Subgroups — C₂×#13114327 #15845522

Order 3328.

682 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718755	C₂	1	2	yes	maximal
2	C₂×#11718340	C₂	1	1	yes	maximal, characteristic
3	#13114327	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718049	C₂	1	1	yes	maximal, characteristic
5	C₂×#11718139	C₂	1	2	yes	maximal
6	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×Q₁₆	—	2	1	no	—
8	C₂²×C₁₃×Q₁₆	—	2	1	no	—
9	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
11	C₁₃×#494	—	2	2	no	—
12	C₁₃×#494	C₂²	1	4	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×#88055	—	2	1	no	—
15	C₂×#88054	—	2	1	no	—
16	#11718755	—	2	4	no	—
17	#11718755	C₂²	1	8	yes	—
18	C₂×#88050	—	2	2	no	—
19	C₂×#88048	—	2	2	no	—
20	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊#1055	C₂²	1	2	yes	—
22	#11718340	—	2	2	no	—
23	#11718340	C₂²	1	4	yes	—
24	#11718139	C₂²	1	8	yes	—
25	#11718049	C₂²	1	4	yes	—
26	C₂²×Q₈×C₁₃	—	2	2	no	—
27	C₂²×Dic₂₆	D₄	1	1	yes	characteristic
28	C₂×C₁₃×Q₁₆	—	2	4	no	—
29	C₂×C₁₃×Q₁₆	—	2	4	no	—
30	C₂×C₁₃×Q₁₆	—	4	4	no	—
31	C₂×C₁₃×Q₁₆	—	4	4	no	—
32	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
34	C₁₃×C₄⋊Q₈	—	2	2	no	—
35	C₁₃×C₄⋊Q₈	—	2	2	no	—
36	C₁₃×C₄⋊Q₈	—	2	2	no	—
37	C₁₃×C₄⋊Q₈	C₂³	1	4	yes	—
38	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
40	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
42	C₂×#6101	—	2	1	no	—
43	C₂×C₄×Dic₁₃	—	2	1	no	—
44	C₂²×#1055	—	2	1	no	—
45	C₂²×#1055	—	2	2	no	—
46	C₄×Dic₂₆	—	2	1	no	—
47	C₄×Dic₂₆	—	2	1	no	—
48	C₄×Dic₂₆	D₄	1	2	yes	—
49	C₄×Dic₂₆	C₂³	1	2	yes	—
50	C₄×C₈×C₁₃	D₄	1	2	yes	—
51	C₄×C₈×C₁₃	C₂³	1	2	yes	—
52	#88055	—	2	4	no	—
53	#88054	—	2	4	no	—
54	#88050	—	2	8	no	—
55	#88048	—	2	8	no	—
56	C₄⋊#1055	—	2	2	no	—
57	C₄⋊#1055	—	2	1	no	—
58	C₄⋊#1055	C₂³	1	4	yes	—
59	C₄⋊#1055	C₂³	1	2	yes	—
60	C₂×#1690	—	13	1	no	maximal, Sylow
61	C₂×Q₈×C₁₃	—	2	4	no	—
62	C₂×Q₈×C₁₃	—	4	2	no	—
63	C₂×Q₈×C₁₃	—	4	4	no	—
64	C₂×Q₈×C₁₃	—	4	4	no	—
65	C₂²×C₄×C₁₃	—	4	2	no	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
69	C₂²×Dic₁₃	—	2	1	no	—
70	C₂²×Dic₁₃	—	2	1	no	—
71	C₂×Dic₂₆	—	2	4	no	—
72	C₂×Dic₂₆	—	2	1	no	—
73	C₂×Dic₂₆	C₂×D₄	1	2	yes	—
74	C₁₃×Q₁₆	—	4	8	no	—
75	C₁₃×Q₁₆	—	4	8	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
78	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
80	C₁₃×C₄⋊C₄	—	2	4	no	—
81	C₁₃×C₄⋊C₄	—	2	4	no	—
82	C₁₃×C₄⋊C₄	—	4	2	no	—
83	C₁₃×C₄⋊C₄	—	4	2	no	—
84	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₁₃	#41	1	1	yes	characteristic
87	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
88	C₄⋊Dic₁₃	—	2	1	no	—
89	C₄⋊Dic₁₃	C₂×D₄	1	2	yes	—
90	#6101	—	2	2	no	—
91	#6101	—	2	2	no	—
92	C₄×Dic₁₃	—	2	2	no	—
93	C₄×Dic₁₃	—	2	2	no	—
94	C₂×#1055	—	2	4	no	—
95	C₂×#1055	—	2	2	no	—
96	C₂×#1055	—	4	2	no	—
97	C₂×#1055	—	4	4	no	—
98	C₂×#494	D₁₃	1	1	yes	characteristic
99	C₂×#475	—	13	2	no	—
100	C₂×#446	—	13	1	no	—
101	#1690	—	13	8	no	—
102	C₂×#334	—	13	1	no	—
103	C₂×#332	—	13	2	no	—
104	C₂³×C₁₃	#121	1	1	yes	characteristic
105	Q₈×C₁₃	—	4	4	no	—
106	Q₈×C₁₃	—	4	4	no	—
107	Q₈×C₁₃	—	8	4	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	4	4	no	—
110	C₂×C₄×C₁₃	—	8	4	no	—
111	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
115	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
116	C₂×C₄×C₁₃	#137	1	4	yes	—
117	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
120	C₂×Dic₁₃	—	2	2	no	—
121	C₂×Dic₁₃	—	2	2	no	—
122	C₂×Dic₁₃	—	4	2	no	—
123	C₂×Dic₁₃	—	4	2	no	—
124	Dic₂₆	—	2	4	no	—
125	Dic₂₆	—	4	2	no	—
126	C₈×C₁₃	—	2	2	no	—
127	C₈×C₁₃	C₂×SD₁₆	1	4	yes	—
128	#1055	—	4	2	no	—
129	#1055	—	4	4	no	—
130	C₂²×Q₁₆	—	2	1	no	—
131	C₂²×Q₁₆	—	2	1	no	—
132	C₂×C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
133	C₂×C₄×Q₈	—	13	1	no	—
134	#494	—	2	2	no	—
135	#494	C₂×D₁₃	1	4	yes	—
136	#475	—	13	8	no	—
137	#475	—	26	4	no	—
138	#446	—	13	4	no	—
139	#446	—	26	2	no	—
140	C₂×#106	—	26	2	no	—
141	C₂×C₄⋊C₈	—	13	2	no	—
142	C₂×C₄⋊C₈	—	13	1	no	—
143	C₂×C₄⋊C₈	—	26	1	no	—
144	C₂×#103	—	26	2	no	—
145	C₂×#97	—	26	1	no	—
146	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
147	#334	—	13	4	no	—
148	#332	—	13	8	no	—
149	C₂²×C₁₃	#460	1	2	yes	—
150	C₂²×C₁₃	#470	1	2	yes	—
151	C₂²×C₁₃	#473	1	2	yes	—
152	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
153	C₄×C₁₃	—	2	1	no	—
154	C₄×C₁₃	—	2	1	no	—
155	C₄×C₁₃	—	8	4	no	—
156	C₄×C₁₃	#460	1	2	yes	—
157	C₄×C₁₃	#470	1	2	yes	—
158	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
159	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
160	C₄×C₁₃	C₂×#137	1	2	yes	—
161	Dic₁₃	—	4	2	no	—
162	Dic₁₃	—	4	2	no	—
163	C₂²×Q₈	—	2	2	no	—
164	C₂²×Q₈	—	13	1	no	—
165	C₂×Q₁₆	—	2	4	no	—
166	C₂×Q₁₆	—	2	4	no	—
167	C₂×Q₁₆	—	4	4	no	—
168	C₂×Q₁₆	—	4	4	no	—
169	C₂²×C₈	—	26	2	no	—
170	C₂²×C₈	—	26	1	no	—
171	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
172	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
173	C₄⋊Q₈	—	2	2	no	—
174	C₄⋊Q₈	—	2	2	no	—
175	C₄⋊Q₈	—	2	2	no	—
176	C₄⋊Q₈	C₂²×D₁₃	1	4	yes	—
177	C₄×Q₈	—	13	2	no	—
178	C₄×Q₈	—	13	2	no	—
179	C₄×Q₈	—	26	1	no	—
180	C₄×Q₈	—	26	1	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	13	1	no	—
184	C₂×C₄⋊C₄	—	26	1	no	—
185	C₂×C₄²	—	26	1	no	—
186	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
187	#106	—	26	8	no	—
188	C₄⋊C₈	—	13	2	no	—
189	C₄⋊C₈	—	13	4	no	—
190	C₄⋊C₈	—	26	4	no	—
191	C₄⋊C₈	—	26	2	no	—
192	C₄⋊C₈	—	26	1	no	—
193	#103	—	26	8	no	—
194	#97	—	26	4	no	—
195	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
196	C₄×C₈	C₂²×D₁₃	1	2	yes	—
197	C₂×C₁₃	#1690	1	4	yes	complemented
198	C₂×C₁₃	C₂×#460	1	1	yes	characteristic
199	C₂×C₁₃	C₂×#470	1	1	yes	characteristic
200	C₂×C₁₃	C₂×#473	1	1	yes	characteristic
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	—	4	2	no	—
203	C₂×Q₈	—	4	4	no	—
204	C₂×Q₈	—	4	4	no	—
205	C₂×Q₈	—	13	2	no	—
206	C₂×Q₈	—	26	1	no	—
207	C₂×Q₈	—	26	4	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	—	26	1	no	—
210	C₂²×C₄	—	26	1	no	—
211	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
212	C₂²×C₄	#6129	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
214	Q₁₆	—	4	8	no	—
215	Q₁₆	—	4	8	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	26	4	no	—
218	C₂×C₈	—	26	2	no	—
219	C₂×C₈	—	52	4	no	—
220	C₂×C₈	—	52	2	no	—
221	C₂×C₈	#6105	1	4	yes	—
222	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
223	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	—	13	2	no	—
229	C₄⋊C₄	—	26	1	no	—
230	C₄⋊C₄	—	26	2	no	—
231	C₄⋊C₄	—	26	2	no	—
232	C₄²	—	26	2	no	—
233	C₄²	—	26	2	no	—
234	C₄²	D₄×D₁₃	1	1	yes	characteristic
235	C₄²	#6129	1	1	yes	characteristic
236	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
237	C₄²	C₂³×D₁₃	1	1	yes	characteristic
238	C₁₃	C₂×#1690	1	1	yes	characteristic, complemented, Sylow
239	C₂³	#88192	1	1	yes	characteristic
240	Q₈	—	4	4	no	—
241	Q₈	—	4	4	no	—
242	Q₈	—	8	4	no	—
243	Q₈	—	26	4	no	—
244	Q₈	—	52	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	8	4	no	—
248	C₂×C₄	—	26	2	no	—
249	C₂×C₄	—	26	2	no	—
250	C₂×C₄	—	52	2	no	—
251	C₂×C₄	—	52	2	no	—
252	C₂×C₄	#88172	1	4	yes	—
253	C₂×C₄	#88186	1	2	yes	—
254	C₂×C₄	C₂×#6105	1	2	yes	—
255	C₂×C₄	#88192	1	1	yes	characteristic
256	C₂×C₄	#88192	1	1	yes	characteristic
257	C₂×C₄	#88192	1	1	yes	characteristic
258	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
259	C₂×C₄	C₂×#6129	1	1	yes	characteristic
260	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	—	52	4	no	—
263	C₈	—	52	2	no	—
264	C₈	C₂×#6105	1	4	yes	—
265	C₂²	#11718736	1	2	yes	—
266	C₂²	#11718850	1	2	yes	—
267	C₂²	#11718853	1	2	yes	—
268	C₂²	C₂×#88192	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	8	4	no	—
272	C₄	—	52	2	no	—
273	C₄	—	52	2	no	—
274	C₄	#11718736	1	2	yes	—
275	C₄	#11718850	1	2	yes	—
276	C₄	C₂×#88172	1	2	yes	—
277	C₄	C₂×#88186	1	1	yes	characteristic
278	C₄	C₂²×#6105	1	1	yes	characteristic
279	C₂	#13114327	1	4	yes	direct product factor
280	C₂	C₂×#11718736	1	1	yes	characteristic
281	C₂	C₂×#11718850	1	1	yes	characteristic
282	C₂	C₂×#11718853	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88192
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#494	C₂
socle, prime radical	C₂³×C₁₃	#121
hypercenter	C₂×#494	D₁₃
hypocenter	C₁₃	C₂×#1690
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #13114327	C₂	direct (N × H)
197. C₂×C₁₃	#1690	semidirect
238. C₁₃	C₂×#1690	semidirect
279. C₂	#13114327	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11718755	2
C₂×#11718340	2
#13114327	2
C₂×#11718049	2
C₂×#11718139	2
C₂×C₁₃×#494	2
C₂×#1690	13

Subgroups — C2×#13114327 #15845522

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114327 #15845522