Subgroups — #16807179

Order 3328.

642 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131805	C₂	1	1	yes	maximal, characteristic, complemented
2	#13131803	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	C₂×#11718270	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718304	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	2	no	—
14	#11719162	C₂²	1	4	yes	complemented
15	C₂×#88106	—	2	2	no	—
16	C₂×#88106	C₂²	1	1	yes	characteristic
17	C₂×#88106	C₂²	1	1	yes	characteristic
18	C₂×#88106	C₂²	1	1	yes	characteristic
19	C₂²×#6092	C₂²	1	1	yes	characteristic
20	#11719096	—	2	2	no	—
21	#11719096	C₂²	1	4	yes	complemented
22	D₁₃⋊M₃₂	—	2	2	no	—
23	C₁₃×#421	C₄	1	1	yes	characteristic
24	#11718785	C₄	1	1	yes	characteristic
25	#11718666	C₄	1	1	yes	characteristic
26	#11718658	C₂²	1	1	yes	characteristic
27	#11718485	C₂²	1	4	yes	—
28	#11718484	C₄	1	4	yes	—
29	#11718304	C₂²	1	2	yes	complemented
30	#11718304	C₂²	1	2	yes	complemented
31	#11718270	C₂²	1	2	yes	complemented
32	#11718270	C₂²	1	2	yes	complemented
33	#11718233	C₂²	1	4	yes	complemented
34	#11718231	C₂²	1	2	yes	complemented
35	C₄×#6092	C₂²	1	2	yes	complemented
36	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
37	C₂×#6127	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
39	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
40	#88205	C₈	1	2	yes	—
41	#88204	C₂×C₄	1	4	yes	—
42	#88203	C₈	1	2	yes	—
43	#88202	—	2	1	no	—
44	#88202	—	2	1	no	—
45	#88202	C₂×C₄	1	2	yes	—
46	#88202	C₂³	1	2	yes	—
47	#88196	C₈	1	2	yes	—
48	#88186	C₈	1	2	yes	—
49	C₂×#6099	—	2	1	no	—
50	C₂×#6099	C₂×C₄	1	1	yes	characteristic
51	C₂×#6099	C₂×C₄	1	1	yes	characteristic
52	C₂²×#1055	—	2	1	no	—
53	C₂²×#1055	C₂³	1	1	yes	characteristic
54	#88161	—	2	1	no	—
55	#88161	—	2	1	no	—
56	#88161	C₂×C₄	1	2	yes	—
57	#88161	C₂³	1	2	yes	—
58	C₂×#6094	—	2	1	no	—
59	C₂×C₈×D₁₃	—	2	1	no	—
60	#88106	—	2	1	no	—
61	#88106	—	2	1	no	—
62	#88106	—	2	4	no	—
63	#88106	D₄	1	2	yes	—
64	#88106	D₄	1	2	yes	—
65	#88106	C₂³	1	2	yes	—
66	#88106	C₂³	1	2	yes	—
67	C₂×#6092	—	2	1	no	—
68	C₂×#6092	—	2	1	no	—
69	C₂×#6092	—	2	1	no	—
70	C₂×#6092	—	2	1	no	—
71	C₂×#6092	—	2	4	no	—
72	C₂×#6092	C₂³	1	1	yes	characteristic
73	C₂×#6092	C₂³	1	1	yes	characteristic
74	C₂×#6092	C₂³	1	2	yes	—
75	#88098	—	2	4	no	—
76	D₁₃⋊C₁₆	—	2	4	no	—
77	C₁₃×#104	C₂×C₄	1	4	yes	—
78	#88062	C₂×C₄	1	4	yes	—
79	#88045	C₂×C₄	1	4	yes	—
80	#88043	C₂×C₄	1	2	yes	—
81	C₄×#1055	C₂³	1	2	yes	—
82	#33442	—	13	1	no	maximal, Sylow
83	C₁₃×#41	—	2	1	no	—
84	C₁₃×#41	—	2	1	no	—
85	C₁₃×#41	C₂×C₈	1	2	yes	—
86	C₁₃×#41	C₂²×C₄	1	2	yes	—
87	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
88	C₂×D₄×C₁₃	—	2	1	no	—
89	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	—	2	1	no	—
91	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₂²⋊D₁₃	—	2	1	no	—
93	#6127	—	2	1	no	—
94	#6127	—	2	1	no	—
95	#6127	C₂×C₈	1	2	yes	—
96	#6127	C₂²×C₄	1	2	yes	—
97	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
98	C₂×C₄×D₁₃	—	2	1	no	—
99	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
100	C₁₃×M₁₆	—	2	1	no	—
101	C₁₃×M₁₆	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₁₃	—	2	1	no	—
103	C₁₃×C₄⋊C₄	—	2	1	no	—
104	C₁₃×C₂²⋊C₄	—	2	1	no	—
105	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
106	#6107	C₂×C₈	1	4	yes	—
107	#6106	C₂×C₈	1	4	yes	—
108	#6105	C₂×C₈	1	4	yes	—
109	#6104	C₂×C₈	1	4	yes	—
110	#6099	—	2	1	no	—
111	#6099	—	2	2	no	—
112	#6099	—	2	2	no	—
113	#6099	C₂×C₈	1	2	yes	—
114	#6099	C₂×C₈	1	2	yes	—
115	#6099	C₂²×C₄	1	2	yes	—
116	C₂×#1055	—	2	1	no	—
117	C₂×#1055	—	2	2	no	—
118	C₂×#1055	—	2	2	no	—
119	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
120	C₂×#1055	C₂²×C₄	1	2	yes	—
121	C₂×#1055	C₂×D₄	1	1	yes	characteristic
122	C₂×#1055	C₂×D₄	1	1	yes	characteristic
123	C₂×#1055	#41	1	1	yes	characteristic
124	C₂×#1055	#41	1	1	yes	characteristic
125	C₂×#1055	C₂⁴	1	1	yes	characteristic
126	#6094	—	2	2	no	—
127	#6094	—	2	2	no	—
128	C₈×D₁₃	—	2	2	no	—
129	C₈×D₁₃	—	2	2	no	—
130	#6092	—	2	2	no	—
131	#6092	—	2	2	no	—
132	#6092	—	2	1	no	—
133	#6092	—	2	1	no	—
134	#6092	C₂×D₄	1	4	yes	—
135	#3407	—	13	1	no	—
136	#3407	—	13	1	no	—
137	#2955	—	13	1	no	—
138	#2171	—	13	4	no	—
139	#2170	—	13	4	no	—
140	#2153	—	13	1	no	—
141	C₂×#350	—	13	1	no	—
142	C₂×#350	—	13	1	no	—
143	#2108	—	13	1	no	—
144	C₂³×C₁₃	—	2	1	no	—
145	C₂³×C₁₃	#131	1	1	yes	characteristic
146	C₂²×D₁₃	—	2	1	no	—
147	Q₈×C₁₃	—	2	1	no	—
148	Q₈×C₁₃	C₂²×C₈	1	2	yes	—
149	D₄×C₁₃	—	2	2	no	—
150	D₄×C₁₃	—	2	2	no	—
151	D₄×C₁₃	—	2	1	no	—
152	D₄×C₁₃	C₂²×C₈	1	2	yes	—
153	C₂×C₄×C₁₃	—	2	1	no	—
154	C₂×C₄×C₁₃	—	2	2	no	—
155	C₂×C₄×C₁₃	—	2	2	no	—
156	C₂×C₄×C₁₃	—	2	2	no	—
157	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
161	C₂×C₄×C₁₃	#131	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	D₅₂	—	2	1	no	—
167	D₅₂	C₂²×C₈	1	2	yes	—
168	C₄×D₁₃	—	2	2	no	—
169	C₄×D₁₃	—	2	2	no	—
170	Dic₂₆	—	2	1	no	—
171	Dic₂₆	C₂²×C₈	1	2	yes	—
172	C₈×C₁₃	—	2	2	no	—
173	#1055	—	2	2	no	—
174	#1055	C₄×D₄	1	1	yes	characteristic
175	#1055	C₄×D₄	1	1	yes	characteristic
176	#1055	C₂²×C₈	1	4	yes	—
177	#1055	C₂²×D₄	1	1	yes	characteristic
178	#1055	C₂×#41	1	1	yes	characteristic
179	#575	—	13	1	no	—
180	C₂×#131	—	13	1	no	—
181	C₂×#131	—	13	1	no	—
182	#504	—	13	4	no	—
183	#504	—	13	4	no	—
184	#504	—	26	2	no	—
185	#504	—	26	2	no	—
186	C₂×M₃₂	—	13	1	no	—
187	C₂×M₃₂	—	13	1	no	—
188	C₂×M₃₂	—	13	1	no	—
189	C₂×M₃₂	—	26	2	no	—
190	C₂×M₃₂	—	26	2	no	—
191	C₂²×C₁₆	—	13	1	no	—
192	#444	—	13	4	no	—
193	#443	—	13	4	no	—
194	#430	—	13	1	no	—
195	#421	—	13	1	no	—
196	#421	#261	1	1	yes	characteristic
197	#405	—	13	1	no	—
198	#364	—	13	4	no	—
199	#350	—	13	2	no	—
200	#350	—	13	2	no	—
201	#350	—	13	2	no	—
202	#350	—	13	2	no	—
203	#346	—	13	2	no	—
204	C₄×C₁₆	—	13	2	no	—
205	C₂²×C₁₃	—	2	2	no	—
206	C₂²×C₁₃	—	4	1	no	—
207	C₂²×C₁₃	—	8	1	no	—
208	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
209	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
210	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
211	C₂×D₁₃	—	2	2	no	—
212	C₂×D₁₃	—	8	1	no	—
213	C₄×C₁₃	—	2	2	no	—
214	C₄×C₁₃	—	4	2	no	—
215	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
216	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
217	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
218	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
219	Dic₁₃	—	2	2	no	—
220	C₂×#41	—	13	1	no	—
221	C₂×#41	C₂×#261	1	1	yes	characteristic
222	#137	—	13	2	no	—
223	#137	—	13	2	no	—
224	AGL(1,ℤ/8)	—	13	2	no	—
225	AGL(1,ℤ/8)	—	13	2	no	—
226	#135	—	13	4	no	—
227	#131	—	13	2	no	—
228	#131	—	13	2	no	—
229	#131	—	13	2	no	—
230	#131	—	13	2	no	—
231	#131	—	26	1	no	—
232	#131	—	26	1	no	—
233	#131	—	26	1	no	—
234	#131	—	26	1	no	—
235	C₂×M₁₆	—	13	1	no	—
236	C₂×M₁₆	—	13	1	no	—
237	C₂×M₁₆	—	26	1	no	—
238	C₂×M₁₆	—	26	1	no	—
239	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
240	C₂²×C₈	—	13	1	no	—
241	C₂²×C₈	—	26	1	no	—
242	C₂²×C₈	—	26	1	no	—
243	#117	C₂×#261	1	1	yes	characteristic
244	M₃₂	—	13	2	no	—
245	M₃₂	—	13	2	no	—
246	M₃₂	—	13	2	no	—
247	M₃₂	—	13	2	no	—
248	M₃₂	—	26	4	no	—
249	M₃₂	—	26	4	no	—
250	M₃₂	—	26	1	no	—
251	M₃₂	—	26	1	no	—
252	C₂×C₁₆	—	13	2	no	—
253	C₂×C₁₆	—	13	1	no	—
254	C₂×C₁₆	—	13	1	no	—
255	C₂×C₁₆	—	26	4	no	—
256	C₂×C₁₆	—	26	4	no	—
257	C₂×C₁₆	—	26	1	no	—
258	C₂×C₁₆	—	26	1	no	—
259	C₂×C₁₆	—	26	1	no	—
260	C₂×C₁₆	—	26	1	no	—
261	#108	—	13	4	no	—
262	#104	—	13	4	no	—
263	#104	C₂×#261	1	4	yes	—
264	#97	—	13	2	no	—
265	C₄×C₈	—	13	2	no	—
266	C₂×C₁₃	—	2	1	no	—
267	C₂×C₁₃	—	2	1	no	—
268	C₂×C₁₃	—	2	1	no	—
269	C₂×C₁₃	—	4	2	no	—
270	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
271	D₁₃	—	4	2	no	—
272	#41	—	2	1	no	—
273	#41	—	2	1	no	—
274	#41	—	13	2	no	—
275	#41	—	13	2	no	—
276	#41	—	26	1	no	—
277	#41	—	26	1	no	—
278	#41	D₁₃⋊C₈	1	2	yes	—
279	#41	C₂²×#261	1	2	yes	—
280	C₂×Q₈	—	13	1	no	—
281	C₂×Q₈	D₁₃⋊C₈	1	1	yes	characteristic
282	C₂×D₄	—	2	1	no	—
283	C₂×D₄	—	13	1	no	—
284	C₂×D₄	—	26	1	no	—
285	C₂×D₄	D₁₃⋊C₈	1	1	yes	characteristic
286	C₂²×C₄	—	2	1	no	—
287	C₂²×C₄	—	26	1	no	—
288	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
289	Q₁₆	—	13	4	no	—
290	SD₁₆	—	13	4	no	—
291	SD₁₆	—	13	4	no	—
292	D₈	—	13	4	no	—
293	M₁₆	—	2	1	no	—
294	M₁₆	—	13	2	no	—
295	M₁₆	—	13	2	no	—
296	M₁₆	—	13	2	no	—
297	M₁₆	—	26	2	no	—
298	M₁₆	—	26	2	no	—
299	M₁₆	—	26	2	no	—
300	M₁₆	—	26	2	no	—
301	M₁₆	—	26	1	no	—
302	M₁₆	C₂²×#261	1	2	yes	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	13	1	no	—
305	C₂×C₈	—	13	1	no	—
306	C₂×C₈	—	13	1	no	—
307	C₂×C₈	—	13	1	no	—
308	C₂×C₈	—	13	1	no	—
309	C₂×C₈	—	13	2	no	—
310	C₂×C₈	—	13	1	no	—
311	C₂×C₈	—	26	2	no	—
312	C₂×C₈	—	26	2	no	—
313	C₂×C₈	—	26	2	no	—
314	C₂×C₈	—	26	2	no	—
315	C₂×C₈	—	26	1	no	—
316	C₄⋊C₄	—	2	1	no	—
317	C₂²⋊C₄	—	2	1	no	—
318	C₄²	C₂²×#261	1	2	yes	—
319	C₁₆	—	13	4	no	—
320	C₁₆	—	26	1	no	—
321	C₁₆	—	26	1	no	—
322	C₁₆	—	26	2	no	—
323	C₁₆	—	26	2	no	—
324	C₁₃	#33442	1	1	yes	characteristic, complemented, Sylow
325	C₂³	—	2	1	no	—
326	C₂³	—	26	1	no	—
327	C₂³	#88238	1	1	yes	characteristic
328	Q₈	—	2	1	no	—
329	Q₈	—	13	2	no	—
330	Q₈	—	26	1	no	—
331	Q₈	C₂×D₁₃⋊C₈	1	2	yes	—
332	D₄	—	2	2	no	—
333	D₄	—	2	2	no	—
334	D₄	—	2	1	no	—
335	D₄	—	13	2	no	—
336	D₄	—	26	2	no	—
337	D₄	—	26	2	no	—
338	D₄	—	26	1	no	—
339	D₄	C₂×D₁₃⋊C₈	1	2	yes	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	26	1	no	—
345	C₂×C₄	—	26	2	no	—
346	C₂×C₄	—	26	2	no	—
347	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
348	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
349	C₂×C₄	#88238	1	1	yes	characteristic
350	C₂×C₄	D₄×#261	1	1	yes	characteristic
351	C₂×C₄	D₄×#261	1	1	yes	characteristic
352	C₂×C₄	C₂³×#261	1	1	yes	characteristic
353	C₈	—	2	2	no	—
354	C₈	—	13	4	no	—
355	C₈	—	13	1	no	—
356	C₈	—	13	1	no	—
357	C₈	—	13	1	no	—
358	C₈	—	13	1	no	—
359	C₈	—	26	2	no	—
360	C₂²	—	2	2	no	—
361	C₂²	—	4	1	no	—
362	C₂²	—	8	1	no	—
363	C₂²	—	26	2	no	—
364	C₂²	—	104	1	no	—
365	C₂²	#11719063	1	1	yes	characteristic
366	C₂²	#11719063	1	1	yes	characteristic
367	C₂²	C₂×#88238	1	1	yes	characteristic
368	C₄	—	2	2	no	—
369	C₄	—	4	2	no	—
370	C₄	—	26	2	no	—
371	C₄	#11719063	1	1	yes	characteristic
372	C₄	#11719063	1	1	yes	characteristic
373	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
374	C₄	C₂×D₄×#261	1	1	yes	characteristic
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	—	2	1	no	—
378	C₂	—	4	2	no	—
379	C₂	—	52	2	no	—
380	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	#33442
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. #13131805	C₂	semidirect
2. #13131803	C₂	semidirect
3. #13131227	C₂	semidirect
4. #13131224	C₂	semidirect
5. #13131179	C₂	semidirect
6. C₂×#11718270	C₂	semidirect
7. C₂×#11718304	C₂	semidirect
8. #13131045	C₂	semidirect
14. #11719162	C₂²	semidirect
21. #11719096	C₂²	semidirect
29. #11718304	C₂²	semidirect
30. #11718304	C₂²	semidirect
31. #11718270	C₂²	semidirect
32. #11718270	C₂²	semidirect
33. #11718233	C₂²	semidirect
34. #11718231	C₂²	semidirect
35. C₄×#6092	C₂²	semidirect
324. C₁₃	#33442	semidirect

Maximal subgroups

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

subgroup	index
#13131805	2
#13131803	2
#13131227	2
#13131224	2
#13131179	2
C₂×#11718270	2
C₂×#11718304	2
#13131045	2
#13123735	2
#33442	13