Subgroups — #16807175

Order 3328.

646 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719096	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11719162	C₂	1	1	yes	maximal, characteristic, complemented
3	#13131227	C₂	1	4	yes	maximal, complemented
4	#13131224	C₂	1	4	yes	maximal, complemented
5	#13131179	C₂	1	1	yes	maximal, characteristic, complemented
6	#13131080	C₂	1	1	yes	maximal, characteristic, complemented
7	#13131078	C₂	1	1	yes	maximal, characteristic, complemented
8	#13131045	C₂	1	1	yes	maximal, characteristic, complemented
9	#13123735	C₂	1	1	yes	maximal, characteristic
10	#11719527	C₄	1	1	yes	characteristic
11	C₂×#88202	C₂²	1	1	yes	characteristic
12	C₂×#88161	C₂²	1	1	yes	characteristic
13	#11719162	—	2	1	no	—
14	#11719162	—	2	1	no	—
15	#11719162	C₂²	1	2	yes	complemented
16	#11719162	C₂²	1	2	yes	complemented
17	C₂×#88106	—	2	1	no	—
18	C₂×#88106	C₂²	1	1	yes	characteristic
19	C₂×#88106	C₂²	1	1	yes	characteristic
20	C₂×#88106	C₂²	1	1	yes	characteristic
21	C₂²×#6092	—	2	1	no	—
22	C₂²×#6092	C₂²	1	1	yes	characteristic
23	#11719096	—	2	1	no	—
24	#11719096	—	2	1	no	—
25	#11719096	C₂²	1	2	yes	complemented
26	#11719096	C₂²	1	2	yes	complemented
27	C₂×#88098	—	2	1	no	—
28	C₂×D₁₃⋊C₁₆	—	2	1	no	—
29	C₁₃×#421	C₄	1	1	yes	characteristic
30	#11718785	C₄	1	1	yes	characteristic
31	#11718666	C₄	1	1	yes	characteristic
32	#11718658	C₂²	1	1	yes	characteristic
33	#11718485	C₂²	1	4	yes	—
34	#11718484	C₄	1	4	yes	—
35	#11718304	C₂²	1	4	yes	complemented
36	#11718270	C₂²	1	4	yes	complemented
37	#11718233	C₂²	1	4	yes	complemented
38	#11718231	C₂²	1	2	yes	complemented
39	C₄×#6092	C₂²	1	2	yes	complemented
40	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
41	C₂×#6127	C₂×C₄	1	1	yes	characteristic
42	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
43	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
44	#88205	C₈	1	2	yes	—
45	#88204	C₂×C₄	1	4	yes	—
46	#88203	C₈	1	2	yes	—
47	#88202	—	2	1	no	—
48	#88202	—	2	1	no	—
49	#88202	C₂×C₄	1	2	yes	—
50	#88202	C₂³	1	2	yes	—
51	#88196	C₈	1	2	yes	—
52	#88186	C₈	1	2	yes	—
53	C₂×#6099	—	2	1	no	—
54	C₂×#6099	C₂×C₄	1	1	yes	characteristic
55	C₂×#6099	C₂×C₄	1	1	yes	characteristic
56	C₂²×#1055	—	2	1	no	—
57	C₂²×#1055	C₂³	1	1	yes	characteristic
58	#88161	—	2	1	no	—
59	#88161	—	2	1	no	—
60	#88161	C₂×C₄	1	2	yes	—
61	#88161	C₂³	1	2	yes	—
62	C₂×#6094	—	2	1	no	—
63	C₂×C₈×D₁₃	—	2	1	no	—
64	#88106	—	2	1	no	—
65	#88106	—	2	1	no	—
66	#88106	—	2	1	no	—
67	#88106	—	2	1	no	—
68	#88106	—	2	2	no	—
69	#88106	—	2	2	no	—
70	#88106	C₂³	1	2	yes	—
71	#88106	C₂³	1	2	yes	—
72	C₂×#6092	—	2	1	no	—
73	C₂×#6092	—	2	1	no	—
74	C₂×#6092	—	2	2	no	—
75	C₂×#6092	—	2	2	no	—
76	C₂×#6092	D₄	1	1	yes	characteristic
77	C₂×#6092	D₄	1	1	yes	characteristic
78	C₂×#6092	D₄	1	1	yes	characteristic
79	C₂×#6092	D₄	1	1	yes	characteristic
80	C₂×#6092	C₂³	1	2	yes	—
81	C₂×#6092	C₂³	1	1	yes	characteristic
82	C₂×#6092	C₂³	1	1	yes	characteristic
83	#88098	—	2	2	no	—
84	#88098	—	2	2	no	—
85	D₁₃⋊C₁₆	—	2	2	no	—
86	D₁₃⋊C₁₆	—	2	2	no	—
87	C₁₃×#104	C₂×C₄	1	4	yes	—
88	#88062	C₂×C₄	1	4	yes	—
89	#88045	C₂×C₄	1	4	yes	—
90	#88043	C₂×C₄	1	2	yes	—
91	C₄×#1055	C₂³	1	2	yes	—
92	#33441	—	13	1	no	maximal, Sylow
93	C₁₃×#41	—	2	1	no	—
94	C₁₃×#41	—	2	1	no	—
95	C₁₃×#41	C₂×C₈	1	2	yes	—
96	C₁₃×#41	C₂²×C₄	1	2	yes	—
97	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
98	C₂×D₄×C₁₃	—	2	1	no	—
99	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
100	C₂²×C₄×C₁₃	—	2	1	no	—
101	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₂²⋊D₁₃	—	2	1	no	—
103	#6127	—	2	1	no	—
104	#6127	—	2	1	no	—
105	#6127	C₂×C₈	1	2	yes	—
106	#6127	C₂²×C₄	1	2	yes	—
107	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
108	C₂×C₄×D₁₃	—	2	1	no	—
109	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
110	C₁₃×M₁₆	—	2	1	no	—
111	C₁₃×M₁₆	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₁₃	—	2	1	no	—
113	C₁₃×C₄⋊C₄	—	2	1	no	—
114	C₁₃×C₂²⋊C₄	—	2	1	no	—
115	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
116	#6107	C₂×C₈	1	4	yes	—
117	#6106	C₂×C₈	1	4	yes	—
118	#6105	C₂×C₈	1	4	yes	—
119	#6104	C₂×C₈	1	4	yes	—
120	#6099	—	2	1	no	—
121	#6099	—	2	2	no	—
122	#6099	—	2	2	no	—
123	#6099	C₂×C₈	1	2	yes	—
124	#6099	C₂×C₈	1	2	yes	—
125	#6099	C₂²×C₄	1	2	yes	—
126	C₂×#1055	—	2	1	no	—
127	C₂×#1055	—	2	2	no	—
128	C₂×#1055	—	2	2	no	—
129	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
130	C₂×#1055	C₂²×C₄	1	2	yes	—
131	C₂×#1055	C₂×D₄	1	1	yes	characteristic
132	C₂×#1055	C₂×D₄	1	1	yes	characteristic
133	C₂×#1055	#41	1	1	yes	characteristic
134	C₂×#1055	#41	1	1	yes	characteristic
135	C₂×#1055	C₂⁴	1	1	yes	characteristic
136	#6094	—	2	2	no	—
137	#6094	—	2	2	no	—
138	C₈×D₁₃	—	2	2	no	—
139	C₈×D₁₃	—	2	2	no	—
140	#6092	—	2	2	no	—
141	#6092	—	2	2	no	—
142	#6092	—	2	2	no	—
143	#6092	C₂×D₄	1	1	yes	characteristic
144	#6092	C₂×D₄	1	1	yes	characteristic
145	#6092	C₂×D₄	1	1	yes	characteristic
146	#6092	C₂×D₄	1	1	yes	characteristic
147	C₂×#504	—	13	1	no	—
148	C₂×#504	—	13	1	no	—
149	#2955	—	13	1	no	—
150	#2171	—	13	4	no	—
151	#2170	—	13	4	no	—
152	#2153	—	13	1	no	—
153	#2117	—	13	1	no	—
154	#2117	—	13	1	no	—
155	#2108	—	13	1	no	—
156	C₂³×C₁₃	—	2	1	no	—
157	C₂³×C₁₃	#131	1	1	yes	characteristic
158	C₂²×D₁₃	—	2	1	no	—
159	Q₈×C₁₃	—	2	1	no	—
160	Q₈×C₁₃	C₂²×C₈	1	2	yes	—
161	D₄×C₁₃	—	2	2	no	—
162	D₄×C₁₃	—	2	2	no	—
163	D₄×C₁₃	—	2	1	no	—
164	D₄×C₁₃	C₂²×C₈	1	2	yes	—
165	C₂×C₄×C₁₃	—	2	1	no	—
166	C₂×C₄×C₁₃	—	2	2	no	—
167	C₂×C₄×C₁₃	—	2	2	no	—
168	C₂×C₄×C₁₃	—	2	2	no	—
169	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
170	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
171	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
172	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
173	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
174	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
175	C₂²⋊D₁₃	—	2	2	no	—
176	C₂²⋊D₁₃	—	2	2	no	—
177	C₂×Dic₁₃	—	2	1	no	—
178	D₅₂	—	2	1	no	—
179	D₅₂	C₂²×C₈	1	2	yes	—
180	C₄×D₁₃	—	2	2	no	—
181	C₄×D₁₃	—	2	2	no	—
182	Dic₂₆	—	2	1	no	—
183	Dic₂₆	C₂²×C₈	1	2	yes	—
184	C₈×C₁₃	—	2	2	no	—
185	#1055	—	2	2	no	—
186	#1055	C₄×D₄	1	1	yes	characteristic
187	#1055	C₄×D₄	1	1	yes	characteristic
188	#1055	C₂²×C₈	1	4	yes	—
189	#1055	C₂²×D₄	1	1	yes	characteristic
190	#1055	C₂×#41	1	1	yes	characteristic
191	#575	—	13	1	no	—
192	C₂×#131	—	13	1	no	—
193	C₂×#131	—	13	1	no	—
194	#504	—	13	2	no	—
195	#504	—	13	2	no	—
196	#504	—	13	2	no	—
197	#504	—	13	2	no	—
198	#504	—	26	1	no	—
199	#504	—	26	1	no	—
200	#504	—	26	1	no	—
201	#504	—	26	1	no	—
202	C₂×M₃₂	—	13	1	no	—
203	C₂×M₃₂	—	13	1	no	—
204	C₂×M₃₂	—	13	1	no	—
205	C₂×M₃₂	—	26	1	no	—
206	C₂×M₃₂	—	26	1	no	—
207	C₂²×C₁₆	—	13	1	no	—
208	C₂²×C₁₆	—	26	1	no	—
209	C₂²×C₁₆	—	26	1	no	—
210	#444	—	13	4	no	—
211	#443	—	13	4	no	—
212	#430	—	13	1	no	—
213	#421	—	13	1	no	—
214	#421	#261	1	1	yes	characteristic
215	#405	—	13	1	no	—
216	#364	—	13	4	no	—
217	#350	—	13	4	no	—
218	#350	—	13	4	no	—
219	#346	—	13	2	no	—
220	C₄×C₁₆	—	13	2	no	—
221	C₂²×C₁₃	—	2	2	no	—
222	C₂²×C₁₃	—	4	1	no	—
223	C₂²×C₁₃	—	4	2	no	—
224	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
225	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
226	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
227	C₂×D₁₃	—	2	2	no	—
228	C₂×D₁₃	—	4	2	no	—
229	C₄×C₁₃	—	2	2	no	—
230	C₄×C₁₃	—	4	2	no	—
231	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
232	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
233	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
234	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
235	Dic₁₃	—	2	2	no	—
236	C₂×#41	—	13	1	no	—
237	C₂×#41	C₂×#261	1	1	yes	characteristic
238	#137	—	13	2	no	—
239	#137	—	13	2	no	—
240	AGL(1,ℤ/8)	—	13	2	no	—
241	AGL(1,ℤ/8)	—	13	2	no	—
242	#135	—	13	4	no	—
243	#131	—	13	2	no	—
244	#131	—	13	2	no	—
245	#131	—	13	2	no	—
246	#131	—	13	2	no	—
247	#131	—	26	1	no	—
248	#131	—	26	1	no	—
249	#131	—	26	1	no	—
250	#131	—	26	1	no	—
251	C₂×M₁₆	—	13	1	no	—
252	C₂×M₁₆	—	13	1	no	—
253	C₂×M₁₆	—	26	1	no	—
254	C₂×M₁₆	—	26	1	no	—
255	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
256	C₂²×C₈	—	13	1	no	—
257	C₂²×C₈	—	26	1	no	—
258	C₂²×C₈	—	26	1	no	—
259	#117	C₂×#261	1	1	yes	characteristic
260	M₃₂	—	13	2	no	—
261	M₃₂	—	13	2	no	—
262	M₃₂	—	26	2	no	—
263	M₃₂	—	26	2	no	—
264	M₃₂	—	26	2	no	—
265	M₃₂	—	26	2	no	—
266	M₃₂	—	26	1	no	—
267	M₃₂	—	26	1	no	—
268	M₃₂	—	26	1	no	—
269	M₃₂	—	26	1	no	—
270	C₂×C₁₆	—	13	1	no	—
271	C₂×C₁₆	—	13	1	no	—
272	C₂×C₁₆	—	13	1	no	—
273	C₂×C₁₆	—	13	1	no	—
274	C₂×C₁₆	—	13	1	no	—
275	C₂×C₁₆	—	13	1	no	—
276	C₂×C₁₆	—	13	2	no	—
277	C₂×C₁₆	—	26	2	no	—
278	C₂×C₁₆	—	26	2	no	—
279	C₂×C₁₆	—	26	2	no	—
280	C₂×C₁₆	—	26	2	no	—
281	C₂×C₁₆	—	26	1	no	—
282	C₂×C₁₆	—	26	1	no	—
283	#108	—	13	4	no	—
284	#104	—	13	4	no	—
285	#104	C₂×#261	1	4	yes	—
286	#97	—	13	2	no	—
287	C₄×C₈	—	13	2	no	—
288	C₂×C₁₃	—	2	1	no	—
289	C₂×C₁₃	—	2	1	no	—
290	C₂×C₁₃	—	2	1	no	—
291	C₂×C₁₃	—	4	2	no	—
292	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
293	D₁₃	—	4	2	no	—
294	#41	—	2	1	no	—
295	#41	—	2	1	no	—
296	#41	—	13	2	no	—
297	#41	—	13	2	no	—
298	#41	—	26	1	no	—
299	#41	—	26	1	no	—
300	#41	D₁₃⋊C₈	1	2	yes	—
301	#41	C₂²×#261	1	2	yes	—
302	C₂×Q₈	—	13	1	no	—
303	C₂×Q₈	D₁₃⋊C₈	1	1	yes	characteristic
304	C₂×D₄	—	2	1	no	—
305	C₂×D₄	—	13	1	no	—
306	C₂×D₄	—	26	1	no	—
307	C₂×D₄	D₁₃⋊C₈	1	1	yes	characteristic
308	C₂²×C₄	—	2	1	no	—
309	C₂²×C₄	—	26	1	no	—
310	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
311	Q₁₆	—	13	4	no	—
312	SD₁₆	—	13	4	no	—
313	SD₁₆	—	13	4	no	—
314	D₈	—	13	4	no	—
315	M₁₆	—	2	1	no	—
316	M₁₆	—	13	2	no	—
317	M₁₆	—	13	2	no	—
318	M₁₆	—	13	2	no	—
319	M₁₆	—	26	2	no	—
320	M₁₆	—	26	2	no	—
321	M₁₆	—	26	2	no	—
322	M₁₆	—	26	2	no	—
323	M₁₆	—	26	1	no	—
324	M₁₆	C₂²×#261	1	2	yes	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	13	1	no	—
327	C₂×C₈	—	13	1	no	—
328	C₂×C₈	—	13	1	no	—
329	C₂×C₈	—	13	1	no	—
330	C₂×C₈	—	13	1	no	—
331	C₂×C₈	—	13	2	no	—
332	C₂×C₈	—	13	1	no	—
333	C₂×C₈	—	26	2	no	—
334	C₂×C₈	—	26	2	no	—
335	C₂×C₈	—	26	2	no	—
336	C₂×C₈	—	26	2	no	—
337	C₂×C₈	—	26	1	no	—
338	C₄⋊C₄	—	2	1	no	—
339	C₂²⋊C₄	—	2	1	no	—
340	C₄²	C₂²×#261	1	2	yes	—
341	C₁₆	—	13	1	no	—
342	C₁₆	—	13	1	no	—
343	C₁₆	—	13	1	no	—
344	C₁₆	—	13	1	no	—
345	C₁₆	—	26	2	no	—
346	C₁₆	—	26	2	no	—
347	C₁₆	—	26	2	no	—
348	C₁₃	#33441	1	1	yes	characteristic, complemented, Sylow
349	C₂³	—	2	1	no	—
350	C₂³	—	26	1	no	—
351	C₂³	#88238	1	1	yes	characteristic
352	Q₈	—	2	1	no	—
353	Q₈	—	13	2	no	—
354	Q₈	—	26	1	no	—
355	Q₈	C₂×D₁₃⋊C₈	1	2	yes	—
356	D₄	—	2	2	no	—
357	D₄	—	2	2	no	—
358	D₄	—	2	1	no	—
359	D₄	—	13	2	no	—
360	D₄	—	26	2	no	—
361	D₄	—	26	2	no	—
362	D₄	—	26	1	no	—
363	D₄	C₂×D₁₃⋊C₈	1	2	yes	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	26	1	no	—
369	C₂×C₄	—	26	2	no	—
370	C₂×C₄	—	26	2	no	—
371	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
372	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
373	C₂×C₄	#88238	1	1	yes	characteristic
374	C₂×C₄	D₄×#261	1	1	yes	characteristic
375	C₂×C₄	D₄×#261	1	1	yes	characteristic
376	C₂×C₄	C₂³×#261	1	1	yes	characteristic
377	C₈	—	2	2	no	—
378	C₈	—	13	4	no	—
379	C₈	—	13	1	no	—
380	C₈	—	13	1	no	—
381	C₈	—	13	1	no	—
382	C₈	—	13	1	no	—
383	C₈	—	26	2	no	—
384	C₂²	—	2	2	no	—
385	C₂²	—	4	1	no	—
386	C₂²	—	4	2	no	—
387	C₂²	—	26	2	no	—
388	C₂²	—	52	2	no	—
389	C₂²	#11719063	1	1	yes	characteristic
390	C₂²	#11719063	1	1	yes	characteristic
391	C₂²	C₂×#88238	1	1	yes	characteristic
392	C₄	—	2	2	no	—
393	C₄	—	4	2	no	—
394	C₄	—	26	2	no	—
395	C₄	#11719063	1	1	yes	characteristic
396	C₄	#11719063	1	1	yes	characteristic
397	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
398	C₄	C₂×D₄×#261	1	1	yes	characteristic
399	C₂	—	2	1	no	—
400	C₂	—	2	1	no	—
401	C₂	—	2	1	no	—
402	C₂	—	4	2	no	—
403	C₂	—	52	2	no	—
404	C₂	C₂×#11719063	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×#261
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₁₃×#421	C₄
socle	C₂×C₁₃	C₂×C₈×D₄
prime radical	#11719527	C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	#421	#261
hypocenter	C₁₃	#33441
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #421

Complements

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

N	H ≅ G/N	type
1. C₂×#11719096	C₂	semidirect
2. C₂×#11719162	C₂	semidirect
3. #13131227	C₂	semidirect
4. #13131224	C₂	semidirect
5. #13131179	C₂	semidirect
6. #13131080	C₂	semidirect
7. #13131078	C₂	semidirect
8. #13131045	C₂	semidirect
15. #11719162	C₂²	semidirect
16. #11719162	C₂²	semidirect
25. #11719096	C₂²	semidirect
26. #11719096	C₂²	semidirect
35. #11718304	C₂²	semidirect
36. #11718270	C₂²	semidirect
37. #11718233	C₂²	semidirect
38. #11718231	C₂²	semidirect
39. C₄×#6092	C₂²	semidirect
348. C₁₃	#33441	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719096	2
C₂×#11719162	2
#13131227	2
#13131224	2
#13131179	2
#13131080	2
#13131078	2
#13131045	2
#13123735	2
#33441	13