Subgroups — #16807254

Order 3328.

506 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131253	C₂	1	2	yes	maximal, complemented
2	#13131252	C₂	1	1	yes	maximal, characteristic, complemented
3	#13131251	C₂	1	1	yes	maximal, characteristic, complemented
4	#13131250	C₂	1	1	yes	maximal, characteristic, complemented
5	#13131250	C₂	1	2	yes	maximal, complemented
6	Q₈×#6092	C₂	1	1	yes	maximal, characteristic, complemented
7	#13131172	C₂	1	2	yes	maximal
8	#13131169	C₂	1	1	yes	maximal, characteristic
9	#13131046	C₂	1	2	yes	maximal
10	C₄×D₁₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
11	#13123767	C₂	1	1	yes	maximal, characteristic
12	#11719283	C₄	1	1	yes	characteristic
13	C₂×#88098	C₂²	1	2	yes	—
14	C₂×D₁₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂×D₁₃⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₁₃×#446	C₄	1	1	yes	characteristic
17	Q₈×#1055	C₂²	1	1	yes	characteristic
18	#11718500	C₄	1	2	yes	—
19	#11718498	C₄	1	1	yes	characteristic
20	#11718491	C₂²	1	1	yes	characteristic
21	#11718491	C₂²	1	2	yes	—
22	#11718369	C₂²	1	2	yes	—
23	#11718362	C₄	1	2	yes	—
24	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
25	#11718340	C₄	1	1	yes	characteristic
26	#11718260	C₂²	1	2	yes	—
27	#11718260	C₂²	1	2	yes	—
28	#11718260	C₂²	1	2	yes	—
29	#11718258	C₂²	1	2	yes	—
30	#11718257	C₂²	1	1	yes	characteristic
31	#11718257	C₂²	1	1	yes	characteristic
32	#11718257	C₂²	1	2	yes	—
33	#11718231	C₂²	1	2	yes	—
34	C₄⋊#6092	C₂²	1	1	yes	characteristic
35	C₄⋊#6092	C₂²	1	1	yes	characteristic
36	C₄⋊#6092	C₂²	1	2	yes	—
37	C₄⋊#6092	C₂²	1	2	yes	—
38	C₄×#6092	C₂²	1	1	yes	characteristic
39	C₄×#6092	C₂²	1	1	yes	characteristic
40	C₄×#6092	C₂²	1	2	yes	—
41	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	#88200	C₈	1	1	yes	characteristic
43	#88198	C₈	1	1	yes	characteristic
44	C₂×#6094	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
47	#88151	C₈	1	1	yes	characteristic
48	#88150	C₈	1	2	yes	—
49	#88143	C₈	1	2	yes	—
50	#88142	C₈	1	1	yes	characteristic
51	#88126	C₂×C₄	1	2	yes	—
52	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
53	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
54	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
55	C₂×#6092	C₂³	1	1	yes	characteristic
56	C₂×#6092	C₂³	1	1	yes	characteristic
57	C₂×#6092	C₂³	1	2	yes	—
58	C₂×#6092	C₂³	1	2	yes	—
59	C₂×#6092	C₂³	1	1	yes	characteristic
60	C₂×#6092	C₂³	1	1	yes	characteristic
61	#88098	—	2	4	no	—
62	D₁₃⋊C₁₆	—	2	2	no	—
63	D₁₃⋊C₁₆	Q₈	1	4	yes	complemented
64	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
65	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
66	C₁₃×#97	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
68	#88055	C₂×C₄	1	2	yes	—
69	#88054	C₂×C₄	1	2	yes	—
70	#88054	C₂×C₄	1	2	yes	—
71	#88054	C₂×C₄	1	2	yes	—
72	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
73	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
74	C₈×Dic₁₃	C₂³	1	2	yes	—
75	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
76	C₄⋊#1055	C₂×C₄	1	2	yes	—
77	C₄×#1055	C₂³	1	1	yes	characteristic
78	C₄×#1055	C₂³	1	2	yes	—
79	#33474	—	13	1	no	maximal, Sylow
80	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
85	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
90	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
91	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
93	#6103	C₂×C₈	1	2	yes	—
94	#6103	C₂×C₈	1	2	yes	—
95	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
96	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
97	#6101	C₂×C₈	1	2	yes	—
98	#6101	C₂×C₈	1	2	yes	—
99	#6101	C₂×C₈	1	2	yes	—
100	#6101	C₂×C₈	1	2	yes	—
101	#6101	C₂×C₈	1	2	yes	—
102	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
103	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
104	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
106	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
107	C₂×#1055	C₂²×C₄	1	2	yes	—
108	C₂×#1055	C₂⁴	1	1	yes	characteristic
109	#6094	—	2	4	no	—
110	C₈×D₁₃	—	2	2	no	—
111	C₈×D₁₃	C₂×Q₈	1	2	yes	—
112	C₈×D₁₃	#41	1	2	yes	—
113	#6092	—	2	1	no	—
114	#6092	—	2	1	no	—
115	#6092	—	2	2	no	—
116	#6092	—	2	2	no	—
117	#6092	C₂×Q₈	1	2	yes	—
118	#6092	C₂×Q₈	1	2	yes	—
119	#2960	—	13	1	no	—
120	#2183	—	13	1	no	—
121	#2183	—	13	1	no	—
122	#2183	—	13	2	no	—
123	Q₈×C₁₆	—	13	1	no	—
124	Q₈×C₁₆	—	13	1	no	—
125	Q₈×C₁₆	—	13	2	no	—
126	#2151	—	13	2	no	—
127	#2150	—	13	1	no	—
128	#2108	—	13	2	no	—
129	C₂×C₄×C₁₆	—	13	1	no	—
130	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
131	Q₈×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
134	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
137	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
138	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
139	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
140	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
141	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
142	C₄×D₁₃	—	2	2	no	—
143	C₄×D₁₃	—	2	2	no	—
144	C₄×D₁₃	C₄×Q₈	1	2	yes	—
145	C₄×D₁₃	#131	1	2	yes	—
146	Dic₂₆	—	2	2	no	—
147	Dic₂₆	—	2	4	no	—
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	—	2	2	no	—
150	C₈×C₁₃	C₄×Q₈	1	2	yes	—
151	#1055	—	2	1	no	—
152	#1055	—	2	2	no	—
153	#1055	C₂²×Q₈	1	1	yes	characteristic
154	#1055	C₂×#41	1	1	yes	characteristic
155	#533	—	13	1	no	—
156	C₂×M₃₂	—	13	2	no	—
157	C₂²×C₁₆	—	13	1	no	—
158	C₂²×C₁₆	—	13	1	no	—
159	#446	—	13	2	no	—
160	#446	—	13	1	no	—
161	#446	#261	1	1	yes	characteristic
162	C₈×Q₈	—	13	1	no	—
163	C₈×Q₈	—	13	2	no	—
164	C₈×Q₈	—	13	1	no	—
165	#424	—	13	2	no	—
166	#423	—	13	1	no	—
167	#405	—	13	2	no	—
168	C₂×C₄×C₈	—	13	1	no	—
169	C₄⋊C₁₆	—	13	1	no	—
170	C₄⋊C₁₆	—	13	1	no	—
171	C₄⋊C₁₆	—	13	2	no	—
172	C₄⋊C₁₆	—	13	2	no	—
173	C₄⋊C₁₆	—	13	2	no	—
174	C₄⋊C₁₆	—	13	2	no	—
175	C₄⋊C₁₆	—	13	2	no	—
176	#346	—	13	2	no	—
177	#346	—	13	2	no	—
178	C₄×C₁₆	—	13	1	no	—
179	C₄×C₁₆	—	13	1	no	—
180	C₄×C₁₆	—	13	1	no	—
181	C₄×C₁₆	—	13	1	no	—
182	C₄×C₁₆	—	13	2	no	—
183	C₄×C₁₆	—	13	2	no	—
184	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
185	C₂×D₁₃	—	2	1	no	—
186	C₂×D₁₃	—	2	1	no	—
187	C₂×D₁₃	C₈×Q₈	1	2	yes	—
188	C₄×C₁₃	—	2	2	no	—
189	C₄×C₁₃	—	2	1	no	—
190	C₄×C₁₃	—	2	2	no	—
191	C₄×C₁₃	#504	1	2	yes	—
192	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
193	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
194	Dic₁₃	—	2	2	no	—
195	Dic₁₃	—	2	2	no	—
196	Dic₁₃	—	2	1	no	—
197	Dic₁₃	C₈×Q₈	1	2	yes	—
198	Dic₁₃	#504	1	2	yes	—
199	C₂×M₁₆	—	13	2	no	—
200	C₂²×C₈	—	13	1	no	—
201	C₂²×C₈	—	13	1	no	—
202	C₄⋊Q₈	—	13	1	no	—
203	C₄⋊Q₈	—	13	1	no	—
204	#125	—	13	2	no	—
205	C₂²⋊Q₈	—	13	1	no	—
206	C₂²⋊Q₈	—	13	2	no	—
207	C₂²⋊Q₈	—	13	1	no	—
208	C₄×Q₈	—	13	2	no	—
209	C₄×Q₈	—	13	1	no	—
210	C₄×Q₈	C₂×#261	1	1	yes	characteristic
211	#117	—	13	2	no	—
212	C₂×C₄²	—	13	1	no	—
213	M₃₂	—	26	4	no	—
214	C₂×C₁₆	—	13	4	no	—
215	C₂×C₁₆	—	13	1	no	—
216	C₂×C₁₆	—	13	1	no	—
217	C₂×C₁₆	—	13	1	no	—
218	C₂×C₁₆	—	13	1	no	—
219	C₂×C₁₆	—	13	2	no	—
220	C₂×C₁₆	—	13	2	no	—
221	C₂×C₁₆	—	26	2	no	—
222	C₄⋊C₈	—	13	2	no	—
223	C₄⋊C₈	—	13	2	no	—
224	C₄⋊C₈	—	13	2	no	—
225	C₄⋊C₈	—	13	1	no	—
226	C₄⋊C₈	—	13	2	no	—
227	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
228	C₄⋊C₈	C₂×#261	1	2	yes	—
229	#97	—	13	2	no	—
230	#97	C₂×#261	1	2	yes	—
231	C₄×C₈	—	13	1	no	—
232	C₄×C₈	—	13	2	no	—
233	C₄×C₈	—	13	1	no	—
234	C₄×C₈	—	13	1	no	—
235	C₄×C₈	—	13	2	no	—
236	C₄×C₈	C₂×#261	1	1	yes	characteristic
237	C₂×C₁₃	C₂×C₈×Q₈	1	1	yes	characteristic
238	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
239	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
240	D₁₃	—	2	2	no	—
241	C₂×Q₈	—	13	2	no	—
242	C₂×Q₈	—	13	1	no	—
243	C₂×Q₈	D₁₃⋊C₈	1	1	yes	characteristic
244	C₂²×C₄	—	13	1	no	—
245	C₂²×C₄	—	13	1	no	—
246	C₂²×C₄	—	13	1	no	—
247	M₁₆	—	26	4	no	—
248	C₂×C₈	—	13	1	no	—
249	C₂×C₈	—	13	2	no	—
250	C₂×C₈	—	13	2	no	—
251	C₂×C₈	—	13	1	no	—
252	C₂×C₈	—	13	2	no	—
253	C₂×C₈	—	26	2	no	—
254	C₂×C₈	C₂²×#261	1	1	yes	characteristic
255	C₂×C₈	C₂²×#261	1	2	yes	—
256	C₂×C₈	C₂²×#261	1	1	yes	characteristic
257	C₄⋊C₄	—	13	2	no	—
258	C₄⋊C₄	—	13	2	no	—
259	C₄⋊C₄	—	13	2	no	—
260	C₄⋊C₄	—	13	2	no	—
261	C₄⋊C₄	—	13	2	no	—
262	C₄⋊C₄	—	13	2	no	—
263	C₄⋊C₄	—	13	1	no	—
264	C₄⋊C₄	D₁₃⋊C₈	1	2	yes	—
265	C₄⋊C₄	D₁₃⋊C₈	1	1	yes	characteristic
266	C₂²⋊C₄	—	13	2	no	—
267	C₂²⋊C₄	—	13	2	no	—
268	C₄²	—	13	1	no	—
269	C₄²	—	13	1	no	—
270	C₄²	—	13	1	no	—
271	C₄²	—	13	2	no	—
272	C₄²	C₂²×#261	1	1	yes	characteristic
273	C₄²	C₂²×#261	1	2	yes	—
274	C₁₆	—	13	2	no	—
275	C₁₆	—	13	2	no	—
276	C₁₆	—	26	1	no	—
277	C₁₆	—	26	1	no	—
278	C₁₆	—	26	2	no	—
279	C₁₆	—	26	2	no	—
280	C₁₃	#33474	1	1	yes	characteristic, complemented, Sylow
281	C₂³	—	13	1	no	—
282	Q₈	—	2	2	no	—
283	Q₈	—	26	4	no	—
284	Q₈	—	26	2	no	—
285	C₂×C₄	—	13	1	no	—
286	C₂×C₄	—	13	1	no	—
287	C₂×C₄	—	13	2	no	—
288	C₂×C₄	—	13	2	no	—
289	C₂×C₄	—	13	1	no	—
290	C₂×C₄	—	13	2	no	—
291	C₂×C₄	—	13	2	no	—
292	C₂×C₄	—	26	2	no	—
293	C₂×C₄	—	26	2	no	—
294	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
295	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
296	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
297	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
298	C₂×C₄	C₂³×#261	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	2	no	—
301	C₈	—	13	1	no	—
302	C₈	—	13	1	no	—
303	C₈	—	26	1	no	—
304	C₈	—	26	2	no	—
305	C₈	Q₈×#261	1	2	yes	—
306	C₂²	—	13	2	no	—
307	C₂²	—	26	1	no	—
308	C₂²	—	26	1	no	—
309	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	13	2	no	—
314	C₄	—	13	2	no	—
315	C₄	—	26	2	no	—
316	C₄	—	26	1	no	—
317	C₄	—	26	2	no	—
318	C₄	#11719096	1	2	yes	—
319	C₄	C₂×Q₈×#261	1	1	yes	characteristic
320	C₄	D₁₃⋊#131	1	1	yes	characteristic
321	C₂	—	26	2	no	—
322	C₂	Q₈×D₁₃⋊C₈	1	1	yes	characteristic
323	C₂	D₁₃⋊#504	1	1	yes	characteristic
324	C₂	C₂×#11719096	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#446	C₄
prime radical	C₂²×D₁₃	C₂²×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	#446	#261
hypocenter	C₁₃	#33474
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
1. #13131253	C₂	semidirect
2. #13131252	C₂	semidirect
3. #13131251	C₂	semidirect
4. #13131250	C₂	semidirect
5. #13131250	C₂	semidirect
6. Q₈×#6092	C₂	semidirect
63. D₁₃⋊C₁₆	Q₈	semidirect
280. C₁₃	#33474	semidirect

Maximal subgroups

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

subgroup	index
#13131253	2
#13131252	2
#13131251	2
#13131250	2
Q₈×#6092	2
#13131172	2
#13131169	2
#13131046	2
C₄×D₁₃⋊C₁₆	2
#13123767	2
#33474	13