Subgroups — C₂×#11719330 #13128202

Order 1664.

1004 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×C₁₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88198	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#88180	C₂	1	1	yes	maximal, characteristic
5	C₂×#88178	C₂	1	1	yes	maximal, characteristic
6	#11719330	C₂	1	16	yes	maximal, direct product factor
7	C₂×#88144	C₂	1	2	yes	maximal, complemented
8	C₂×#88144	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#88143	C₂	1	2	yes	maximal, complemented
10	C₂×#88141	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#88133	C₂	1	2	yes	maximal
12	C₂×#88133	C₂	1	2	yes	maximal
13	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
16	C₁₃×C₂²⋊Q₈	C₂²	1	8	yes	—
17	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₂²⋊Dic₁₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊Dic₁₃	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊Dic₁₃	C₂²	1	1	yes	characteristic
23	Q₈×Dic₁₃	C₂²	1	8	yes	complemented
24	#88198	C₂²	1	8	yes	complemented
25	#88180	C₂²	1	8	yes	—
26	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
27	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
28	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
31	#88178	C₂²	1	8	yes	—
32	C₂×#6101	C₂²	1	2	yes	—
33	C₂×#6101	C₂²	1	2	yes	—
34	C₂×#6101	C₂²	1	2	yes	—
35	C₂×#6101	C₂²	1	2	yes	—
36	C₂×#6101	C₂²	1	2	yes	—
37	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
38	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
39	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
41	#88144	C₂²	1	16	yes	complemented
42	#88144	C₂²	1	8	yes	complemented
43	#88143	C₂²	1	16	yes	complemented
44	#88141	C₂²	1	8	yes	complemented
45	#88133	C₂²	1	16	yes	—
46	#88133	C₂²	1	16	yes	—
47	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×Q₈×C₁₃	—	2	4	no	—
49	C₂×Q₈×C₁₃	C₂³	1	4	yes	—
50	C₂²×C₄×C₁₃	—	2	4	no	—
51	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
55	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
56	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
57	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
58	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
59	C₂²×Dic₁₃	C₂³	1	2	yes	—
60	C₂²×Dic₁₃	C₂³	1	2	yes	—
61	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
62	C₂×Dic₂₆	—	2	4	no	—
63	C₂×Dic₂₆	C₂³	1	4	yes	—
64	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
65	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
66	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
67	C₂²⋊Dic₁₃	C₂³	1	8	yes	—
68	C₂²⋊Dic₁₃	C₂³	1	4	yes	—
69	C₂²⋊Dic₁₃	C₂³	1	4	yes	—
70	C₄⋊Dic₁₃	C₂³	1	8	yes	—
71	C₄⋊Dic₁₃	C₂³	1	8	yes	—
72	C₄⋊Dic₁₃	C₂³	1	4	yes	—
73	C₄⋊Dic₁₃	C₂³	1	4	yes	—
74	C₄⋊Dic₁₃	C₂³	1	4	yes	—
75	#6101	C₂³	1	8	yes	—
76	#6101	C₂³	1	8	yes	—
77	#6101	C₂³	1	8	yes	—
78	#6101	C₂³	1	8	yes	—
79	#6101	C₂³	1	8	yes	—
80	C₄×Dic₁₃	C₂³	1	8	yes	—
81	C₄×Dic₁₃	C₂³	1	8	yes	—
82	C₄×Dic₁₃	C₂³	1	4	yes	—
83	C₄×Dic₁₃	C₂³	1	4	yes	—
84	C₂×#541	—	13	1	no	maximal, Sylow
85	C₂³×C₁₃	—	2	2	no	—
86	C₂³×C₁₃	—	2	2	no	—
87	C₂³×C₁₃	—	2	2	no	—
88	C₂³×C₁₃	C₂⁴	1	2	yes	—
89	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
90	Q₈×C₁₃	—	2	8	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	4	no	—
93	C₂×C₄×C₁₃	—	2	4	no	—
94	C₂×C₄×C₁₃	—	2	8	no	—
95	C₂×C₄×C₁₃	#41	1	4	yes	—
96	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
97	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
98	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
99	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
100	C₂×Dic₁₃	—	2	4	no	—
101	C₂×Dic₁₃	—	2	2	no	—
102	C₂×Dic₁₃	—	2	4	no	—
103	C₂×Dic₁₃	—	2	2	no	—
104	C₂×Dic₁₃	—	2	2	no	—
105	C₂×Dic₁₃	—	2	2	no	—
106	C₂×Dic₁₃	C₂⁴	1	2	yes	—
107	C₂×Dic₁₃	C₂⁴	1	2	yes	—
108	C₂×Dic₁₃	C₂⁴	1	2	yes	—
109	C₂×Dic₁₃	C₂⁴	1	4	yes	—
110	C₂×Dic₁₃	C₂⁴	1	4	yes	—
111	C₂×Dic₁₃	C₂⁴	1	2	yes	—
112	Dic₂₆	—	2	8	no	—
113	#541	—	13	16	no	—
114	C₂×C₄⋊Q₈	—	13	1	no	—
115	C₂×#126	—	13	2	no	—
116	C₂×#126	—	13	2	no	—
117	C₂×#125	—	13	2	no	—
118	C₂×#125	—	13	1	no	—
119	C₂×#125	—	13	2	no	—
120	C₂×C₂²⋊Q₈	—	13	1	no	—
121	C₂×C₂²⋊Q₈	D₁₃	1	1	yes	characteristic
122	C₂×C₄×Q₈	—	13	1	no	—
123	C₂×C₄×Q₈	—	13	1	no	—
124	C₂×#117	—	13	1	no	—
125	C₂²×C₁₃	—	2	2	no	—
126	C₂²×C₁₃	—	2	2	no	—
127	C₂²×C₁₃	—	2	2	no	—
128	C₂²×C₁₃	—	4	4	no	—
129	C₂²×C₁₃	C₂×#41	1	2	yes	—
130	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
131	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
132	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
133	C₄×C₁₃	—	2	2	no	—
134	C₄×C₁₃	—	2	4	no	—
135	C₄×C₁₃	—	2	4	no	—
136	C₄×C₁₃	C₂×#41	1	4	yes	—
137	Dic₁₃	—	2	4	no	—
138	Dic₁₃	—	2	2	no	—
139	Dic₁₃	—	2	4	no	—
140	Dic₁₃	—	2	2	no	—
141	Dic₁₃	—	2	2	no	—
142	Dic₁₃	—	2	2	no	—
143	C₂²×Q₈	—	13	1	no	—
144	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
145	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
146	C₄⋊Q₈	—	13	8	no	—
147	#126	—	13	16	no	—
148	#126	—	13	16	no	—
149	#125	—	13	16	no	—
150	#125	—	13	8	no	—
151	#125	—	13	16	no	—
152	C₂²⋊Q₈	—	13	8	no	—
153	C₂²⋊Q₈	C₂×D₁₃	1	8	yes	—
154	C₄×Q₈	—	13	8	no	—
155	C₄×Q₈	—	13	8	no	—
156	#117	—	13	8	no	—
157	C₂×C₄⋊C₄	—	13	1	no	—
158	C₂×C₄⋊C₄	—	13	1	no	—
159	C₂×C₄⋊C₄	—	13	1	no	—
160	C₂×C₄⋊C₄	—	13	2	no	—
161	C₂×C₄⋊C₄	—	13	2	no	—
162	C₂×C₄⋊C₄	—	13	2	no	—
163	C₂×C₄⋊C₄	—	13	2	no	—
164	C₂×C₄⋊C₄	—	13	2	no	—
165	C₂×C₄⋊C₄	—	13	2	no	—
166	C₂×C₄⋊C₄	—	13	2	no	—
167	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
168	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
169	C₂×C₂²⋊C₄	—	13	1	no	—
170	C₂×C₂²⋊C₄	—	13	1	no	—
171	C₂×C₂²⋊C₄	—	13	2	no	—
172	C₂×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
173	C₂×C₄²	—	13	1	no	—
174	C₂×C₄²	—	13	1	no	—
175	C₂×C₄²	—	13	2	no	—
176	C₂×C₄²	—	13	2	no	—
177	C₂×C₁₃	—	4	2	no	—
178	C₂×C₁₃	#541	1	4	yes	complemented
179	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
180	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
181	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
182	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
183	C₂×Q₈	—	2	4	no	—
184	C₂×Q₈	—	13	4	no	—
185	C₂×Q₈	—	26	4	no	—
186	C₂×Q₈	C₂²×D₁₃	1	4	yes	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	13	1	no	—
189	C₂²×C₄	—	13	1	no	—
190	C₂²×C₄	—	13	1	no	—
191	C₂²×C₄	—	13	2	no	—
192	C₂²×C₄	—	13	1	no	—
193	C₂²×C₄	—	13	2	no	—
194	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
195	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
196	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
197	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
199	C₄⋊C₄	—	13	4	no	—
200	C₄⋊C₄	—	13	4	no	—
201	C₄⋊C₄	—	13	4	no	—
202	C₄⋊C₄	—	13	8	no	—
203	C₄⋊C₄	—	13	8	no	—
204	C₄⋊C₄	—	13	8	no	—
205	C₄⋊C₄	—	13	8	no	—
206	C₄⋊C₄	—	13	8	no	—
207	C₄⋊C₄	—	13	8	no	—
208	C₄⋊C₄	—	13	8	no	—
209	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
210	C₄⋊C₄	C₂²×D₁₃	1	8	yes	—
211	C₂²⋊C₄	—	13	4	no	—
212	C₂²⋊C₄	—	13	4	no	—
213	C₂²⋊C₄	—	13	8	no	—
214	C₂²⋊C₄	C₂²×D₁₃	1	8	yes	—
215	C₄²	—	13	4	no	—
216	C₄²	—	13	4	no	—
217	C₄²	—	13	8	no	—
218	C₄²	—	13	8	no	—
219	C₁₃	C₂×#541	1	1	yes	characteristic, complemented, Sylow
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	C₂³×D₁₃	1	2	yes	—
224	C₂³	C₂³×D₁₃	1	1	yes	characteristic
225	Q₈	—	2	8	no	—
226	Q₈	—	26	8	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	13	2	no	—
232	C₂×C₄	—	13	2	no	—
233	C₂×C₄	—	13	2	no	—
234	C₂×C₄	—	13	4	no	—
235	C₂×C₄	—	13	2	no	—
236	C₂×C₄	—	13	4	no	—
237	C₂×C₄	—	26	4	no	—
238	C₂×C₄	—	26	2	no	—
239	C₂×C₄	—	26	2	no	—
240	C₂×C₄	—	26	4	no	—
241	C₂×C₄	—	26	2	no	—
242	C₂×C₄	—	26	2	no	—
243	C₂×C₄	#6129	1	4	yes	—
244	C₂×C₄	C₂³×D₁₃	1	4	yes	—
245	C₂×C₄	C₂³×D₁₃	1	4	yes	—
246	C₂×C₄	C₂³×D₁₃	1	2	yes	—
247	C₂×C₄	C₂³×D₁₃	1	2	yes	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	4	4	no	—
252	C₂²	C₂×#6129	1	2	yes	—
253	C₂²	#88254	1	2	yes	—
254	C₂²	#88257	1	2	yes	—
255	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	2	4	no	—
258	C₄	—	2	4	no	—
259	C₄	—	26	4	no	—
260	C₄	—	26	2	no	—
261	C₄	—	26	2	no	—
262	C₄	—	26	4	no	—
263	C₄	—	26	2	no	—
264	C₄	—	26	2	no	—
265	C₄	C₂×#6129	1	4	yes	—
266	C₂	—	4	2	no	—
267	C₂	#11719330	1	4	yes	direct product factor
268	C₂	C₂²×#6129	1	1	yes	characteristic
269	C₂	C₂×#88254	1	1	yes	characteristic
270	C₂	C₂×#88257	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃
Fitting	C₂×C₁₃×C₂²⋊Q₈	C₂
socle	C₂³×C₁₃	C₂⁴
prime radical	C₂⁴×C₁₃	C₂³
hypercenter	C₂×C₂²⋊Q₈	D₁₃
hypocenter	C₁₃	C₂×#541
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×Q₈×Dic₁₃	C₂	semidirect
3. C₂×#88198	C₂	semidirect
6. #11719330	C₂	direct (N × H)
7. C₂×#88144	C₂	semidirect
8. C₂×#88144	C₂	semidirect
9. C₂×#88143	C₂	semidirect
10. C₂×#88141	C₂	semidirect
23. Q₈×Dic₁₃	C₂²	semidirect
24. #88198	C₂²	semidirect
41. #88144	C₂²	semidirect
42. #88144	C₂²	semidirect
43. #88143	C₂²	semidirect
44. #88141	C₂²	semidirect
178. C₂×C₁₃	#541	semidirect
219. C₁₃	C₂×#541	semidirect
267. C₂	#11719330	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×C₁₃×C₂²⋊Q₈	2
C₂×Q₈×Dic₁₃	2
C₂×#88198	2
C₂×#88180	2
C₂×#88178	2
#11719330	2
C₂×#88144	2
C₂×#88143	2
C₂×#88141	2
C₂×#88133	2
C₂×#541	13

Subgroups — C2×#11719330 #13128202

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11719330 #13128202