Subgroups — #16807417

Order 3328.

502 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131251	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#6092	C₂	1	1	yes	maximal, characteristic, complemented
3	#13131220	C₂	1	1	yes	maximal, characteristic, complemented
4	#13131216	C₂	1	2	yes	maximal, complemented
5	D₄×#6092	C₂	1	2	yes	maximal, complemented
6	D₄×#6092	C₂	1	1	yes	maximal, characteristic, complemented
7	#13131201	C₂	1	2	yes	maximal, complemented
8	#13131193	C₂	1	1	yes	maximal, characteristic, complemented
9	#13131171	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊#6092	C₂	1	1	yes	maximal, characteristic, complemented
11	#13123812	C₂	1	1	yes	maximal, characteristic
12	C₁₃×#529	C₄	1	1	yes	characteristic
13	C₂×#88106	C₂²	1	2	yes	—
14	C₂²×#6092	C₂²	1	2	yes	—
15	C₂²×#6092	C₂²	1	2	yes	—
16	#11718685	C₄	1	1	yes	characteristic
17	Q₈×#1055	C₂²	1	1	yes	characteristic
18	#11718673	C₄	1	1	yes	characteristic
19	#11718672	C₄	1	2	yes	—
20	D₄×#1055	C₂²	1	2	yes	—
21	D₄×#1055	C₂²	1	1	yes	characteristic
22	#11718661	C₄	1	2	yes	—
23	#11718658	C₂²	1	2	yes	—
24	#11718587	C₄	1	1	yes	characteristic
25	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
26	C₂²⋊#6092	C₂²	1	1	yes	characteristic
27	C₂²⋊#6092	C₂²	1	1	yes	characteristic
28	C₂²⋊#6092	C₂²	1	2	yes	—
29	C₂²⋊#6092	C₂²	1	2	yes	—
30	#11718231	C₂²	1	2	yes	—
31	C₄⋊#6092	C₂²	1	2	yes	—
32	C₄⋊#6092	C₂²	1	1	yes	characteristic
33	C₄⋊#6092	C₂²	1	1	yes	characteristic
34	C₄⋊#6092	C₂²	1	1	yes	characteristic
35	C₄⋊#6092	C₂²	1	2	yes	—
36	C₄⋊#6092	C₂²	1	2	yes	—
37	C₄⋊#6092	C₂²	1	1	yes	characteristic
38	C₄×#6092	C₂²	1	1	yes	characteristic
39	C₄×#6092	C₂²	1	1	yes	characteristic
40	C₄×#6092	C₂²	1	2	yes	—
41	C₁₃×#126	C₈	1	2	yes	—
42	C₁₃×#125	C₈	1	1	yes	characteristic
43	C₁₃×#124	C₈	1	1	yes	characteristic
44	C₁₃×#123	C₈	1	2	yes	—
45	C₁₃×C₂²⋊Q₈	C₈	1	1	yes	characteristic
46	C₁₃×#121	C₈	1	1	yes	characteristic
47	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
49	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
50	C₁₃×#117	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
52	C₂×#6099	C₂×C₄	1	2	yes	—
53	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
54	C₂²×#1055	C₂³	1	2	yes	—
55	C₂²×#1055	C₂³	1	1	yes	characteristic
56	#88106	—	4	2	no	—
57	C₂×#6092	—	4	2	no	—
58	C₂×#6092	—	4	2	no	—
59	C₂×#6092	C₂³	1	1	yes	characteristic
60	C₂×#6092	C₂³	1	1	yes	characteristic
61	C₂×#6092	C₂³	1	2	yes	—
62	C₂×#6092	C₂³	1	1	yes	characteristic
63	C₂×#6092	C₂³	1	1	yes	characteristic
64	C₂×#6092	C₂³	1	2	yes	—
65	C₂²⋊#1055	C₂×C₄	1	2	yes	—
66	C₂²⋊#1055	C₂×C₄	1	2	yes	—
67	C₂²⋊#1055	C₂×C₄	1	2	yes	—
68	C₄⋊#1055	C₂×C₄	1	2	yes	—
69	C₄⋊#1055	C₂×C₄	1	2	yes	—
70	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
71	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
72	#88043	C₂×C₄	1	2	yes	—
73	#88043	C₂×C₄	1	2	yes	—
74	C₄×#1055	C₂×C₄	1	1	yes	characteristic
75	C₄×#1055	C₂×C₄	1	1	yes	characteristic
76	C₄×#1055	C₂³	1	1	yes	characteristic
77	C₄×#1055	C₂³	1	2	yes	—
78	C₄×#1055	C₂³	1	1	yes	characteristic
79	#33508	—	13	1	no	maximal, Sylow
80	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
81	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
82	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
87	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
88	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
89	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
90	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
91	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
92	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
93	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
94	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
95	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
96	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
97	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
98	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
99	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
100	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
103	#6099	—	4	2	no	—
104	C₂×#1055	—	2	1	no	—
105	C₂×#1055	—	2	2	no	—
106	C₂×#1055	—	2	2	no	—
107	C₂×#1055	—	2	1	no	—
108	C₂×#1055	—	4	1	no	—
109	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
110	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
111	C₂×#1055	C₂²×C₄	1	2	yes	—
112	C₂×#1055	C₂²×C₄	1	2	yes	—
113	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
114	C₂×#1055	C₂⁴	1	1	yes	characteristic
115	#6092	—	2	1	no	—
116	#6092	—	2	2	no	—
117	#6092	—	2	2	no	—
118	#6092	—	2	1	no	—
119	#6092	—	2	1	no	—
120	#6092	—	2	1	no	—
121	#2966	—	13	1	no	—
122	#2183	—	13	1	no	—
123	Q₈×C₁₆	—	13	1	no	—
124	#2169	—	13	1	no	—
125	#2168	—	13	2	no	—
126	D₄×C₁₆	—	13	1	no	—
127	D₄×C₁₆	—	13	2	no	—
128	#2164	—	13	2	no	—
129	#2162	—	13	1	no	—
130	#2151	—	13	2	no	—
131	C₂×C₄⋊C₁₆	—	13	1	no	—
132	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
133	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
134	Q₈×C₁₃	—	2	2	no	—
135	D₄×C₁₃	—	2	4	no	—
136	D₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	2	1	no	—
139	C₂×C₄×C₁₃	—	2	2	no	—
140	C₂×C₄×C₁₃	—	2	1	no	—
141	C₂×C₄×C₁₃	—	2	2	no	—
142	C₂×C₄×C₁₃	—	2	2	no	—
143	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
148	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
149	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
150	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
152	#1055	—	2	1	no	—
153	#1055	—	2	2	no	—
154	#1055	—	2	1	no	—
155	#1055	—	2	1	no	—
156	#1055	—	2	2	no	—
157	#1055	H(5,𝔽₂)	1	1	yes	characteristic
158	#1055	2₋¹⁺⁴	1	1	yes	characteristic
159	#529	#261	1	1	yes	characteristic
160	C₂×M₃₂	—	13	2	no	—
161	C₂²×C₁₆	—	13	2	no	—
162	C₂²×C₁₆	—	13	2	no	—
163	#446	—	13	1	no	—
164	C₈×Q₈	—	13	1	no	—
165	#436	—	13	1	no	—
166	#435	—	13	2	no	—
167	C₈×D₄	—	13	2	no	—
168	C₈×D₄	—	13	1	no	—
169	#433	—	13	2	no	—
170	#432	—	13	1	no	—
171	#405	—	13	2	no	—
172	C₂×C₄×C₈	—	13	1	no	—
173	C₄⋊C₁₆	—	13	2	no	—
174	C₄⋊C₁₆	—	13	1	no	—
175	C₄⋊C₁₆	—	13	1	no	—
176	C₄⋊C₁₆	—	13	1	no	—
177	C₄⋊C₁₆	—	13	2	no	—
178	C₄⋊C₁₆	—	13	2	no	—
179	C₄⋊C₁₆	—	13	1	no	—
180	C₂²⋊C₁₆	—	13	1	no	—
181	C₂²⋊C₁₆	—	13	1	no	—
182	C₂²⋊C₁₆	—	13	2	no	—
183	C₂²⋊C₁₆	—	13	2	no	—
184	#346	—	13	2	no	—
185	C₄×C₁₆	—	13	1	no	—
186	C₄×C₁₆	—	13	1	no	—
187	C₄×C₁₆	—	13	2	no	—
188	C₂²×C₁₃	—	2	1	no	—
189	C₂²×C₁₃	—	2	1	no	—
190	C₂²×C₁₃	—	2	2	no	—
191	C₂²×C₁₃	—	2	2	no	—
192	C₂²×C₁₃	—	2	2	no	—
193	C₂²×C₁₃	#504	1	2	yes	—
194	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
195	C₄×C₁₃	—	2	1	no	—
196	C₄×C₁₃	—	2	1	no	—
197	C₄×C₁₃	—	2	1	no	—
198	C₄×C₁₃	—	2	2	no	—
199	C₄×C₁₃	—	2	2	no	—
200	C₄×C₁₃	—	2	1	no	—
201	C₄×C₁₃	—	2	2	no	—
202	C₄×C₁₃	#504	1	2	yes	—
203	C₄×C₁₃	#520	1	1	yes	characteristic
204	C₄×C₁₃	#568	1	1	yes	characteristic
205	C₂×M₁₆	—	13	2	no	—
206	C₂²×C₈	—	13	1	no	—
207	C₂²×C₈	—	13	2	no	—
208	C₂²×C₈	—	13	1	no	—
209	#126	#1057	1	2	yes	—
210	#125	#1057	1	1	yes	characteristic
211	#124	#1057	1	1	yes	characteristic
212	#123	#1057	1	2	yes	—
213	C₂²⋊Q₈	#1057	1	1	yes	characteristic
214	#121	#1057	1	1	yes	characteristic
215	C₄×Q₈	C₂×#261	1	1	yes	characteristic
216	C₄×D₄	C₂×#261	1	2	yes	—
217	C₄×D₄	C₂×#261	1	1	yes	characteristic
218	#117	C₂×#261	1	2	yes	—
219	C₂×C₄²	C₂×#261	1	1	yes	characteristic
220	M₃₂	—	52	2	no	—
221	C₂×C₁₆	—	13	1	no	—
222	C₂×C₁₆	—	13	1	no	—
223	C₂×C₁₆	—	13	1	no	—
224	C₂×C₁₆	—	13	2	no	—
225	C₂×C₁₆	—	13	1	no	—
226	C₂×C₁₆	—	13	2	no	—
227	C₂×C₁₆	—	52	2	no	—
228	C₂×C₁₆	—	52	2	no	—
229	C₄⋊C₈	—	13	1	no	—
230	C₄⋊C₈	—	13	1	no	—
231	C₄⋊C₈	—	13	2	no	—
232	C₄⋊C₈	—	13	2	no	—
233	C₂²⋊C₈	—	13	2	no	—
234	C₂²⋊C₈	—	13	2	no	—
235	C₂²⋊C₈	—	13	2	no	—
236	#97	—	13	2	no	—
237	#97	—	13	2	no	—
238	C₄×C₈	—	13	1	no	—
239	C₄×C₈	—	13	2	no	—
240	C₄×C₈	—	13	1	no	—
241	C₄×C₈	—	13	1	no	—
242	C₄×C₈	—	13	1	no	—
243	C₂×C₁₃	—	2	2	no	—
244	C₂×C₁₃	—	4	2	no	—
245	C₂×C₁₃	#2988	1	1	yes	characteristic
246	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
247	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
248	C₂×Q₈	C₂×#1057	1	1	yes	characteristic
249	C₂×D₄	C₂×#1057	1	2	yes	—
250	C₂×D₄	C₂×#1057	1	1	yes	characteristic
251	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
252	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
253	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
254	C₂²×C₄	C₂²×#261	1	2	yes	—
255	M₁₆	—	52	2	no	—
256	C₂×C₈	—	13	1	no	—
257	C₂×C₈	—	13	1	no	—
258	C₂×C₈	—	13	1	no	—
259	C₂×C₈	—	13	1	no	—
260	C₂×C₈	—	13	2	no	—
261	C₂×C₈	—	13	2	no	—
262	C₂×C₈	—	26	1	no	—
263	C₂×C₈	—	26	2	no	—
264	C₂×C₈	—	26	1	no	—
265	C₂×C₈	—	26	2	no	—
266	C₂×C₈	—	52	1	no	—
267	C₄⋊C₄	C₂×#1057	1	2	yes	—
268	C₄⋊C₄	C₂×#1057	1	2	yes	—
269	C₄⋊C₄	C₂×#1057	1	1	yes	characteristic
270	C₄⋊C₄	C₂×#1057	1	2	yes	—
271	C₄⋊C₄	C₂×#1057	1	1	yes	characteristic
272	C₄⋊C₄	C₂×#1057	1	2	yes	—
273	C₂²⋊C₄	C₂×#1057	1	2	yes	—
274	C₂²⋊C₄	C₂×#1057	1	2	yes	—
275	C₂²⋊C₄	C₂×#1057	1	2	yes	—
276	C₂²⋊C₄	C₂×#1057	1	2	yes	—
277	C₂²⋊C₄	C₂×#1057	1	2	yes	—
278	C₄²	C₂×#1057	1	1	yes	characteristic
279	C₄²	C₂×#1057	1	1	yes	characteristic
280	C₄²	C₂²×#261	1	1	yes	characteristic
281	C₄²	C₂²×#261	1	1	yes	characteristic
282	C₄²	C₂²×#261	1	2	yes	—
283	C₁₆	—	26	1	no	—
284	C₁₆	—	26	2	no	—
285	C₁₆	—	26	2	no	—
286	C₁₆	—	26	1	no	—
287	C₁₆	—	26	1	no	—
288	C₁₆	—	26	1	no	—
289	C₁₃	#33508	1	1	yes	characteristic, complemented, Sylow
290	C₂³	C₂²×#1057	1	2	yes	—
291	C₂³	C₂²×#1057	1	1	yes	characteristic
292	Q₈	—	2	2	no	—
293	D₄	—	2	4	no	—
294	D₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
302	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
303	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
304	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
305	C₂×C₄	C₂²×#1057	1	2	yes	—
306	C₂×C₄	C₂²×#1057	1	2	yes	—
307	C₂×C₄	C₂²×#1057	1	2	yes	—
308	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
309	C₂×C₄	C₂³×#261	1	1	yes	characteristic
310	C₈	—	13	1	no	—
311	C₈	—	13	1	no	—
312	C₈	—	26	1	no	—
313	C₈	—	26	1	no	—
314	C₈	—	26	2	no	—
315	C₈	—	26	1	no	—
316	C₈	—	26	2	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	#11719162	1	2	yes	—
323	C₂²	C₂³×#1057	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	2	no	—
331	C₄	#11719162	1	2	yes	—
332	C₄	#11719629	1	1	yes	characteristic
333	C₄	#11719631	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	4	2	no	—
336	C₂	#13131751	1	1	yes	characteristic
337	C₂	C₂×#11719162	1	1	yes	characteristic
338	C₂	C₂×#11719162	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#529	C₄
prime radical	C₁₃×#121	C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	#529	#261
hypocenter	C₁₃	#33508
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #529

Complements

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

N	H ≅ G/N	type
1. #13131251	C₂	semidirect
2. Q₈×#6092	C₂	semidirect
3. #13131220	C₂	semidirect
4. #13131216	C₂	semidirect
5. D₄×#6092	C₂	semidirect
6. D₄×#6092	C₂	semidirect
7. #13131201	C₂	semidirect
8. #13131193	C₂	semidirect
9. #13131171	C₂	semidirect
10. C₂×C₄⋊#6092	C₂	semidirect
289. C₁₃	#33508	semidirect

Maximal subgroups

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

subgroup	index
#13131251	2
Q₈×#6092	2
#13131220	2
#13131216	2
D₄×#6092	2
#13131201	2
#13131193	2
#13131171	2
C₂×C₄⋊#6092	2
#13123812	2
#33508	13