Subgroups — #16798248

Order 3328.

616 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13130767	C₂	1	2	yes	maximal, complemented
2	#13130755	C₂	1	2	yes	maximal, complemented
3	#13130746	C₂	1	2	yes	maximal
4	#13130720	C₂	1	1	yes	maximal, characteristic
5	#13130718	C₂	1	1	yes	maximal, characteristic
6	#13130638	C₂	1	2	yes	maximal, complemented
7	#13130592	C₂	1	2	yes	maximal
8	#13130558	C₂	1	2	yes	maximal, complemented
9	#13123258	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×C₈×Dic₁₃	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	#11718368	C₄	1	1	yes	characteristic, complemented
25	#11718368	C₂²	1	1	yes	characteristic
26	#11718350	C₄	1	1	yes	characteristic, complemented
27	#11718350	C₂²	1	1	yes	characteristic
28	C₄×C₈×D₁₃	C₄	1	1	yes	characteristic, complemented
29	C₄×C₈×D₁₃	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	2	no	—
47	#88155	—	2	2	no	—
48	C₂×#6094	C₂×C₄	1	1	yes	characteristic
49	C₂×#6094	C₂×C₄	1	1	yes	characteristic
50	C₂×#6094	C₂³	1	1	yes	characteristic
51	C₂×#6094	C₂³	1	1	yes	characteristic
52	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
55	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
56	#88147	—	2	1	no	—
57	#88141	—	2	1	no	—
58	#88135	—	2	1	no	—
59	#88135	—	2	1	no	—
60	#88126	C₂×C₄	1	1	yes	characteristic
61	#88126	C₂³	1	1	yes	characteristic
62	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
63	C₄²×D₁₃	C₂³	1	1	yes	characteristic
64	#88121	—	2	2	no	—
65	#88119	—	2	2	no	—
66	#88113	—	2	2	no	—
67	#88107	—	2	2	no	—
68	#88104	—	2	2	no	—
69	#88103	—	2	2	no	—
70	#88102	—	2	2	no	—
71	#88101	—	2	2	no	—
72	#88100	—	2	4	no	—
73	C₈×#261	—	2	4	no	—
74	C₁₃×#97	C₂×C₄	1	1	yes	characteristic
75	C₁₃×#97	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
78	#88055	C₂×C₄	1	1	yes	characteristic
79	#88055	C₂×C₄	1	1	yes	characteristic
80	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
82	C₈×Dic₁₃	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	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
86	#88043	C₂×C₄	1	1	yes	characteristic
87	#88043	C₂×C₄	1	1	yes	characteristic
88	C₄×#1055	C₂×C₄	1	1	yes	characteristic
89	C₄×#1055	C₂×C₄	1	1	yes	characteristic
90	#20787	—	13	1	no	maximal, Sylow
91	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₂²⋊D₁₃	#41	1	1	yes	characteristic
93	C₂×C₂²⋊D₁₃	#41	1	1	yes	characteristic
94	C₂²×Dic₁₃	—	2	1	no	—
95	#6127	—	2	2	no	—
96	#6127	—	2	2	no	—
97	C₂×D₅₂	#41	1	1	yes	characteristic
98	C₂×C₄×D₁₃	—	2	1	no	—
99	C₂×C₄×D₁₃	—	2	1	no	—
100	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×D₁₃	C₂⁴	1	1	yes	characteristic
102	C₂×Dic₂₆	#41	1	1	yes	characteristic
103	C₂²⋊#261	—	2	4	no	—
104	#6122	—	2	4	no	—
105	C₄⋊#261	—	2	4	no	—
106	C₄×#261	—	2	4	no	—
107	#6118	—	2	4	no	—
108	D₁₃⋊C₈	—	2	4	no	—
109	C₁₃×M₁₆	—	2	2	no	—
110	C₂×C₈×C₁₃	—	2	1	no	—
111	C₂×C₈×C₁₃	—	2	1	no	—
112	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
116	C₁₃×C₄⋊C₄	—	2	1	no	—
117	C₁₃×C₂²⋊C₄	—	2	1	no	—
118	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
119	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
120	#6103	—	2	1	no	—
121	#6103	—	2	1	no	—
122	#6101	—	2	1	no	—
123	#6101	—	2	1	no	—
124	C₄×Dic₁₃	—	2	1	no	—
125	C₄×Dic₁₃	—	2	1	no	—
126	C₄×Dic₁₃	—	2	1	no	—
127	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
128	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
129	#6099	—	2	2	no	—
130	C₂×#1055	—	2	1	no	—
131	C₂×#1055	—	2	1	no	—
132	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
133	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
134	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
135	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
136	#6094	—	2	2	no	—
137	#6094	—	2	2	no	—
138	#6094	—	2	2	no	—
139	#6094	—	2	2	no	—
140	C₈×D₁₃	—	2	1	no	—
141	C₈×D₁₃	—	2	1	no	—
142	C₈×D₁₃	—	2	1	no	—
143	C₈×D₁₃	—	2	1	no	—
144	C₈×D₁₃	#41	1	2	yes	—
145	C₈×D₁₃	#41	1	2	yes	—
146	C₈×D₁₃	#41	1	2	yes	—
147	C₈×D₁₃	#41	1	2	yes	—
148	C₄×#131	—	13	1	no	—
149	#1841	—	13	2	no	—
150	#1833	—	13	2	no	—
151	#1827	—	13	2	no	—
152	#1813	—	13	1	no	—
153	#1812	—	13	1	no	—
154	#1772	—	13	2	no	—
155	#1757	—	13	2	no	—
156	#1747	—	13	2	no	—
157	C₂³×C₁₃	#117	1	1	yes	characteristic
158	C₂²×D₁₃	#117	1	1	yes	characteristic
159	C₂²×D₁₃	C₂×#41	1	1	yes	characteristic
160	C₂×#261	—	4	4	no	—
161	C₂×C₄×C₁₃	—	2	1	no	—
162	C₂×C₄×C₁₃	—	2	1	no	—
163	C₂×C₄×C₁₃	—	2	1	no	—
164	C₂×C₄×C₁₃	—	2	1	no	—
165	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
166	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
167	C₂²⋊D₁₃	—	2	2	no	—
168	C₂²⋊D₁₃	—	2	2	no	—
169	C₂×Dic₁₃	—	2	1	no	—
170	C₂×Dic₁₃	—	2	1	no	—
171	C₂×Dic₁₃	—	4	2	no	—
172	C₂×Dic₁₃	#117	1	1	yes	characteristic
173	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
174	D₅₂	—	2	2	no	—
175	C₄×D₁₃	—	2	1	no	—
176	C₄×D₁₃	—	2	1	no	—
177	C₄×D₁₃	—	4	2	no	—
178	C₄×D₁₃	—	4	2	no	—
179	C₄×D₁₃	C₂×#41	1	2	yes	—
180	C₄×D₁₃	C₂×#41	1	2	yes	—
181	Dic₂₆	—	2	2	no	—
182	#1057	—	4	4	no	—
183	C₈×C₁₃	—	2	1	no	—
184	C₈×C₁₃	—	2	1	no	—
185	C₈×C₁₃	#117	1	1	yes	characteristic
186	C₈×C₁₃	#117	1	1	yes	characteristic
187	C₈×C₁₃	#117	1	2	yes	—
188	#1055	—	2	1	no	—
189	#1055	—	2	1	no	—
190	#1055	#117	1	1	yes	characteristic
191	#1055	#117	1	1	yes	characteristic
192	#1055	#117	1	2	yes	—
193	C₂×#131	—	13	1	no	—
194	C₂×#131	—	13	1	no	—
195	C₄×#41	—	13	1	no	—
196	C₂×#108	—	13	2	no	—
197	#427	—	13	2	no	—
198	#413	—	13	2	no	—
199	#410	—	13	2	no	—
200	#405	—	13	1	no	—
201	#405	—	13	1	no	—
202	#405	—	13	1	no	—
203	#405	—	13	1	no	—
204	#405	—	13	1	no	—
205	#405	—	13	4	no	—
206	#405	#261	1	1	yes	characteristic, complemented
207	C₄×M₁₆	—	13	1	no	—
208	C₄×M₁₆	—	13	1	no	—
209	C₄×M₁₆	—	13	1	no	—
210	C₂×C₄×C₈	—	13	1	no	—
211	C₂×C₄×C₈	—	13	1	no	—
212	C₂×C₄×C₈	—	13	1	no	—
213	#344	—	13	4	no	—
214	#339	—	13	4	no	—
215	#338	—	13	2	no	—
216	H(3,ℤ/4)	—	13	2	no	—
217	H(3,ℤ/4)	—	13	4	no	—
218	C₂²×C₁₃	—	2	1	no	—
219	C₂²×C₁₃	—	4	1	no	—
220	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
221	C₂×D₁₃	—	2	1	no	—
222	C₂×D₁₃	—	4	1	no	—
223	C₂×D₁₃	—	4	1	no	—
224	C₂×D₁₃	#529	1	2	yes	—
225	#261	—	8	4	no	—
226	C₄×C₁₃	—	2	1	no	—
227	C₄×C₁₃	—	4	1	no	—
228	C₄×C₁₃	—	4	1	no	—
229	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
230	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
231	Dic₁₃	—	2	2	no	—
232	Dic₁₃	—	2	1	no	—
233	Dic₁₃	—	4	1	no	—
234	Dic₁₃	#529	1	2	yes	—
235	C₂×#41	—	13	1	no	—
236	#131	—	26	2	no	—
237	#131	—	26	2	no	—
238	#131	—	26	2	no	—
239	#131	—	26	2	no	—
240	C₂×M₁₆	—	13	1	no	—
241	C₂×M₁₆	—	13	1	no	—
242	C₂×M₁₆	—	13	1	no	—
243	C₂×M₁₆	—	13	1	no	—
244	C₂×M₁₆	—	13	1	no	—
245	C₂×M₁₆	—	13	4	no	—
246	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
247	C₂²×C₈	—	13	1	no	—
248	C₂²×C₈	—	13	1	no	—
249	C₂²×C₈	—	13	1	no	—
250	C₂²×C₈	—	13	1	no	—
251	C₂²×C₈	—	13	1	no	—
252	C₂²×C₈	C₂×#261	1	1	yes	characteristic
253	C₄×Q₈	—	26	1	no	—
254	C₄×D₄	—	26	1	no	—
255	C₄×D₄	—	26	1	no	—
256	C₄×D₄	—	26	1	no	—
257	#117	—	13	1	no	—
258	#117	—	13	1	no	—
259	#117	—	13	4	no	—
260	#117	C₂×#261	1	1	yes	characteristic
261	C₂×C₄²	—	13	1	no	—
262	C₂×C₄²	—	13	1	no	—
263	C₂×C₄²	—	13	1	no	—
264	#108	—	26	2	no	—
265	#108	—	26	2	no	—
266	#107	—	26	2	no	—
267	#106	—	26	2	no	—
268	#101	—	26	2	no	—
269	#100	—	26	2	no	—
270	#99	—	26	2	no	—
271	#99	—	26	2	no	—
272	#97	—	13	1	no	—
273	#97	—	13	1	no	—
274	#97	—	13	1	no	—
275	#97	—	13	1	no	—
276	#97	—	26	4	no	—
277	#97	C₂×#261	1	1	yes	characteristic
278	#97	C₂×#261	1	1	yes	characteristic
279	C₄×C₈	—	13	1	no	—
280	C₄×C₈	—	13	1	no	—
281	C₄×C₈	—	13	1	no	—
282	C₄×C₈	—	13	1	no	—
283	C₄×C₈	—	13	1	no	—
284	C₄×C₈	—	13	1	no	—
285	C₄×C₈	—	13	1	no	—
286	C₄×C₈	—	13	1	no	—
287	C₄×C₈	—	26	4	no	—
288	C₄×C₈	C₂×#261	1	1	yes	characteristic
289	C₄×C₈	C₂×#261	1	1	yes	characteristic
290	C₂×C₁₃	—	2	1	no	—
291	C₂×C₁₃	—	4	1	no	—
292	C₂×C₁₃	#2283	1	1	yes	characteristic
293	D₁₃	—	2	2	no	—
294	D₁₃	—	4	1	no	—
295	#41	—	26	2	no	—
296	#41	—	26	2	no	—
297	C₂×Q₈	—	13	1	no	—
298	C₂×D₄	—	13	1	no	—
299	C₂×D₄	—	13	1	no	—
300	C₂×D₄	—	13	1	no	—
301	C₂²×C₄	—	13	1	no	—
302	C₂²×C₄	—	13	1	no	—
303	C₂²×C₄	—	26	1	no	—
304	C₂²×C₄	—	26	1	no	—
305	C₂²×C₄	—	26	1	no	—
306	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
307	M₁₆	—	2	2	no	—
308	M₁₆	—	26	2	no	—
309	M₁₆	—	26	2	no	—
310	M₁₆	—	26	2	no	—
311	M₁₆	—	26	2	no	—
312	M₁₆	—	26	2	no	—
313	M₁₆	—	26	4	no	—
314	M₁₆	—	26	4	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	13	1	no	—
318	C₂×C₈	—	13	1	no	—
319	C₂×C₈	—	13	2	no	—
320	C₂×C₈	—	13	2	no	—
321	C₂×C₈	—	13	2	no	—
322	C₂×C₈	—	13	2	no	—
323	C₂×C₈	—	13	1	no	—
324	C₂×C₈	—	13	1	no	—
325	C₂×C₈	—	26	1	no	—
326	C₂×C₈	—	26	1	no	—
327	C₂×C₈	—	26	1	no	—
328	C₂×C₈	—	26	1	no	—
329	C₂×C₈	—	26	1	no	—
330	C₂×C₈	—	26	1	no	—
331	C₂×C₈	—	26	4	no	—
332	C₂×C₈	C₂²×#261	1	1	yes	characteristic
333	C₂×C₈	C₂²×#261	1	1	yes	characteristic
334	C₂×C₈	C₂²×#261	1	1	yes	characteristic
335	C₂×C₈	C₂²×#261	1	1	yes	characteristic
336	C₄⋊C₄	—	2	1	no	—
337	C₄⋊C₄	—	26	1	no	—
338	C₄⋊C₄	—	26	1	no	—
339	C₄⋊C₄	—	26	4	no	—
340	C₂²⋊C₄	—	2	1	no	—
341	C₂²⋊C₄	—	26	1	no	—
342	C₂²⋊C₄	—	26	1	no	—
343	C₂²⋊C₄	—	26	4	no	—
344	C₄²	—	13	1	no	—
345	C₄²	—	13	1	no	—
346	C₄²	—	26	1	no	—
347	C₄²	—	26	1	no	—
348	C₄²	—	26	4	no	—
349	C₄²	—	26	1	no	—
350	C₄²	C₂²×#261	1	1	yes	characteristic
351	C₄²	C₂²×#261	1	1	yes	characteristic
352	C₁₃	#20787	1	1	yes	characteristic, complemented, Sylow
353	C₂³	—	13	1	no	—
354	C₂³	—	13	1	no	—
355	C₂³	#88237	1	1	yes	characteristic
356	Q₈	—	26	2	no	—
357	D₄	—	26	2	no	—
358	D₄	—	26	2	no	—
359	D₄	—	26	2	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	13	1	no	—
365	C₂×C₄	—	13	2	no	—
366	C₂×C₄	—	13	2	no	—
367	C₂×C₄	—	13	1	no	—
368	C₂×C₄	—	26	1	no	—
369	C₂×C₄	—	26	1	no	—
370	C₂×C₄	—	26	1	no	—
371	C₂×C₄	—	26	1	no	—
372	C₂×C₄	—	52	2	no	—
373	C₂×C₄	—	52	2	no	—
374	C₂×C₄	—	52	2	no	—
375	C₂×C₄	—	52	4	no	—
376	C₂×C₄	#88237	1	1	yes	characteristic
377	C₂×C₄	C₂³×#261	1	1	yes	characteristic
378	C₈	—	2	1	no	—
379	C₈	—	2	1	no	—
380	C₈	—	13	2	no	—
381	C₈	—	13	1	no	—
382	C₈	—	13	1	no	—
383	C₈	—	26	1	no	—
384	C₈	—	26	1	no	—
385	C₈	—	52	4	no	—
386	C₈	#88237	1	1	yes	characteristic
387	C₈	#88237	1	1	yes	characteristic
388	C₈	#88237	1	2	yes	—
389	C₂²	—	2	1	no	—
390	C₂²	—	4	1	no	—
391	C₂²	—	13	2	no	—
392	C₂²	—	26	1	no	—
393	C₂²	—	52	1	no	—
394	C₂²	—	52	1	no	—
395	C₂²	C₂×#88237	1	1	yes	characteristic
396	C₄	—	2	1	no	—
397	C₄	—	4	1	no	—
398	C₄	—	4	1	no	—
399	C₄	—	13	2	no	—
400	C₄	—	26	2	no	—
401	C₄	—	26	1	no	—
402	C₄	—	52	1	no	—
403	C₄	—	104	4	no	—
404	C₄	C₂×#88237	1	1	yes	characteristic
405	C₄	C₂×#88237	1	1	yes	characteristic
406	C₂	—	2	1	no	—
407	C₂	—	4	1	no	—
408	C₂	—	26	2	no	—
409	C₂	—	52	1	no	—
410	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	#20787
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. #13130767	C₂	semidirect
2. #13130755	C₂	semidirect
6. #13130638	C₂	semidirect
8. #13130558	C₂	semidirect
18. C₁₃×#405	C₄	semidirect
19. M₁₆×Dic₁₃	C₄	semidirect
20. C₂×C₈×Dic₁₃	C₄	semidirect
21. #11718658	C₄	semidirect
22. #11718369	C₄	semidirect
24. #11718368	C₄	semidirect
26. #11718350	C₄	semidirect
28. C₄×C₈×D₁₃	C₄	semidirect
206. #405	#261	semidirect
352. C₁₃	#20787	semidirect

Maximal subgroups

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

subgroup	index
#13130767	2
#13130755	2
#13130746	2
#13130720	2
#13130718	2
#13130638	2
#13130592	2
#13130558	2
#13123258	2
#20787	13