Subgroups — #16785030

Order 3328.

528 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#1057	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#1057	C₂	1	1	yes	maximal, characteristic
3	#13129672	C₂	1	4	yes	maximal
4	C₄⋊Q₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×Q₈×Dic₁₃	—	2	1	no	—
7	C₂×C₄⋊#1057	—	2	1	no	—
8	C₂×C₄⋊#1057	C₂²	1	2	yes	—
9	C₂×C₄×#1057	—	2	1	no	—
10	C₂×C₄×#1057	C₂²	1	2	yes	—
11	#11718891	—	2	1	no	—
12	#11718646	—	2	1	no	—
13	#11718646	C₄	1	2	yes	—
14	C₄⋊C₄×Dic₁₃	—	2	1	no	—
15	C₄⋊C₄×Dic₁₃	C₂²	1	2	yes	—
16	#11718596	C₄	1	1	yes	characteristic
17	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
18	#11718290	—	2	1	no	—
19	#11718290	—	2	1	no	—
20	C₂²×Q₈×C₁₃	—	2	1	no	—
21	C₂²×#1057	—	2	2	no	—
22	C₂²×#1057	—	2	2	no	—
23	C₁₃×C₄⋊Q₈	—	2	2	no	—
24	C₁₃×C₄⋊Q₈	C₈	1	4	yes	complemented
25	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
26	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	Q₈×Dic₁₃	—	4	2	no	—
29	Q₈×Dic₁₃	—	4	2	no	—
30	C₂×C₄⋊Dic₁₃	—	2	1	no	—
31	C₂×C₄⋊Dic₁₃	—	2	1	no	—
32	C₂×C₄⋊Dic₁₃	—	2	2	no	—
33	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
34	C₂×C₄×Dic₁₃	—	2	2	no	—
35	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
37	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
38	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
39	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
40	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
41	C₄⋊#1057	—	2	4	no	—
42	C₄⋊#1057	—	2	2	no	—
43	C₄⋊#1057	Q₈	1	4	yes	complemented
44	C₄×#1057	—	2	4	no	—
45	C₄×#1057	—	4	2	no	—
46	#88071	—	2	2	no	—
47	#88071	—	2	2	no	—
48	#9865	—	13	1	no	maximal, Sylow
49	C₂×Q₈×C₁₃	—	2	4	no	—
50	C₂×Q₈×C₁₃	—	4	2	no	—
51	C₂×Q₈×C₁₃	—	4	2	no	—
52	C₂²×C₄×C₁₃	—	2	2	no	—
53	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₁₃	—	2	2	no	—
57	C₂²×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×#1057	—	2	4	no	—
62	C₂×#1057	—	2	4	no	—
63	C₂×#1057	—	4	2	no	—
64	C₂×#1057	—	4	2	no	—
65	C₂×#1057	—	4	2	no	—
66	C₂×#1057	—	4	2	no	—
67	C₁₃×C₄⋊C₄	—	2	4	no	—
68	C₁₃×C₄⋊C₄	—	2	2	no	—
69	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
70	C₄²×C₁₃	C₂×C₈	1	2	yes	—
71	C₄²×C₁₃	M₁₆	1	2	yes	—
72	C₄⋊Dic₁₃	—	2	4	no	—
73	C₄⋊Dic₁₃	—	4	2	no	—
74	C₄⋊Dic₁₃	—	4	2	no	—
75	C₄⋊Dic₁₃	—	4	4	no	—
76	C₄×Dic₁₃	—	2	2	no	—
77	C₄×Dic₁₃	—	2	2	no	—
78	C₄×Dic₁₃	—	2	2	no	—
79	C₄×Dic₁₃	—	4	1	no	—
80	C₄×Dic₁₃	—	4	2	no	—
81	C₄×Dic₁₃	—	4	2	no	—
82	C₄×Dic₁₃	D₈	1	2	yes	—
83	C₄×Dic₁₃	SD₁₆	1	2	yes	—
84	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
85	C₄×Dic₁₃	#41	1	2	yes	—
86	C₄×C₄⋊Q₈	—	13	1	no	—
87	C₄×C₄⋊C₈	—	13	1	no	—
88	C₄×C₄⋊C₈	—	13	1	no	—
89	#1279	—	13	4	no	—
90	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	Q₈×C₁₃	—	4	8	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	2	4	no	—
95	C₂×C₄×C₁₃	—	4	2	no	—
96	C₂×C₄×C₁₃	—	4	2	no	—
97	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
98	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
99	C₂×C₄×C₁₃	#102	1	2	yes	—
100	C₂×C₄×C₁₃	#104	1	2	yes	—
101	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
102	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₁₃	#131	1	2	yes	—
105	C₂×Dic₁₃	—	2	1	no	—
106	C₂×Dic₁₃	—	2	1	no	—
107	C₂×Dic₁₃	—	2	1	no	—
108	C₂×Dic₁₃	—	2	1	no	—
109	C₂×Dic₁₃	—	2	1	no	—
110	C₂×Dic₁₃	—	2	1	no	—
111	C₂×Dic₁₃	—	2	1	no	—
112	C₂×Dic₁₃	—	4	2	no	—
113	C₂×Dic₁₃	—	4	2	no	—
114	C₂×Dic₁₃	—	4	2	no	—
115	C₂×Dic₁₃	#102	1	2	yes	—
116	C₂×Dic₁₃	C₂²⋊Q₈	1	2	yes	—
117	C₂×Dic₁₃	#124	1	1	yes	characteristic
118	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
119	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
120	C₂×Dic₁₃	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×Dic₁₃	#137	1	2	yes	—
122	#1057	—	4	4	no	—
123	#1057	—	4	4	no	—
124	C₂×C₄⋊Q₈	#261	1	1	yes	characteristic
125	C₂×C₄×Q₈	—	26	1	no	—
126	C₂×C₄⋊C₈	—	13	2	no	—
127	C₂×C₄⋊C₈	—	26	1	no	—
128	C₂×C₄×C₈	—	13	2	no	—
129	C₂×C₄×C₈	—	26	1	no	—
130	#391	—	26	1	no	—
131	#389	—	13	2	no	—
132	#389	—	26	1	no	—
133	#384	—	13	1	no	—
134	C₄×C₄⋊C₄	—	13	2	no	—
135	C₄×C₄⋊C₄	—	26	1	no	—
136	C₄³	—	13	1	no	—
137	#336	—	26	1	no	—
138	#336	—	26	1	no	—
139	C₂²×C₁₃	#391	1	1	yes	characteristic
140	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
141	C₂²×C₁₃	#408	1	1	yes	characteristic
142	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
143	C₂²×C₁₃	#419	1	1	yes	characteristic
144	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
145	C₂²×C₁₃	#421	1	1	yes	characteristic
146	C₄×C₁₃	—	2	2	no	—
147	C₄×C₁₃	—	4	4	no	—
148	C₄×C₁₃	D₄⋊C₈	1	4	yes	—
149	C₄×C₁₃	C₈×Q₈	1	2	yes	—
150	C₄×C₁₃	#446	1	2	yes	—
151	Dic₁₃	—	2	1	no	—
152	Dic₁₃	—	2	1	no	—
153	Dic₁₃	—	2	2	no	—
154	Dic₁₃	—	4	1	no	—
155	Dic₁₃	—	8	2	no	—
156	Dic₁₃	#474	1	2	yes	—
157	Dic₁₃	#475	1	2	yes	—
158	C₂²×Q₈	—	2	1	no	—
159	C₂²×C₈	—	26	2	no	—
160	C₂²×C₈	—	26	2	no	—
161	C₄⋊Q₈	—	2	2	no	—
162	C₄⋊Q₈	#1057	1	4	yes	complemented
163	C₄×Q₈	—	52	2	no	—
164	C₄×Q₈	—	52	2	no	—
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₄⋊C₄	—	13	2	no	—
167	C₂×C₄⋊C₄	—	26	2	no	—
168	C₂×C₄⋊C₄	—	26	1	no	—
169	C₂×C₄⋊C₄	—	26	1	no	—
170	C₂×C₄⋊C₄	C₂×#261	1	2	yes	—
171	C₂×C₄²	—	13	1	no	—
172	C₂×C₄²	—	13	1	no	—
173	C₂×C₄²	—	13	1	no	—
174	C₂×C₄²	—	13	1	no	—
175	C₂×C₄²	—	13	1	no	—
176	C₂×C₄²	—	13	1	no	—
177	C₂×C₄²	—	26	2	no	—
178	C₂×C₄²	C₂×#261	1	1	yes	characteristic
179	C₄⋊C₈	—	13	4	no	—
180	C₄⋊C₈	—	26	2	no	—
181	C₄⋊C₈	—	26	4	no	—
182	C₄×C₈	—	26	4	no	—
183	C₄×C₈	—	52	2	no	—
184	#95	—	26	2	no	—
185	#95	—	26	2	no	—
186	C₂×C₁₃	C₂×D₄⋊C₈	1	1	yes	characteristic
187	C₂×C₁₃	#1479	1	1	yes	characteristic
188	C₂×C₁₃	#1480	1	1	yes	characteristic
189	C₂×C₁₃	#1485	1	1	yes	characteristic
190	C₂×C₁₃	#1984	1	1	yes	characteristic
191	C₂×C₁₃	#1986	1	1	yes	characteristic
192	C₂×C₁₃	#1988	1	1	yes	characteristic
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	—	4	2	no	—
195	C₂×Q₈	—	4	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	13	1	no	—
198	C₂²×C₄	—	13	1	no	—
199	C₂²×C₄	—	13	1	no	—
200	C₂²×C₄	—	13	1	no	—
201	C₂²×C₄	—	26	2	no	—
202	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
203	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
204	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
205	C₂×C₈	—	26	4	no	—
206	C₂×C₈	—	26	4	no	—
207	C₂×C₈	—	52	2	no	—
208	C₂×C₈	—	52	2	no	—
209	C₂×C₈	—	52	2	no	—
210	C₂×C₈	—	52	2	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	26	4	no	—
214	C₄⋊C₄	—	52	2	no	—
215	C₄⋊C₄	—	52	4	no	—
216	C₄⋊C₄	—	52	2	no	—
217	C₄⋊C₄	C₂×#1057	1	4	yes	—
218	C₄²	—	13	2	no	—
219	C₄²	—	13	2	no	—
220	C₄²	—	13	2	no	—
221	C₄²	—	13	2	no	—
222	C₄²	—	26	2	no	—
223	C₄²	—	26	2	no	—
224	C₄²	—	26	2	no	—
225	C₄²	—	52	1	no	—
226	C₄²	—	52	2	no	—
227	C₄²	—	52	2	no	—
228	C₄²	C₂×#1057	1	2	yes	—
229	C₄²	#6122	1	2	yes	—
230	C₁₃	#9865	1	1	yes	characteristic, complemented, Sylow
231	C₂³	C₂×C₂²⋊#261	1	1	yes	characteristic
232	Q₈	—	4	8	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	13	1	no	—
239	C₂×C₄	—	13	1	no	—
240	C₂×C₄	—	13	2	no	—
241	C₂×C₄	—	13	2	no	—
242	C₂×C₄	—	13	2	no	—
243	C₂×C₄	—	13	1	no	—
244	C₂×C₄	—	13	1	no	—
245	C₂×C₄	—	26	1	no	—
246	C₂×C₄	—	26	1	no	—
247	C₂×C₄	—	26	1	no	—
248	C₂×C₄	—	26	1	no	—
249	C₂×C₄	—	26	1	no	—
250	C₂×C₄	—	26	1	no	—
251	C₂×C₄	—	26	1	no	—
252	C₂×C₄	—	52	2	no	—
253	C₂×C₄	—	52	2	no	—
254	C₂×C₄	—	52	2	no	—
255	C₂×C₄	#88115	1	2	yes	—
256	C₂×C₄	#88116	1	2	yes	—
257	C₂×C₄	C₂²⋊#1057	1	2	yes	—
258	C₂×C₄	C₂²⋊#1057	1	2	yes	—
259	C₂×C₄	#88240	1	2	yes	—
260	C₂×C₄	Q₈×#261	1	2	yes	—
261	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
262	C₂×C₄	C₂×#6122	1	1	yes	characteristic
263	C₈	—	52	4	no	—
264	C₈	—	52	4	no	—
265	C₂²	C₂×#88115	1	1	yes	characteristic
266	C₂²	C₂×#88116	1	1	yes	characteristic
267	C₂²	#11719153	1	1	yes	characteristic
268	C₂²	#11719155	1	1	yes	characteristic
269	C₂²	#11719159	1	1	yes	characteristic
270	C₂²	(C₂×#261)⋊Q₈	1	1	yes	characteristic
271	C₂²	C₂×C₂²⋊#1057	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	4	4	no	—
274	C₄	—	13	2	no	—
275	C₄	—	13	2	no	—
276	C₄	—	26	1	no	—
277	C₄	—	26	1	no	—
278	C₄	—	26	2	no	—
279	C₄	—	52	1	no	—
280	C₄	—	104	2	no	—
281	C₄	D₄⋊#1057	1	4	yes	—
282	C₄	Q₈×#1057	1	2	yes	—
283	C₄	#11719158	1	2	yes	—
284	C₂	C₂×D₄⋊#1057	1	1	yes	characteristic
285	C₂	#13130138	1	1	yes	characteristic
286	C₂	#13130143	1	1	yes	characteristic
287	C₂	#13130160	1	1	yes	characteristic
288	C₂	#13131005	1	1	yes	characteristic
289	C₂	#13131012	1	1	yes	characteristic
290	C₂	#13131015	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₁₃×C₄⋊Q₈	C₄
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	C₂×C₄⋊Q₈	#261
hypocenter	C₁₃	#9865
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
24. C₁₃×C₄⋊Q₈	C₈	semidirect
43. C₄⋊#1057	Q₈	semidirect
162. C₄⋊Q₈	#1057	semidirect
230. C₁₃	#9865	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#1057	2
#13129672	2
C₄⋊Q₈×Dic₁₃	2
#9865	13