Subgroups — #16787347

Order 3328.

380 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718261	C₂	1	1	yes	maximal, characteristic, complemented
2	#13130039	C₂	1	1	yes	maximal, characteristic, complemented
3	#13129800	C₂	1	4	yes	maximal, complemented
4	C₂×#11718857	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#135	C₄	1	1	yes	characteristic
6	C₂×#88202	—	2	1	no	—
7	#11719119	—	2	1	no	—
8	C₂×#88106	—	2	1	no	—
9	C₂×#88077	—	2	1	no	—
10	#11718857	—	2	1	no	—
11	#11718857	—	2	1	no	—
12	#11718857	C₄	1	2	yes	—
13	#11718857	C₂²	1	2	yes	—
14	C₂×#88059	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
16	#11718261	—	2	1	no	—
17	#11718261	C₂²	1	2	yes	complemented
18	#11718251	C₂²	1	2	yes	complemented
19	C₈×#1057	—	2	1	no	—
20	C₂×C₁₃×#41	—	2	1	no	—
21	C₁₃×#135	—	2	1	no	—
22	C₁₃×#135	—	2	1	no	—
23	C₁₃×#135	C₈	1	2	yes	complemented
24	C₁₃×#135	C₂×C₄	1	2	yes	—
25	C₂×C₁₃×Q₁₆	C₈	1	1	yes	characteristic, complemented
26	C₂×C₁₃×SD₁₆	—	2	1	no	—
27	C₂×C₁₃×D₈	C₈	1	1	yes	characteristic, complemented
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	#88202	—	2	2	no	—
30	#88202	—	2	2	no	—
31	#88202	—	4	1	no	—
32	#88202	—	4	1	no	—
33	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
34	C₂×#6099	—	2	1	no	—
35	C₂×#6099	—	4	1	no	—
36	C₂²×#1055	—	4	1	no	—
37	C₂²×#1055	D₄	1	1	yes	characteristic
38	C₂²⋊#1057	—	2	2	no	—
39	C₄⋊#1057	—	2	2	no	—
40	C₄×#1057	—	2	2	no	—
41	#88106	—	2	2	no	—
42	#88106	—	4	1	no	—
43	C₂×#6092	—	4	1	no	—
44	#88077	—	2	2	no	—
45	#88077	—	2	2	no	—
46	#88059	—	2	1	no	—
47	#88059	C₂×C₄	1	2	yes	—
48	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
49	C₈×Dic₁₃	D₄	1	1	yes	characteristic
50	C₈×Dic₁₃	D₄	1	1	yes	characteristic
51	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
52	#11604	—	13	1	no	maximal, Sylow
53	C₁₃×#41	—	2	2	no	—
54	C₁₃×#41	—	2	2	no	—
55	C₁₃×#41	—	4	1	no	—
56	C₁₃×#41	—	4	1	no	—
57	C₂×Q₈×C₁₃	—	2	1	no	—
58	C₂×D₄×C₁₃	—	2	1	no	—
59	C₂×D₄×C₁₃	—	4	1	no	—
60	C₂²×C₄×C₁₃	—	4	1	no	—
61	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×Dic₁₃	—	2	1	no	—
63	C₂×#1057	—	2	4	no	—
64	C₁₃×Q₁₆	—	2	1	no	—
65	C₁₃×Q₁₆	C₂×C₈	1	2	yes	—
66	C₁₃×SD₁₆	—	2	2	no	—
67	C₁₃×SD₁₆	—	2	2	no	—
68	C₁₃×D₈	—	2	1	no	—
69	C₁₃×D₈	C₂×C₈	1	2	yes	—
70	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
72	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
73	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
74	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₄×Dic₁₃	—	2	1	no	—
77	C₄×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
79	#6099	—	2	2	no	—
80	#6099	—	4	1	no	—
81	#6099	—	4	2	no	—
82	#6099	—	4	2	no	—
83	C₂×#1055	—	4	1	no	—
84	C₂×#1055	—	4	2	no	—
85	C₂×#1055	—	4	2	no	—
86	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×#1055	D₈	1	1	yes	characteristic
88	C₂×#1055	D₈	1	1	yes	characteristic
89	C₂×#1055	SD₁₆	1	1	yes	characteristic
90	C₂×#1055	SD₁₆	1	1	yes	characteristic
91	C₂×#1055	C₂×D₄	1	1	yes	characteristic
92	#6092	—	4	2	no	—
93	C₂×#443	—	13	1	no	—
94	C₂×#364	—	13	1	no	—
95	#1450	—	13	1	no	—
96	#1336	—	13	4	no	—
97	C₂³×C₁₃	—	4	1	no	—
98	C₂³×C₁₃	#104	1	1	yes	characteristic
99	Q₈×C₁₃	—	2	2	no	—
100	Q₈×C₁₃	—	4	1	no	—
101	D₄×C₁₃	—	2	2	no	—
102	D₄×C₁₃	—	4	2	no	—
103	D₄×C₁₃	—	4	2	no	—
104	D₄×C₁₃	—	4	1	no	—
105	C₂×C₄×C₁₃	—	4	1	no	—
106	C₂×C₄×C₁₃	—	4	2	no	—
107	C₂×C₄×C₁₃	—	4	2	no	—
108	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
111	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#104	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×Dic₁₃	—	2	1	no	—
115	C₂×Dic₁₃	—	2	1	no	—
116	C₂×Dic₁₃	—	4	2	no	—
117	#1057	—	4	4	no	—
118	C₈×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
119	C₈×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
120	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
121	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
122	#1055	—	4	2	no	—
123	#1055	#102	1	1	yes	characteristic
124	#1055	#102	1	1	yes	characteristic
125	#1055	C₂×D₈	1	1	yes	characteristic
126	#1055	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×#135	#261	1	1	yes	characteristic
128	C₂×#131	—	26	1	no	—
129	C₂×M₃₂	—	26	1	no	—
130	#443	—	13	2	no	—
131	#443	—	13	2	no	—
132	#443	—	26	1	no	—
133	#443	—	26	1	no	—
134	#431	—	26	1	no	—
135	C₂×#108	—	13	1	no	—
136	C₂×#104	—	26	1	no	—
137	C₂×C₄×C₈	—	13	1	no	—
138	#364	—	13	2	no	—
139	#364	—	26	1	no	—
140	#322	—	13	2	no	—
141	C₈²	—	26	1	no	—
142	C₂²×C₁₃	—	4	1	no	—
143	C₂²×C₁₃	—	4	2	no	—
144	C₂²×C₁₃	—	8	2	no	—
145	C₂²×C₁₃	D₄⋊C₈	1	1	yes	characteristic
146	C₂²×C₁₃	D₄⋊C₈	1	1	yes	characteristic
147	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
148	C₄×C₁₃	—	4	2	no	—
149	C₄×C₁₃	D₄⋊C₈	1	1	yes	characteristic
150	C₄×C₁₃	D₄⋊C₈	1	1	yes	characteristic
151	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
152	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
153	Dic₁₃	—	2	2	no	—
154	Dic₁₃	—	4	1	no	—
155	C₂×#41	—	2	1	no	—
156	#135	—	2	1	no	—
157	#135	—	2	1	no	—
158	#135	#1057	1	2	yes	complemented
159	#135	C₂×#261	1	2	yes	—
160	C₂×Q₁₆	#1057	1	1	yes	characteristic, complemented
161	C₂×SD₁₆	—	2	1	no	—
162	C₂×D₈	#1057	1	1	yes	characteristic, complemented
163	#131	—	26	2	no	—
164	#131	—	26	2	no	—
165	#131	—	52	1	no	—
166	#131	—	52	1	no	—
167	C₂×M₁₆	—	26	1	no	—
168	C₂×M₁₆	—	52	1	no	—
169	C₂²×C₈	—	13	1	no	—
170	C₂²×C₈	—	52	1	no	—
171	C₂²×C₈	C₂×#261	1	1	yes	characteristic
172	C₂×C₄²	—	13	1	no	—
173	M₃₂	—	26	2	no	—
174	M₃₂	—	52	1	no	—
175	C₂×C₁₆	—	52	1	no	—
176	#108	—	13	2	no	—
177	#108	—	26	1	no	—
178	C₄⋊C₈	—	26	2	no	—
179	#104	—	26	2	no	—
180	#104	—	26	2	no	—
181	C₂²⋊C₈	—	26	2	no	—
182	C₄×C₈	—	13	1	no	—
183	C₄×C₈	—	13	1	no	—
184	C₄×C₈	—	13	1	no	—
185	C₄×C₈	—	13	1	no	—
186	C₄×C₈	—	26	2	no	—
187	C₂×C₁₃	—	2	1	no	—
188	C₂×C₁₃	—	2	1	no	—
189	C₂×C₁₃	—	2	1	no	—
190	C₂×C₁₃	—	8	2	no	—
191	C₂×C₁₃	C₂×D₄⋊C₈	1	1	yes	characteristic
192	#41	—	2	2	no	—
193	#41	—	2	2	no	—
194	#41	—	4	1	no	—
195	#41	—	4	1	no	—
196	C₂×Q₈	—	2	1	no	—
197	C₂×D₄	—	2	1	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	—	26	1	no	—
201	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
202	Q₁₆	—	2	1	no	—
203	Q₁₆	C₂×#1057	1	2	yes	—
204	SD₁₆	—	2	2	no	—
205	SD₁₆	—	2	2	no	—
206	D₈	—	2	1	no	—
207	D₈	C₂×#1057	1	2	yes	—
208	M₁₆	—	26	2	no	—
209	M₁₆	—	52	2	no	—
210	M₁₆	—	52	2	no	—
211	M₁₆	—	52	1	no	—
212	C₂×C₈	—	13	1	no	—
213	C₂×C₈	—	13	1	no	—
214	C₂×C₈	—	13	1	no	—
215	C₂×C₈	—	13	1	no	—
216	C₂×C₈	—	13	1	no	—
217	C₂×C₈	—	13	1	no	—
218	C₂×C₈	—	26	4	no	—
219	C₂×C₈	—	52	2	no	—
220	C₂×C₈	—	52	2	no	—
221	C₂×C₈	—	52	1	no	—
222	C₂×C₈	C₂×#1057	1	1	yes	characteristic
223	C₂×C₈	C₂×#1057	1	1	yes	characteristic
224	C₂×C₈	#6122	1	1	yes	characteristic
225	C₂×C₈	#6122	1	1	yes	characteristic
226	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
227	C₂×C₈	C₂²×#261	1	1	yes	characteristic
228	C₄²	—	13	1	no	—
229	C₄²	—	13	1	no	—
230	C₄²	—	26	1	no	—
231	C₁₆	—	52	2	no	—
232	C₁₃	#11604	1	1	yes	characteristic, complemented, Sylow
233	C₂³	—	4	1	no	—
234	C₂³	#88116	1	1	yes	characteristic
235	Q₈	—	2	2	no	—
236	Q₈	—	4	1	no	—
237	D₄	—	2	2	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	2	no	—
240	D₄	—	4	1	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	26	1	no	—
245	C₂×C₄	—	26	1	no	—
246	C₂×C₄	—	52	2	no	—
247	C₂×C₄	#88115	1	1	yes	characteristic
248	C₂×C₄	#88115	1	1	yes	characteristic
249	C₂×C₄	#88116	1	1	yes	characteristic
250	C₂×C₄	C₂²⋊#1057	1	1	yes	characteristic
251	C₂×C₄	C₂²⋊#1057	1	1	yes	characteristic
252	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
253	C₈	—	13	1	no	—
254	C₈	—	13	1	no	—
255	C₈	—	13	1	no	—
256	C₈	—	13	1	no	—
257	C₈	—	52	4	no	—
258	C₈	—	52	2	no	—
259	C₈	C₂²⋊#1057	1	1	yes	characteristic
260	C₈	C₂²⋊#1057	1	1	yes	characteristic
261	C₈	C₂²×#1057	1	1	yes	characteristic
262	C₈	C₂×#6122	1	1	yes	characteristic
263	C₂²	—	4	1	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	D₄⋊#1057	1	1	yes	characteristic
267	C₂²	D₄⋊#1057	1	1	yes	characteristic
268	C₂²	C₂×#88116	1	1	yes	characteristic
269	C₄	—	4	2	no	—
270	C₄	—	26	2	no	—
271	C₄	—	52	1	no	—
272	C₄	D₄⋊#1057	1	1	yes	characteristic
273	C₄	D₄⋊#1057	1	1	yes	characteristic
274	C₄	C₂×#88115	1	1	yes	characteristic
275	C₄	C₂×C₂²⋊#1057	1	1	yes	characteristic
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	2	1	no	—
279	C₂	—	8	2	no	—
280	C₂	C₂×D₄⋊#1057	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#88115
commutator	C₈×C₁₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#261
Fitting, prime radical	C₂×C₁₃×#135	C₄
socle	C₂×C₁₃	C₂×D₄⋊C₈
cosocle	C₈×Dic₁₃	C₂³
hypercenter	C₂×#135	#261
hypocenter	C₁₃	#11604
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#11718261	C₂	semidirect
2. #13130039	C₂	semidirect
3. #13129800	C₂	semidirect
17. #11718261	C₂²	semidirect
18. #11718251	C₂²	semidirect
23. C₁₃×#135	C₈	semidirect
25. C₂×C₁₃×Q₁₆	C₈	semidirect
27. C₂×C₁₃×D₈	C₈	semidirect
158. #135	#1057	semidirect
160. C₂×Q₁₆	#1057	semidirect
162. C₂×D₈	#1057	semidirect
232. C₁₃	#11604	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718261	2
#13130039	2
#13129800	2
C₂×#11718857	2
#11604	13