Subgroups — C₂×#11719328 #13128235

Order 1664.

1080 conjugacy classes of subgroups.

The subgroup lattice has 252 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×#88198	C₂	1	2	yes	maximal, complemented
3	C₂×#88178	C₂	1	1	yes	maximal, characteristic
4	C₂²×#6101	C₂	1	1	yes	maximal, characteristic
5	#11719328	C₂	1	16	yes	maximal, direct product factor
6	C₂×#88143	C₂	1	2	yes	maximal, complemented
7	C₂×#88143	C₂	1	2	yes	maximal, complemented
8	C₂×#88142	C₂	1	2	yes	maximal, complemented
9	C₂×#88132	C₂	1	2	yes	maximal
10	C₂×#88131	C₂	1	2	yes	maximal
11	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
13	C₂³×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂³×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₂₆	C₂²	1	2	yes	—
16	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
17	C₁₃×C₂²⋊Q₈	C₂²	1	8	yes	—
18	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₂²⋊Dic₁₃	C₂²	1	2	yes	—
22	#88198	C₂²	1	16	yes	complemented
23	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
24	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
25	#88178	C₂²	1	8	yes	—
26	C₂×#6101	C₂²	1	2	yes	—
27	C₂×#6101	C₂²	1	2	yes	—
28	C₂×#6101	C₂²	1	2	yes	—
29	C₂×#6101	C₂²	1	2	yes	—
30	C₂×#6101	C₂²	1	2	yes	—
31	C₂×#6101	C₂²	1	2	yes	—
32	C₂×#6101	C₂²	1	2	yes	—
33	C₂×#6101	C₂²	1	8	yes	—
34	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
35	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
36	#88143	C₂²	1	16	yes	complemented
37	#88143	C₂²	1	16	yes	complemented
38	#88142	C₂²	1	16	yes	complemented
39	#88132	C₂²	1	16	yes	—
40	#88131	C₂²	1	16	yes	—
41	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×Q₈×C₁₃	—	4	2	no	—
43	C₂×Q₈×C₁₃	C₂³	1	4	yes	—
44	C₂²×C₄×C₁₃	—	4	2	no	—
45	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
50	C₂²×Dic₁₃	—	4	2	no	—
51	C₂²×Dic₁₃	Q₈	1	8	yes	complemented
52	C₂²×Dic₁₃	C₂³	1	2	yes	—
53	C₂²×Dic₁₃	C₂³	1	2	yes	—
54	C₂²×Dic₁₃	C₂³	1	2	yes	—
55	C₂²×Dic₁₃	C₂³	1	4	yes	—
56	C₂²×Dic₁₃	C₂³	1	2	yes	—
57	C₂²×Dic₁₃	C₂³	1	4	yes	—
58	C₂×Dic₂₆	—	4	2	no	—
59	C₂×Dic₂₆	—	4	4	no	—
60	C₂×Dic₂₆	C₂³	1	8	yes	—
61	C₂×Dic₂₆	C₂³	1	4	yes	—
62	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
63	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
64	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
65	C₂²⋊Dic₁₃	C₂³	1	8	yes	—
66	C₄⋊Dic₁₃	C₂³	1	8	yes	—
67	C₄⋊Dic₁₃	C₂³	1	4	yes	—
68	#6101	C₂³	1	8	yes	—
69	#6101	C₂³	1	8	yes	—
70	#6101	C₂³	1	8	yes	—
71	#6101	C₂³	1	8	yes	—
72	#6101	C₂³	1	8	yes	—
73	#6101	C₂³	1	8	yes	—
74	#6101	C₂³	1	8	yes	—
75	C₄×Dic₁₃	C₂³	1	8	yes	—
76	C₄×Dic₁₃	C₂³	1	8	yes	—
77	C₂×#544	—	13	1	no	maximal, Sylow
78	C₂³×C₁₃	—	2	2	no	—
79	C₂³×C₁₃	—	2	2	no	—
80	C₂³×C₁₃	C₂×Q₈	1	4	yes	—
81	C₂³×C₁₃	C₂⁴	1	2	yes	—
82	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
83	Q₈×C₁₃	—	4	4	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	2	4	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	2	4	no	—
88	C₂×C₄×C₁₃	—	4	4	no	—
89	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
90	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
91	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
92	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
93	C₂×Dic₁₃	—	2	4	no	—
94	C₂×Dic₁₃	—	2	4	no	—
95	C₂×Dic₁₃	—	2	4	no	—
96	C₂×Dic₁₃	—	4	4	no	—
97	C₂×Dic₁₃	C₂×Q₈	1	8	yes	—
98	C₂×Dic₁₃	C₂×Q₈	1	16	yes	complemented
99	C₂×Dic₁₃	C₂⁴	1	4	yes	—
100	C₂×Dic₁₃	C₂⁴	1	4	yes	—
101	C₂×Dic₁₃	C₂⁴	1	4	yes	—
102	C₂×Dic₁₃	C₂⁴	1	4	yes	—
103	Dic₂₆	—	4	4	no	—
104	Dic₂₆	—	4	8	no	—
105	#544	—	13	16	no	—
106	C₂×C₄⋊Q₈	—	13	2	no	—
107	C₂×C₄⋊Q₈	—	13	2	no	—
108	C₂×#125	—	13	2	no	—
109	C₂×#125	—	13	2	no	—
110	C₂×C₂²⋊Q₈	—	13	1	no	—
111	C₂×C₂²⋊Q₈	—	13	2	no	—
112	C₂×C₂²⋊Q₈	D₁₃	1	1	yes	characteristic
113	C₂×#117	—	13	2	no	—
114	C₂²×C₄⋊C₄	—	13	1	no	—
115	C₂²×C₁₃	—	2	8	no	—
116	C₂²×C₁₃	—	2	2	no	—
117	C₂²×C₁₃	—	2	2	no	—
118	C₂²×C₁₃	C₂²×Q₈	1	4	yes	—
119	C₂²×C₁₃	C₂²×Q₈	1	2	yes	—
120	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
121	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
122	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
123	C₄×C₁₃	—	2	2	no	—
124	C₄×C₁₃	—	2	4	no	—
125	C₄×C₁₃	—	2	2	no	—
126	C₄×C₁₃	—	2	4	no	—
127	Dic₁₃	—	2	4	no	—
128	Dic₁₃	—	2	4	no	—
129	Dic₁₃	—	2	4	no	—
130	Dic₁₃	C₂²×Q₈	1	8	yes	—
131	C₂²×Q₈	—	13	1	no	—
132	C₂²×Q₈	—	13	2	no	—
133	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
134	C₂³×C₄	—	13	1	no	—
135	C₂³×C₄	—	13	1	no	—
136	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
137	C₄⋊Q₈	—	13	16	no	—
138	C₄⋊Q₈	—	13	16	no	—
139	#125	—	13	16	no	—
140	#125	—	13	16	no	—
141	C₂²⋊Q₈	—	13	8	no	—
142	C₂²⋊Q₈	—	13	16	no	—
143	C₂²⋊Q₈	C₂×D₁₃	1	8	yes	—
144	#117	—	13	16	no	—
145	C₂×C₄⋊C₄	—	13	2	no	—
146	C₂×C₄⋊C₄	—	13	2	no	—
147	C₂×C₄⋊C₄	—	13	8	no	—
148	C₂×C₄⋊C₄	—	13	2	no	—
149	C₂×C₄⋊C₄	—	13	2	no	—
150	C₂×C₄⋊C₄	—	13	2	no	—
151	C₂×C₄⋊C₄	—	13	1	no	—
152	C₂×C₄⋊C₄	—	13	2	no	—
153	C₂×C₄⋊C₄	—	13	2	no	—
154	C₂×C₄⋊C₄	—	13	2	no	—
155	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
156	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
157	C₂×C₂²⋊C₄	—	13	2	no	—
158	C₂×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
159	C₂×C₄²	—	13	2	no	—
160	C₂×C₄²	—	13	2	no	—
161	C₂×C₁₃	—	2	4	no	—
162	C₂×C₁₃	#544	1	4	yes	complemented
163	C₂×C₁₃	C₂³×Q₈	1	1	yes	characteristic
164	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
165	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
166	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
167	C₂×Q₈	—	4	2	no	—
168	C₂×Q₈	—	13	4	no	—
169	C₂×Q₈	—	13	8	no	—
170	C₂×Q₈	—	52	2	no	—
171	C₂×Q₈	—	52	4	no	—
172	C₂×Q₈	C₂²×D₁₃	1	4	yes	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	—	13	8	no	—
175	C₂²×C₄	—	13	4	no	—
176	C₂²×C₄	—	13	2	no	—
177	C₂²×C₄	—	13	2	no	—
178	C₂²×C₄	—	13	2	no	—
179	C₂²×C₄	—	13	2	no	—
180	C₂²×C₄	—	13	4	no	—
181	C₂²×C₄	—	52	2	no	—
182	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
183	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
184	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
185	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
186	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
187	C₄⋊C₄	—	13	8	no	—
188	C₄⋊C₄	—	13	8	no	—
189	C₄⋊C₄	—	13	8	no	—
190	C₄⋊C₄	—	13	8	no	—
191	C₄⋊C₄	—	13	8	no	—
192	C₄⋊C₄	—	13	4	no	—
193	C₄⋊C₄	—	13	8	no	—
194	C₄⋊C₄	—	13	8	no	—
195	C₄⋊C₄	—	13	8	no	—
196	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
197	C₄⋊C₄	C₂²×D₁₃	1	8	yes	—
198	C₂²⋊C₄	—	13	8	no	—
199	C₂²⋊C₄	C₂²×D₁₃	1	8	yes	—
200	C₄²	—	13	8	no	—
201	C₄²	—	13	8	no	—
202	C₁₃	C₂×#544	1	1	yes	characteristic, complemented, Sylow
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	Q₈×D₁₃	1	4	yes	—
206	C₂³	C₂³×D₁₃	1	2	yes	—
207	C₂³	C₂³×D₁₃	1	1	yes	characteristic
208	Q₈	—	4	4	no	—
209	Q₈	—	52	4	no	—
210	Q₈	—	52	8	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	13	8	no	—
217	C₂×C₄	—	13	16	no	—
218	C₂×C₄	—	13	4	no	—
219	C₂×C₄	—	13	4	no	—
220	C₂×C₄	—	13	4	no	—
221	C₂×C₄	—	13	4	no	—
222	C₂×C₄	—	26	4	no	—
223	C₂×C₄	—	26	4	no	—
224	C₂×C₄	—	26	4	no	—
225	C₂×C₄	—	52	4	no	—
226	C₂×C₄	C₂³×D₁₃	1	4	yes	—
227	C₂×C₄	C₂³×D₁₃	1	4	yes	—
228	C₂×C₄	C₂³×D₁₃	1	2	yes	—
229	C₂×C₄	C₂³×D₁₃	1	2	yes	—
230	C₂²	—	2	8	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	2	no	—
233	C₂²	#88251	1	2	yes	—
234	C₂²	C₂×Q₈×D₁₃	1	4	yes	—
235	C₂²	C₂×Q₈×D₁₃	1	2	yes	—
236	C₂²	#88257	1	2	yes	—
237	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	2	4	no	—
241	C₄	—	2	4	no	—
242	C₄	—	13	8	no	—
243	C₄	—	26	4	no	—
244	C₄	—	26	4	no	—
245	C₄	—	26	4	no	—
246	C₂	—	2	4	no	—
247	C₂	#11719328	1	4	yes	direct product factor
248	C₂	C₂×#88251	1	1	yes	characteristic
249	C₂	C₂²×Q₈×D₁₃	1	1	yes	characteristic
250	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×#544
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×#88198	C₂	semidirect
5. #11719328	C₂	direct (N × H)
6. C₂×#88143	C₂	semidirect
7. C₂×#88143	C₂	semidirect
8. C₂×#88142	C₂	semidirect
22. #88198	C₂²	semidirect
36. #88143	C₂²	semidirect
37. #88143	C₂²	semidirect
38. #88142	C₂²	semidirect
51. C₂²×Dic₁₃	Q₈	semidirect
98. C₂×Dic₁₃	C₂×Q₈	semidirect
162. C₂×C₁₃	#544	semidirect
202. C₁₃	C₂×#544	semidirect
247. C₂	#11719328	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×#88198	2
C₂×#88178	2
C₂²×#6101	2
#11719328	2
C₂×#88143	2
C₂×#88142	2
C₂×#88132	2
C₂×#88131	2
C₂×#544	13

Subgroups — C2×#11719328 #13128235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11719328 #13128235