Subgroups — #16785056

Order 3328.

490 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13130751	C₂	1	1	yes	maximal, characteristic
2	#13130550	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	C₂²	1	2	yes	—
8	C₂×C₄×#1057	C₂²	1	2	yes	—
9	#11718891	—	2	1	no	—
10	#11718646	—	2	1	no	—
11	#11718646	C₄	1	2	yes	—
12	C₄⋊C₄×Dic₁₃	—	2	1	no	—
13	C₄⋊C₄×Dic₁₃	C₂²	1	2	yes	—
14	#11718596	C₄	1	1	yes	characteristic
15	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
16	#11718290	—	2	1	no	—
17	#11718290	—	2	1	no	—
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	4	no	—
29	C₂×C₄⋊Dic₁₃	—	2	1	no	—
30	C₂×C₄⋊Dic₁₃	—	2	1	no	—
31	C₂×C₄⋊Dic₁₃	—	2	2	no	—
32	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
33	C₂×C₄×Dic₁₃	—	2	2	no	—
34	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₁₃	D₄	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	C₂³	1	1	yes	characteristic
40	C₄⋊#1057	—	2	4	no	—
41	C₄×#1057	—	2	2	no	—
42	C₄×#1057	Q₈	1	4	yes	complemented
43	#88071	—	2	2	no	—
44	#88071	—	2	2	no	—
45	#9873	—	13	1	no	maximal, Sylow
46	C₂×Q₈×C₁₃	—	2	4	no	—
47	C₂×Q₈×C₁₃	—	4	2	no	—
48	C₂×Q₈×C₁₃	—	4	2	no	—
49	C₂²×C₄×C₁₃	—	2	2	no	—
50	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×Dic₁₃	—	2	2	no	—
54	C₂²×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂×#1057	—	2	4	no	—
59	C₂×#1057	—	4	2	no	—
60	C₂×#1057	—	4	2	no	—
61	C₂×#1057	—	4	2	no	—
62	C₂×#1057	—	4	2	no	—
63	C₂×#1057	—	4	2	no	—
64	C₁₃×C₄⋊C₄	—	2	4	no	—
65	C₁₃×C₄⋊C₄	—	2	2	no	—
66	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
67	C₄²×C₁₃	C₂×C₈	1	2	yes	—
68	C₄²×C₁₃	M₁₆	1	2	yes	—
69	C₄⋊Dic₁₃	—	2	4	no	—
70	C₄⋊Dic₁₃	—	4	4	no	—
71	C₄⋊Dic₁₃	—	4	2	no	—
72	C₄⋊Dic₁₃	—	4	2	no	—
73	C₄×Dic₁₃	—	2	2	no	—
74	C₄×Dic₁₃	—	2	2	no	—
75	C₄×Dic₁₃	—	2	2	no	—
76	C₄×Dic₁₃	—	4	1	no	—
77	C₄×Dic₁₃	—	4	1	no	—
78	C₄×Dic₁₃	—	4	4	no	—
79	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
80	C₄×Dic₁₃	#41	1	2	yes	—
81	C₄×C₄⋊Q₈	—	13	1	no	—
82	#1831	—	13	1	no	—
83	#1744	—	13	1	no	—
84	#1279	—	13	4	no	—
85	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	Q₈×C₁₃	—	4	8	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	4	no	—
92	C₂×C₄×C₁₃	—	4	2	no	—
93	C₂×C₄×C₁₃	—	4	2	no	—
94	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
95	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
96	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
97	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₁₃	#131	1	2	yes	—
100	C₂×Dic₁₃	—	2	1	no	—
101	C₂×Dic₁₃	—	2	1	no	—
102	C₂×Dic₁₃	—	2	1	no	—
103	C₂×Dic₁₃	—	2	1	no	—
104	C₂×Dic₁₃	—	2	1	no	—
105	C₂×Dic₁₃	—	2	1	no	—
106	C₂×Dic₁₃	—	2	1	no	—
107	C₂×Dic₁₃	—	2	1	no	—
108	C₂×Dic₁₃	—	4	2	no	—
109	C₂×Dic₁₃	—	4	2	no	—
110	C₂×Dic₁₃	—	4	2	no	—
111	C₂×Dic₁₃	C₂²⋊Q₈	1	2	yes	—
112	C₂×Dic₁₃	#124	1	1	yes	characteristic
113	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×Dic₁₃	AGL(1,ℤ/8)	1	2	yes	—
115	C₂×Dic₁₃	#137	1	2	yes	—
116	#1057	—	4	4	no	—
117	#1057	—	8	2	no	—
118	C₂×C₄⋊Q₈	#261	1	1	yes	characteristic
119	C₂×C₄×Q₈	—	26	1	no	—
120	C₂×C₄⋊C₈	—	13	2	no	—
121	C₂×C₄×C₈	—	13	2	no	—
122	#391	—	26	1	no	—
123	#389	—	13	2	no	—
124	#389	—	26	1	no	—
125	#384	—	13	1	no	—
126	C₄×C₄⋊C₄	—	13	2	no	—
127	C₄×C₄⋊C₄	—	26	1	no	—
128	C₄³	—	13	1	no	—
129	#336	—	26	1	no	—
130	#336	—	26	1	no	—
131	#336	—	26	1	no	—
132	#336	—	26	1	no	—
133	C₂²×C₁₃	#391	1	1	yes	characteristic
134	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
135	C₂²×C₁₃	#408	1	1	yes	characteristic
136	C₂²×C₁₃	#418	1	1	yes	characteristic
137	C₂²×C₁₃	#419	1	1	yes	characteristic
138	C₂²×C₁₃	#421	1	1	yes	characteristic
139	C₂²×C₁₃	#421	1	1	yes	characteristic
140	C₄×C₁₃	—	2	2	no	—
141	C₄×C₁₃	—	2	2	no	—
142	C₄×C₁₃	—	4	4	no	—
143	C₄×C₁₃	C₈×Q₈	1	2	yes	—
144	C₄×C₁₃	#446	1	2	yes	—
145	Dic₁₃	—	2	2	no	—
146	Dic₁₃	—	4	1	no	—
147	Dic₁₃	—	4	1	no	—
148	Dic₁₃	—	8	2	no	—
149	Dic₁₃	#489	1	2	yes	—
150	Dic₁₃	#501	1	2	yes	—
151	C₂²×Q₈	—	2	1	no	—
152	C₂²×C₈	—	26	2	no	—
153	C₂²×C₈	—	26	2	no	—
154	C₄⋊Q₈	—	2	2	no	—
155	C₄⋊Q₈	#1057	1	4	yes	complemented
156	C₄×Q₈	—	52	4	no	—
157	C₂×C₄⋊C₄	—	2	1	no	—
158	C₂×C₄⋊C₄	—	13	2	no	—
159	C₂×C₄⋊C₄	—	26	2	no	—
160	C₂×C₄⋊C₄	—	26	1	no	—
161	C₂×C₄⋊C₄	—	26	1	no	—
162	C₂×C₄⋊C₄	C₂×#261	1	2	yes	—
163	C₂×C₄²	—	13	1	no	—
164	C₂×C₄²	—	13	1	no	—
165	C₂×C₄²	—	13	1	no	—
166	C₂×C₄²	—	13	1	no	—
167	C₂×C₄²	—	13	1	no	—
168	C₂×C₄²	—	13	1	no	—
169	C₂×C₄²	—	26	2	no	—
170	C₂×C₄²	C₂×#261	1	1	yes	characteristic
171	C₄⋊C₈	—	26	4	no	—
172	C₄×C₈	—	13	4	no	—
173	C₄×C₈	—	26	2	no	—
174	#95	—	26	2	no	—
175	#95	—	26	2	no	—
176	C₂×C₁₃	#1478	1	1	yes	characteristic
177	C₂×C₁₃	#1479	1	1	yes	characteristic
178	C₂×C₁₃	#1496	1	1	yes	characteristic
179	C₂×C₁₃	#1497	1	1	yes	characteristic
180	C₂×C₁₃	#1984	1	1	yes	characteristic
181	C₂×C₁₃	#1993	1	1	yes	characteristic
182	C₂×C₁₃	#1995	1	1	yes	characteristic
183	C₂×Q₈	—	2	4	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	13	1	no	—
188	C₂²×C₄	—	13	1	no	—
189	C₂²×C₄	—	13	1	no	—
190	C₂²×C₄	—	13	1	no	—
191	C₂²×C₄	—	26	2	no	—
192	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
193	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
194	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
195	C₂×C₈	—	26	4	no	—
196	C₂×C₈	—	52	2	no	—
197	C₂×C₈	—	52	2	no	—
198	C₂×C₈	—	52	2	no	—
199	C₂×C₈	—	52	2	no	—
200	C₂×C₈	—	52	2	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	26	4	no	—
204	C₄⋊C₄	—	52	2	no	—
205	C₄⋊C₄	—	52	2	no	—
206	C₄⋊C₄	—	52	4	no	—
207	C₄⋊C₄	C₂×#1057	1	4	yes	—
208	C₄²	—	13	2	no	—
209	C₄²	—	13	2	no	—
210	C₄²	—	26	2	no	—
211	C₄²	—	26	2	no	—
212	C₄²	—	26	2	no	—
213	C₄²	—	52	1	no	—
214	C₄²	—	52	1	no	—
215	C₄²	—	52	4	no	—
216	C₄²	C₂×#1057	1	2	yes	—
217	C₄²	#6122	1	2	yes	—
218	C₁₃	#9873	1	1	yes	characteristic, complemented, Sylow
219	C₂³	C₂×C₂²⋊#261	1	1	yes	characteristic
220	Q₈	—	4	8	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	13	2	no	—
229	C₂×C₄	—	13	2	no	—
230	C₂×C₄	—	13	1	no	—
231	C₂×C₄	—	13	1	no	—
232	C₂×C₄	—	13	2	no	—
233	C₂×C₄	—	26	1	no	—
234	C₂×C₄	—	26	1	no	—
235	C₂×C₄	—	26	1	no	—
236	C₂×C₄	—	26	1	no	—
237	C₂×C₄	—	26	1	no	—
238	C₂×C₄	—	26	1	no	—
239	C₂×C₄	—	26	1	no	—
240	C₂×C₄	—	26	1	no	—
241	C₂×C₄	—	52	2	no	—
242	C₂×C₄	—	52	2	no	—
243	C₂×C₄	—	52	2	no	—
244	C₂×C₄	C₂²⋊#1057	1	2	yes	—
245	C₂×C₄	C₂²⋊#1057	1	2	yes	—
246	C₂×C₄	#88240	1	2	yes	—
247	C₂×C₄	Q₈×#261	1	2	yes	—
248	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
249	C₂×C₄	C₂×#6122	1	1	yes	characteristic
250	C₈	—	52	4	no	—
251	C₈	—	104	2	no	—
252	C₂²	#11719138	1	1	yes	characteristic
253	C₂²	#11719140	1	1	yes	characteristic
254	C₂²	#11719153	1	1	yes	characteristic
255	C₂²	#11719155	1	1	yes	characteristic
256	C₂²	#11719159	1	1	yes	characteristic
257	C₂²	(C₂×#261)⋊Q₈	1	1	yes	characteristic
258	C₂²	C₂×C₂²⋊#1057	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	4	4	no	—
262	C₄	—	13	2	no	—
263	C₄	—	13	2	no	—
264	C₄	—	26	2	no	—
265	C₄	—	52	1	no	—
266	C₄	—	52	1	no	—
267	C₄	—	104	2	no	—
268	C₄	Q₈×#1057	1	2	yes	—
269	C₄	#11719158	1	2	yes	—
270	C₂	#13130135	1	1	yes	characteristic
271	C₂	#13130138	1	1	yes	characteristic
272	C₂	#13130197	1	1	yes	characteristic
273	C₂	#13130200	1	1	yes	characteristic
274	C₂	#13131005	1	1	yes	characteristic
275	C₂	#13131020	1	1	yes	characteristic
276	C₂	#13131024	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	#9873
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
42. C₄×#1057	Q₈	semidirect
155. C₄⋊Q₈	#1057	semidirect
218. C₁₃	#9873	semidirect

Maximal subgroups

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

subgroup	index
#13130751	2
#13130550	2
#13129672	2
C₄⋊Q₈×Dic₁₃	2
#9873	13