Subgroups — C₂×#13114361 #15845550

Order 3328.

618 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718752	C₂	1	2	yes	maximal
2	C₂×#11718340	C₂	1	1	yes	maximal, characteristic
3	#13114361	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718049	C₂	1	1	yes	maximal, characteristic
5	C₂×#11718138	C₂	1	2	yes	maximal
6	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
9	C₁₃×#500	—	2	2	no	—
10	C₁₃×#500	C₂²	1	4	yes	—
11	C₂×C₁₃×#107	—	2	1	no	—
12	C₂×C₁₃×#107	—	2	1	no	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×#88055	—	2	1	no	—
15	C₂×#88054	—	2	1	no	—
16	#11718752	—	2	4	no	—
17	#11718752	C₂²	1	8	yes	—
18	C₂×#88050	—	2	2	no	—
19	C₂×#88047	—	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	#11718138	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	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₁₃×C₄⋊Q₈	—	2	2	no	—
31	C₁₃×C₄⋊Q₈	—	2	2	no	—
32	C₁₃×C₄⋊Q₈	—	2	2	no	—
33	C₁₃×C₄⋊Q₈	C₂³	1	4	yes	—
34	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
38	C₂×#6101	—	2	1	no	—
39	C₂×C₄×Dic₁₃	—	2	1	no	—
40	C₂²×#1055	—	2	1	no	—
41	C₂²×#1055	—	2	2	no	—
42	C₄×Dic₂₆	—	2	1	no	—
43	C₄×Dic₂₆	—	2	1	no	—
44	C₄×Dic₂₆	D₄	1	2	yes	—
45	C₄×Dic₂₆	C₂³	1	2	yes	—
46	C₁₃×#107	—	2	4	no	—
47	C₁₃×#107	—	2	4	no	—
48	C₄×C₈×C₁₃	D₄	1	2	yes	—
49	C₄×C₈×C₁₃	C₂³	1	2	yes	—
50	#88055	—	2	4	no	—
51	#88054	—	2	4	no	—
52	#88050	—	2	8	no	—
53	#88047	—	2	8	no	—
54	C₄⋊#1055	—	2	2	no	—
55	C₄⋊#1055	—	2	1	no	—
56	C₄⋊#1055	C₂³	1	4	yes	—
57	C₄⋊#1055	C₂³	1	2	yes	—
58	C₂×#1696	—	13	1	no	maximal, Sylow
59	C₂×Q₈×C₁₃	—	2	4	no	—
60	C₂×Q₈×C₁₃	—	4	2	no	—
61	C₂×Q₈×C₁₃	—	8	4	no	—
62	C₂²×C₄×C₁₃	—	4	2	no	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
66	C₂²×Dic₁₃	—	2	1	no	—
67	C₂²×Dic₁₃	—	2	1	no	—
68	C₂×Dic₂₆	—	2	4	no	—
69	C₂×Dic₂₆	—	2	1	no	—
70	C₂×Dic₂₆	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	Q₁₆	1	4	yes	—
73	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
75	C₁₃×C₄⋊C₄	—	2	4	no	—
76	C₁₃×C₄⋊C₄	—	2	4	no	—
77	C₁₃×C₄⋊C₄	—	4	2	no	—
78	C₁₃×C₄⋊C₄	—	4	2	no	—
79	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₁₃	#41	1	1	yes	characteristic
82	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
83	C₄⋊Dic₁₃	—	2	1	no	—
84	C₄⋊Dic₁₃	C₂×D₄	1	2	yes	—
85	#6101	—	2	2	no	—
86	#6101	—	2	2	no	—
87	C₄×Dic₁₃	—	2	2	no	—
88	C₄×Dic₁₃	—	2	2	no	—
89	C₂×#1055	—	2	4	no	—
90	C₂×#1055	—	2	2	no	—
91	C₂×#1055	—	4	4	no	—
92	C₂×#1055	—	4	2	no	—
93	C₂×#500	D₁₃	1	1	yes	characteristic
94	C₂×#477	—	13	2	no	—
95	C₂×#446	—	13	1	no	—
96	#1696	—	13	8	no	—
97	C₂×#334	—	13	1	no	—
98	C₂×#333	—	13	2	no	—
99	C₂³×C₁₃	#121	1	1	yes	characteristic
100	Q₈×C₁₃	—	8	4	no	—
101	Q₈×C₁₃	—	8	4	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	4	4	no	—
104	C₂×C₄×C₁₃	—	8	4	no	—
105	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
106	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
109	C₂×C₄×C₁₃	#137	1	2	yes	—
110	C₂×C₄×C₁₃	#137	1	4	yes	—
111	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
114	C₂×Dic₁₃	—	2	2	no	—
115	C₂×Dic₁₃	—	2	2	no	—
116	C₂×Dic₁₃	—	4	2	no	—
117	C₂×Dic₁₃	—	4	2	no	—
118	Dic₂₆	—	2	4	no	—
119	Dic₂₆	—	4	2	no	—
120	C₈×C₁₃	—	2	2	no	—
121	C₈×C₁₃	C₂×Q₁₆	1	4	yes	—
122	#1055	—	4	4	no	—
123	#1055	—	4	2	no	—
124	C₂×C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
125	C₂×C₄×Q₈	—	13	1	no	—
126	#500	—	2	2	no	—
127	#500	C₂×D₁₃	1	4	yes	—
128	#477	—	13	8	no	—
129	#477	—	26	4	no	—
130	#446	—	13	4	no	—
131	#446	—	26	2	no	—
132	C₂×#107	—	2	1	no	—
133	C₂×#107	—	2	1	no	—
134	C₂×#107	—	26	2	no	—
135	C₂×C₄⋊C₈	—	13	2	no	—
136	C₂×C₄⋊C₈	—	13	1	no	—
137	C₂×C₄⋊C₈	—	26	1	no	—
138	C₂×#103	—	26	2	no	—
139	C₂×#97	—	26	1	no	—
140	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
141	#334	—	13	4	no	—
142	#333	—	13	8	no	—
143	C₂²×C₁₃	#462	1	2	yes	—
144	C₂²×C₁₃	#470	1	2	yes	—
145	C₂²×C₁₃	#472	1	2	yes	—
146	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
147	C₄×C₁₃	—	2	1	no	—
148	C₄×C₁₃	—	2	1	no	—
149	C₄×C₁₃	—	8	4	no	—
150	C₄×C₁₃	#462	1	2	yes	—
151	C₄×C₁₃	#470	1	2	yes	—
152	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
153	C₄×C₁₃	C₂×#137	1	2	yes	—
154	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
155	Dic₁₃	—	4	2	no	—
156	Dic₁₃	—	4	2	no	—
157	C₂²×Q₈	—	2	2	no	—
158	C₂²×Q₈	—	13	1	no	—
159	C₂²×C₈	—	26	2	no	—
160	C₂²×C₈	—	26	1	no	—
161	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
162	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
163	C₄⋊Q₈	—	2	2	no	—
164	C₄⋊Q₈	—	2	2	no	—
165	C₄⋊Q₈	—	2	2	no	—
166	C₄⋊Q₈	C₂²×D₁₃	1	4	yes	—
167	C₄×Q₈	—	13	2	no	—
168	C₄×Q₈	—	13	2	no	—
169	C₄×Q₈	—	26	1	no	—
170	C₄×Q₈	—	26	1	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	13	1	no	—
174	C₂×C₄⋊C₄	—	26	1	no	—
175	C₂×C₄²	—	26	1	no	—
176	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
177	#107	—	2	4	no	—
178	#107	—	2	4	no	—
179	#107	—	26	8	no	—
180	C₄⋊C₈	—	13	2	no	—
181	C₄⋊C₈	—	13	4	no	—
182	C₄⋊C₈	—	26	4	no	—
183	C₄⋊C₈	—	26	2	no	—
184	C₄⋊C₈	—	26	1	no	—
185	#103	—	26	8	no	—
186	#97	—	26	4	no	—
187	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
188	C₄×C₈	C₂²×D₁₃	1	2	yes	—
189	C₂×C₁₃	#1696	1	4	yes	complemented
190	C₂×C₁₃	C₂×#462	1	1	yes	characteristic
191	C₂×C₁₃	C₂×#470	1	1	yes	characteristic
192	C₂×C₁₃	C₂×#472	1	1	yes	characteristic
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	—	4	2	no	—
195	C₂×Q₈	—	8	4	no	—
196	C₂×Q₈	—	13	2	no	—
197	C₂×Q₈	—	26	1	no	—
198	C₂×Q₈	—	26	4	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	26	1	no	—
201	C₂²×C₄	—	26	1	no	—
202	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
203	C₂²×C₄	#6129	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	26	4	no	—
207	C₂×C₈	—	26	2	no	—
208	C₂×C₈	—	52	4	no	—
209	C₂×C₈	—	52	2	no	—
210	C₂×C₈	#6107	1	4	yes	—
211	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
212	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	13	2	no	—
218	C₄⋊C₄	—	26	1	no	—
219	C₄⋊C₄	—	26	2	no	—
220	C₄⋊C₄	—	26	2	no	—
221	C₄²	—	26	2	no	—
222	C₄²	—	26	2	no	—
223	C₄²	D₄×D₁₃	1	1	yes	characteristic
224	C₄²	#6129	1	1	yes	characteristic
225	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
226	C₄²	C₂³×D₁₃	1	1	yes	characteristic
227	C₁₃	C₂×#1696	1	1	yes	characteristic, complemented, Sylow
228	C₂³	#88192	1	1	yes	characteristic
229	Q₈	—	8	4	no	—
230	Q₈	—	8	4	no	—
231	Q₈	—	26	4	no	—
232	Q₈	—	52	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	8	4	no	—
236	C₂×C₄	—	26	2	no	—
237	C₂×C₄	—	26	2	no	—
238	C₂×C₄	—	52	2	no	—
239	C₂×C₄	—	52	2	no	—
240	C₂×C₄	#88172	1	4	yes	—
241	C₂×C₄	#88192	1	1	yes	characteristic
242	C₂×C₄	#88192	1	1	yes	characteristic
243	C₂×C₄	#88192	1	1	yes	characteristic
244	C₂×C₄	#88196	1	2	yes	—
245	C₂×C₄	C₂×#6107	1	2	yes	—
246	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
247	C₂×C₄	C₂×#6129	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	52	4	no	—
251	C₈	—	52	2	no	—
252	C₈	C₂×#6107	1	4	yes	—
253	C₂²	#11718751	1	2	yes	—
254	C₂²	#11718821	1	2	yes	—
255	C₂²	#11718850	1	2	yes	—
256	C₂²	C₂×#88192	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	8	4	no	—
260	C₄	—	52	2	no	—
261	C₄	—	52	2	no	—
262	C₄	#11718751	1	2	yes	—
263	C₄	#11718850	1	2	yes	—
264	C₄	C₂×#88172	1	2	yes	—
265	C₄	C₂×#88196	1	1	yes	characteristic
266	C₄	C₂²×#6107	1	1	yes	characteristic
267	C₂	#13114361	1	4	yes	direct product factor
268	C₂	C₂×#11718751	1	1	yes	characteristic
269	C₂	C₂×#11718850	1	1	yes	characteristic
270	C₂	C₂×#11718821	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88192
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#500	C₂
socle, prime radical	C₂³×C₁₃	#121
hypercenter	C₂×#500	D₁₃
hypocenter	C₁₃	C₂×#1696
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #13114361	C₂	direct (N × H)
189. C₂×C₁₃	#1696	semidirect
227. C₁₃	C₂×#1696	semidirect
267. C₂	#13114361	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×#11718752	2
C₂×#11718340	2
#13114361	2
C₂×#11718049	2
C₂×#11718138	2
C₂×C₁₃×#500	2
C₂×#1696	13

Subgroups — C2×#13114361 #15845550

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114361 #15845550