Subgroups — #16798199

Order 3328.

684 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13130767	C₂	1	4	yes	maximal, complemented
2	#13130766	C₂	1	4	yes	maximal, complemented
3	#13130546	C₂	1	2	yes	maximal
4	#13130517	C₂	1	2	yes	maximal
5	C₂×#11718269	C₂	1	1	yes	maximal, characteristic
6	C₂×#11718267	C₂	1	1	yes	maximal, characteristic
7	#13123252	C₂	1	1	yes	maximal, characteristic
8	C₂×#88237	C₂²	1	2	yes	—
9	C₂×D₁₃⋊M₁₆	C₂²	1	2	yes	—
10	C₂×M₁₆×D₁₃	C₂²	1	1	yes	characteristic
11	C₂²×#6094	C₂²	1	1	yes	characteristic
12	D₁₃×#117	C₂²	1	1	yes	characteristic
13	C₁₃×#405	C₄	1	1	yes	characteristic, complemented
14	#11718783	C₄	1	1	yes	characteristic, complemented
15	#11718762	C₄	1	1	yes	characteristic, complemented
16	#11718658	C₄	1	1	yes	characteristic, complemented
17	#11718369	C₄	1	2	yes	complemented
18	#11718368	C₂²	1	2	yes	—
19	#11718350	C₄	1	2	yes	complemented
20	C₄×#6094	C₂²	1	2	yes	—
21	#11718269	C₂²	1	2	yes	—
22	#11718269	C₂²	1	2	yes	—
23	#11718267	C₂²	1	2	yes	—
24	#11718267	C₂²	1	2	yes	—
25	#11718266	C₂²	1	8	yes	—
26	#11718234	C₂²	1	8	yes	—
27	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₂²⋊#261	—	2	2	no	—
29	C₂×#6122	—	2	2	no	—
30	#88237	—	2	2	no	—
31	#88237	—	2	2	no	—
32	#88237	—	2	2	no	—
33	#88237	C₂³	1	4	yes	—
34	C₂×C₄⋊#261	—	2	2	no	—
35	C₂×C₄×#261	—	2	2	no	—
36	D₁₃⋊M₁₆	—	2	2	no	—
37	D₁₃⋊M₁₆	—	2	2	no	—
38	D₁₃⋊M₁₆	—	2	2	no	—
39	D₁₃⋊M₁₆	C₂³	1	4	yes	—
40	C₂×#6118	—	2	2	no	—
41	C₂×D₁₃⋊C₈	—	2	2	no	—
42	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
45	#88180	C₂×C₄	1	1	yes	characteristic
46	C₂×#6099	C₂×C₄	1	1	yes	characteristic
47	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
48	M₁₆×D₁₃	—	2	2	no	—
49	M₁₆×D₁₃	—	2	2	no	—
50	C₂×#6094	—	2	2	no	—
51	C₂×#6094	—	2	2	no	—
52	C₂×#6094	C₂×C₄	1	1	yes	characteristic
53	C₂×#6094	C₂×C₄	1	1	yes	characteristic
54	C₂×#6094	C₂³	1	1	yes	characteristic
55	C₂×#6094	C₂³	1	1	yes	characteristic
56	C₂×#6094	C₂³	1	2	yes	—
57	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
58	#88146	—	2	1	no	—
59	C₄⋊C₄×D₁₃	—	2	1	no	—
60	C₂²⋊C₄×D₁₃	—	2	1	no	—
61	#88131	—	2	1	no	—
62	#88126	C₂×C₄	1	2	yes	—
63	C₄²×D₁₃	C₂³	1	2	yes	—
64	C₁₃×#97	C₂×C₄	1	2	yes	complemented
65	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	complemented
66	#88055	C₂×C₄	1	2	yes	complemented
67	#88055	C₂×C₄	1	2	yes	complemented
68	C₈×Dic₁₃	C₂×C₄	1	2	yes	complemented
69	C₈×Dic₁₃	C₂×C₄	1	2	yes	complemented
70	#88043	C₂×C₄	1	2	yes	complemented
71	C₄×#1055	C₂×C₄	1	2	yes	complemented
72	#20776	—	13	1	no	maximal, Sylow
73	C₂³×D₁₃	#41	1	1	yes	characteristic
74	C₂²×#261	—	4	2	no	—
75	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×Dic₁₃	#41	1	1	yes	characteristic
77	C₂×C₄×D₁₃	—	2	2	no	—
78	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×D₁₃	#41	1	2	yes	—
80	C₂×C₄×D₁₃	#41	1	2	yes	—
81	C₂×C₄×D₁₃	#41	1	2	yes	—
82	C₂×C₄×D₁₃	#41	1	2	yes	—
83	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
84	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
85	C₂×C₄×D₁₃	C₂⁴	1	1	yes	characteristic
86	C₂²⋊#261	—	2	4	no	—
87	C₂²⋊#261	—	4	2	no	—
88	#6122	—	2	4	no	—
89	#6122	—	4	2	no	—
90	C₂×#1057	—	4	2	no	—
91	C₄⋊#261	—	2	4	no	—
92	C₄⋊#261	—	4	2	no	—
93	C₄×#261	—	2	4	no	—
94	C₄×#261	—	4	2	no	—
95	#6118	—	2	4	no	—
96	#6118	—	4	2	no	—
97	D₁₃⋊C₈	—	2	4	no	—
98	D₁₃⋊C₈	—	4	2	no	—
99	C₁₃×M₁₆	—	2	1	no	—
100	C₁₃×M₁₆	—	2	1	no	—
101	C₂×C₈×C₁₃	—	2	1	no	—
102	C₂×C₈×C₁₃	—	2	1	no	—
103	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₁₃×C₄⋊C₄	—	2	1	no	—
107	C₁₃×C₂²⋊C₄	—	2	1	no	—
108	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
109	C₂²⋊Dic₁₃	—	2	1	no	—
110	#6103	—	2	2	no	—
111	C₄⋊Dic₁₃	—	2	1	no	—
112	#6101	—	2	2	no	—
113	C₄×Dic₁₃	—	2	2	no	—
114	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
115	#6099	—	2	1	no	—
116	#6099	—	2	1	no	—
117	C₂×#1055	—	2	1	no	—
118	C₂×#1055	—	2	1	no	—
119	C₂×#1055	C₂²×C₄	1	2	yes	—
120	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
121	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
122	#6094	—	2	2	no	—
123	#6094	—	2	2	no	—
124	#6094	—	2	2	no	—
125	#6094	—	2	2	no	—
126	#6094	—	2	4	no	—
127	C₈×D₁₃	—	2	4	no	—
128	#2873	—	13	1	no	—
129	#1841	—	13	4	no	—
130	#1841	—	13	4	no	—
131	C₂×#344	—	13	2	no	—
132	#1733	—	13	2	no	—
133	C₂×#338	—	13	1	no	—
134	C₂×H(3,ℤ/4)	—	13	1	no	—
135	C₂³×C₁₃	#117	1	1	yes	characteristic
136	C₂²×D₁₃	—	8	1	no	—
137	C₂²×D₁₃	#117	1	1	yes	characteristic
138	C₂²×D₁₃	#126	1	2	yes	—
139	C₂²×D₁₃	#126	1	2	yes	—
140	C₂²×D₁₃	C₂×#41	1	1	yes	characteristic
141	C₂×#261	—	4	4	no	—
142	C₂×#261	—	8	4	no	—
143	C₂×C₄×C₁₃	—	2	2	no	—
144	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
145	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
146	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
147	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
148	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
150	C₂×Dic₁₃	—	2	2	no	—
151	C₂×Dic₁₃	#117	1	1	yes	characteristic
152	C₂×Dic₁₃	#126	1	2	yes	—
153	C₂×Dic₁₃	#126	1	2	yes	—
154	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
155	C₄×D₁₃	—	8	2	no	—
156	C₄×D₁₃	#117	1	2	yes	—
157	C₄×D₁₃	#117	1	2	yes	—
158	C₄×D₁₃	#126	1	2	yes	—
159	C₄×D₁₃	#126	1	2	yes	—
160	C₄×D₁₃	#126	1	2	yes	—
161	C₄×D₁₃	#126	1	2	yes	—
162	C₄×D₁₃	C₂×#41	1	2	yes	—
163	C₄×D₁₃	C₂×#41	1	2	yes	—
164	#1057	—	4	4	no	—
165	C₈×C₁₃	—	2	2	no	—
166	C₈×C₁₃	—	2	1	no	—
167	C₈×C₁₃	—	2	1	no	—
168	#1055	—	2	2	no	—
169	#1055	—	2	1	no	—
170	#1055	—	2	1	no	—
171	C₂²×M₁₆	—	13	1	no	—
172	C₂²×M₁₆	—	13	1	no	—
173	C₂²×M₁₆	—	13	2	no	—
174	C₂×#117	—	13	1	no	—
175	C₂×#117	—	13	2	no	—
176	#405	—	13	1	no	—
177	#405	—	13	1	no	—
178	#405	—	13	1	no	—
179	#405	—	13	2	no	—
180	#405	—	13	2	no	—
181	#405	#261	1	1	yes	characteristic, complemented
182	C₄×M₁₆	—	13	2	no	—
183	C₄×M₁₆	—	13	2	no	—
184	#344	—	13	8	no	—
185	#339	—	13	8	no	—
186	#338	—	13	2	no	—
187	#338	—	13	2	no	—
188	H(3,ℤ/4)	—	13	2	no	—
189	H(3,ℤ/4)	—	13	2	no	—
190	C₂²×C₁₃	—	4	1	no	—
191	C₂²×C₁₃	#383	1	1	yes	characteristic
192	C₂²×C₁₃	#383	1	1	yes	characteristic
193	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
194	C₂×D₁₃	—	4	1	no	—
195	C₂×D₁₃	—	4	2	no	—
196	C₂×D₁₃	—	4	1	no	—
197	C₂×D₁₃	—	4	2	no	—
198	C₂×D₁₃	#383	1	2	yes	—
199	C₂×D₁₃	C₂×#126	1	2	yes	—
200	#261	—	8	4	no	—
201	C₄×C₁₃	—	4	2	no	—
202	C₄×C₁₃	#383	1	1	yes	characteristic
203	C₄×C₁₃	#383	1	1	yes	characteristic
204	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
205	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
206	Dic₁₃	—	4	2	no	—
207	Dic₁₃	#383	1	2	yes	—
208	Dic₁₃	C₂×#126	1	2	yes	—
209	C₂³×C₄	—	13	1	no	—
210	C₂×M₁₆	—	13	1	no	—
211	C₂×M₁₆	—	13	1	no	—
212	C₂×M₁₆	—	13	1	no	—
213	C₂×M₁₆	—	13	2	no	—
214	C₂×M₁₆	—	13	1	no	—
215	C₂×M₁₆	—	13	1	no	—
216	C₂×M₁₆	—	13	4	no	—
217	C₂×M₁₆	—	26	2	no	—
218	C₂×M₁₆	—	26	2	no	—
219	C₂×M₁₆	—	26	2	no	—
220	C₂×M₁₆	—	26	2	no	—
221	C₂×M₁₆	—	26	2	no	—
222	C₂×M₁₆	—	26	2	no	—
223	C₂×M₁₆	—	26	2	no	—
224	C₂×M₁₆	—	26	2	no	—
225	C₂×M₁₆	—	26	2	no	—
226	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
227	C₂²×C₈	—	13	1	no	—
228	C₂²×C₈	—	13	2	no	—
229	C₂²×C₈	—	26	2	no	—
230	C₂²×C₈	C₂×#261	1	1	yes	characteristic
231	#117	—	13	1	no	—
232	#117	—	13	2	no	—
233	#117	—	13	4	no	—
234	#117	—	26	1	no	—
235	#117	—	26	1	no	—
236	#117	—	26	2	no	—
237	#117	—	26	2	no	—
238	#117	—	26	2	no	—
239	#117	C₂×#261	1	1	yes	characteristic
240	C₂×C₄⋊C₄	—	26	1	no	—
241	C₂×C₄⋊C₄	—	26	2	no	—
242	C₂×C₂²⋊C₄	—	26	1	no	—
243	C₂×C₂²⋊C₄	—	26	2	no	—
244	C₂×C₄²	—	13	2	no	—
245	C₂×C₄²	—	26	2	no	—
246	#97	—	13	2	no	—
247	#97	—	13	2	no	—
248	#97	—	13	2	no	—
249	#97	C₂×#261	1	2	yes	complemented
250	C₄×C₈	—	13	2	no	—
251	C₄×C₈	—	13	2	no	—
252	C₄×C₈	—	13	2	no	—
253	C₄×C₈	C₂×#261	1	2	yes	complemented
254	C₂×C₁₃	—	2	1	no	—
255	C₂×C₁₃	—	2	1	no	—
256	C₂×C₁₃	—	2	1	no	—
257	C₂×C₁₃	C₂×#383	1	1	yes	characteristic
258	D₁₃	—	2	2	no	—
259	D₁₃	—	2	2	no	—
260	C₂⁴	—	13	1	no	—
261	C₂²×C₄	—	13	1	no	—
262	C₂²×C₄	—	13	2	no	—
263	C₂²×C₄	—	13	1	no	—
264	C₂²×C₄	—	13	2	no	—
265	C₂²×C₄	—	13	2	no	—
266	C₂²×C₄	—	13	2	no	—
267	C₂²×C₄	—	13	1	no	—
268	C₂²×C₄	—	13	1	no	—
269	C₂²×C₄	—	13	1	no	—
270	C₂²×C₄	—	26	2	no	—
271	C₂²×C₄	—	52	2	no	—
272	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
273	M₁₆	—	2	1	no	—
274	M₁₆	—	2	1	no	—
275	M₁₆	—	26	2	no	—
276	M₁₆	—	26	2	no	—
277	M₁₆	—	26	2	no	—
278	M₁₆	—	26	2	no	—
279	M₁₆	—	26	4	no	—
280	M₁₆	—	26	1	no	—
281	M₁₆	—	26	1	no	—
282	M₁₆	—	26	4	no	—
283	M₁₆	—	26	4	no	—
284	M₁₆	—	52	2	no	—
285	M₁₆	—	52	2	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	13	1	no	—
289	C₂×C₈	—	13	1	no	—
290	C₂×C₈	—	13	2	no	—
291	C₂×C₈	—	26	4	no	—
292	C₂×C₈	—	26	1	no	—
293	C₂×C₈	—	26	1	no	—
294	C₂×C₈	—	26	4	no	—
295	C₂×C₈	—	52	2	no	—
296	C₂×C₈	—	52	2	no	—
297	C₂×C₈	C₂²×#261	1	1	yes	characteristic
298	C₂×C₈	C₂²×#261	1	2	yes	—
299	C₂×C₈	C₂²×#261	1	1	yes	characteristic
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	26	2	no	—
302	C₄⋊C₄	—	26	4	no	—
303	C₄⋊C₄	—	26	1	no	—
304	C₄⋊C₄	—	52	2	no	—
305	C₂²⋊C₄	—	2	1	no	—
306	C₂²⋊C₄	—	26	2	no	—
307	C₂²⋊C₄	—	26	1	no	—
308	C₂²⋊C₄	—	26	4	no	—
309	C₂²⋊C₄	—	52	2	no	—
310	C₄²	—	13	2	no	—
311	C₄²	—	26	2	no	—
312	C₄²	—	26	4	no	—
313	C₄²	—	52	2	no	—
314	C₄²	C₂²×#261	1	2	yes	—
315	C₁₃	#20776	1	1	yes	characteristic, complemented, Sylow
316	C₂³	—	13	2	no	—
317	C₂³	—	13	2	no	—
318	C₂³	—	13	1	no	—
319	C₂³	—	13	1	no	—
320	C₂³	—	104	1	no	—
321	C₂³	#88237	1	1	yes	characteristic
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	13	2	no	—
324	C₂×C₄	—	13	2	no	—
325	C₂×C₄	—	13	1	no	—
326	C₂×C₄	—	13	1	no	—
327	C₂×C₄	—	13	2	no	—
328	C₂×C₄	—	13	2	no	—
329	C₂×C₄	—	13	2	no	—
330	C₂×C₄	—	13	2	no	—
331	C₂×C₄	—	13	2	no	—
332	C₂×C₄	—	13	2	no	—
333	C₂×C₄	—	13	2	no	—
334	C₂×C₄	—	13	2	no	—
335	C₂×C₄	—	26	2	no	—
336	C₂×C₄	—	52	4	no	—
337	C₂×C₄	—	104	2	no	—
338	C₂×C₄	—	104	4	no	—
339	C₂×C₄	#88237	1	1	yes	characteristic
340	C₂×C₄	#88237	1	1	yes	characteristic
341	C₂×C₄	#88237	1	1	yes	characteristic
342	C₂×C₄	#88237	1	1	yes	characteristic
343	C₂×C₄	#88237	1	1	yes	characteristic
344	C₂×C₄	C₂³×#261	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	2	1	no	—
347	C₈	—	2	2	no	—
348	C₈	—	26	1	no	—
349	C₈	—	26	1	no	—
350	C₈	—	26	2	no	—
351	C₈	—	52	4	no	—
352	C₂²	—	4	1	no	—
353	C₂²	—	13	2	no	—
354	C₂²	—	13	2	no	—
355	C₂²	—	52	1	no	—
356	C₂²	—	52	2	no	—
357	C₂²	—	52	1	no	—
358	C₂²	—	52	2	no	—
359	C₂²	#11719061	1	1	yes	characteristic
360	C₂²	#11719061	1	1	yes	characteristic
361	C₂²	C₂×#88237	1	1	yes	characteristic
362	C₄	—	4	2	no	—
363	C₄	—	13	2	no	—
364	C₄	—	13	2	no	—
365	C₄	—	52	2	no	—
366	C₄	—	104	4	no	—
367	C₄	#11719061	1	1	yes	characteristic
368	C₄	#11719061	1	1	yes	characteristic
369	C₄	C₂×#88237	1	1	yes	characteristic
370	C₄	C₂×#88237	1	1	yes	characteristic
371	C₂	—	2	1	no	—
372	C₂	—	2	1	no	—
373	C₂	—	2	1	no	—
374	C₂	—	26	2	no	—
375	C₂	—	26	2	no	—
376	C₂	C₂×#11719061	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	C₂×#383
prime radical	C₂³×D₁₃	#41
cosocle	C₂×C₄×D₁₃	C₂⁴
hypercenter	#405	#261
hypocenter	C₁₃	#20776
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
1. #13130767	C₂	semidirect
2. #13130766	C₂	semidirect
13. C₁₃×#405	C₄	semidirect
14. #11718783	C₄	semidirect
15. #11718762	C₄	semidirect
16. #11718658	C₄	semidirect
17. #11718369	C₄	semidirect
19. #11718350	C₄	semidirect
64. C₁₃×#97	C₂×C₄	semidirect
65. C₄×C₈×C₁₃	C₂×C₄	semidirect
66. #88055	C₂×C₄	semidirect
67. #88055	C₂×C₄	semidirect
68. C₈×Dic₁₃	C₂×C₄	semidirect
69. C₈×Dic₁₃	C₂×C₄	semidirect
70. #88043	C₂×C₄	semidirect
71. C₄×#1055	C₂×C₄	semidirect
181. #405	#261	semidirect
249. #97	C₂×#261	semidirect
253. C₄×C₈	C₂×#261	semidirect
315. C₁₃	#20776	semidirect

Maximal subgroups

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

subgroup	index
#13130767	2
#13130766	2
#13130546	2
#13130517	2
C₂×#11718269	2
C₂×#11718267	2
#13123252	2
#20776	13