336	C₄²	—	13	1	no	—
337	C₄²	—	26	1	no	—
338	C₄²	—	26	1	no	—
339	C₄²	—	26	4	no	—
340	C₄²	—	26	1	no	—
341	C₄²	C₂²×#261	1	1	yes	characteristic
342	C₄²	C₂²×#261	1	1	yes	characteristic
343	C₁₃	#20788	1	1	yes	characteristic, complemented, Sylow
344	C₂³	—	13	1	no	—
345	C₂³	—	13	1	no	—
346	C₂³	#88237	1	1	yes	characteristic
347	Q₈	—	26	2	no	—
348	D₄	—	26	2	no	—
349	D₄	—	26	2	no	—
350	D₄	—	26	2	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	13	1	no	—
356	C₂×C₄	—	13	2	no	—
357	C₂×C₄	—	13	2	no	—
358	C₂×C₄	—	13	1	no	—
359	C₂×C₄	—	26	1	no	—
360	C₂×C₄	—	26	1	no	—
361	C₂×C₄	—	26	1	no	—
362	C₂×C₄	—	26	1	no	—
363	C₂×C₄	—	52	4	no	—
364	C₂×C₄	—	104	1	no	—
365	C₂×C₄	—	104	1	no	—
366	C₂×C₄	—	104	1	no	—
367	C₂×C₄	#88237	1	1	yes	characteristic
368	C₂×C₄	C₂³×#261	1	1	yes	characteristic
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	13	1	no	—
372	C₈	—	13	1	no	—
373	C₈	—	13	2	no	—
374	C₈	—	26	1	no	—
375	C₈	—	26	1	no	—
376	C₈	—	52	4	no	—
377	C₈	#88237	1	2	yes	—
378	C₈	#88237	1	2	yes	—
379	C₂²	—	2	1	no	—
380	C₂²	—	4	1	no	—
381	C₂²	—	13	2	no	—
382	C₂²	—	26	1	no	—
383	C₂²	—	52	1	no	—
384	C₂²	—	52	1	no	—
385	C₂²	C₂×#88237	1	1	yes	characteristic
386	C₄	—	2	1	no	—
387	C₄	—	4	1	no	—
388	C₄	—	4	1	no	—
389	C₄	—	13	2	no	—
390	C₄	—	26	1	no	—
391	C₄	—	52	1	no	—
392	C₄	—	52	1	no	—
393	C₄	—	104	4	no	—
394	C₄	C₂×#88237	1	1	yes	characteristic
395	C₄	C₂×#88237	1	1	yes	characteristic
396	C₂	—	2	1	no	—
397	C₂	—	4	1	no	—
398	C₂	—	26	2	no	—
399	C₂	—	52	1	no	—
400	C₂	#13131283	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#88237
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₁₃×#405	C₄
socle	C₂×C₁₃	#2283
prime radical	C₂×#6127	C₂³
cosocle	C₂×C₄×D₁₃	C₂⁴
hypercenter	#405	#261
hypocenter	C₁₃	#20788
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ #97 ◁ #405

Complements

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

N	H ≅ G/N	type
1. #13130766	C₂	semidirect
2. #13130755	C₂	semidirect
6. #13130642	C₂	semidirect
8. #13130558	C₂	semidirect
18. C₁₃×#405	C₄	semidirect
19. M₁₆×Dic₁₃	C₄	semidirect
20. C₂×#88055	C₄	semidirect
21. #11718658	C₄	semidirect
22. #11718369	C₄	semidirect
24. D₁₃×#97	C₄	semidirect
26. #11718350	C₄	semidirect
28. C₄×#6094	C₄	semidirect
201. #405	#261	semidirect
343. C₁₃	#20788	semidirect

Maximal subgroups

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

subgroup	index
#13130766	2
#13130755	2
#13130746	2
#13130721	2
#13130719	2
#13130642	2
#13130591	2
#13130558	2
#13123263	2
#20788	13