Subgroups — C₂×#11719288 #13128200

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×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88200	C₂	1	1	yes	maximal, characteristic
3	C₂×#88198	C₂	1	1	yes	maximal, characteristic, complemented
4	#11719288	C₂	1	16	yes	maximal, direct product factor
5	C₂×#88152	C₂	1	2	yes	maximal
6	C₂×#88151	C₂	1	1	yes	maximal, characteristic
7	C₂×#88144	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#88143	C₂	1	2	yes	maximal, complemented
9	C₂×#88130	C₂	1	2	yes	maximal
10	C₂×#88126	C₂	1	1	yes	maximal, characteristic
11	C₂×#88124	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
16	C₄×Q₈×C₁₃	C₂²	1	8	yes	complemented
17	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
21	#88200	C₂²	1	8	yes	—
22	#88198	C₂²	1	8	yes	complemented
23	C₂×#6103	C₂²	1	1	yes	characteristic
24	C₂×#6103	C₂²	1	1	yes	characteristic
25	C₂×#6103	C₂²	1	2	yes	—
26	C₂×#6103	C₂²	1	2	yes	—
27	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
29	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
30	C₂×#6101	C₂²	1	2	yes	—
31	C₂×#6101	C₂²	1	1	yes	characteristic
32	C₂×#6101	C₂²	1	2	yes	—
33	C₂×#6101	C₂²	1	2	yes	—
34	C₂×#6101	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
39	#88152	C₂²	1	16	yes	—
40	#88151	C₂²	1	8	yes	—
41	#88144	C₂²	1	8	yes	complemented
42	#88143	C₂²	1	16	yes	complemented
43	#88130	C₂²	1	16	yes	—
44	#88126	C₂²	1	8	yes	—
45	#88124	C₂²	1	16	yes	complemented
46	C₄×Dic₂₆	C₂²	1	8	yes	complemented
47	C₂³×D₁₃	C₂³	1	1	yes	characteristic
48	C₂×Q₈×C₁₃	—	2	4	no	—
49	C₂×Q₈×C₁₃	C₂³	1	4	yes	—
50	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
55	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
56	C₂²×Dic₁₃	C₂³	1	2	yes	—
57	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
58	C₂²×Dic₁₃	C₂³	1	2	yes	—
59	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
60	C₂×C₄×D₁₃	—	2	4	no	—
61	C₂×C₄×D₁₃	C₂³	1	4	yes	—
62	C₂×Dic₂₆	—	2	4	no	—
63	C₂×Dic₂₆	C₂³	1	4	yes	—
64	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
65	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
66	C₄²×C₁₃	C₂³	1	4	yes	—
67	C₄²×C₁₃	C₂³	1	8	yes	—
68	#6103	C₂³	1	4	yes	—
69	#6103	C₂³	1	4	yes	—
70	#6103	C₂³	1	8	yes	—
71	#6103	C₂³	1	8	yes	—
72	C₄⋊Dic₁₃	C₂³	1	4	yes	—
73	C₄⋊Dic₁₃	C₂³	1	8	yes	—
74	C₄⋊Dic₁₃	C₂³	1	8	yes	—
75	#6101	C₂³	1	8	yes	—
76	#6101	C₂³	1	4	yes	—
77	#6101	C₂³	1	8	yes	—
78	#6101	C₂³	1	8	yes	—
79	#6101	C₂³	1	4	yes	—
80	#6101	C₂³	1	8	yes	—
81	#6101	C₂³	1	8	yes	—
82	C₄×Dic₁₃	C₂³	1	8	yes	—
83	C₄×Dic₁₃	C₂³	1	4	yes	—
84	C₂×#541	—	13	1	no	maximal, Sylow
85	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
86	C₂²×D₁₃	—	2	2	no	—
87	C₂²×D₁₃	—	2	2	no	—
88	C₂²×D₁₃	—	2	2	no	—
89	C₂²×D₁₃	C₂⁴	1	2	yes	—
90	Q₈×C₁₃	—	2	8	no	—
91	C₂×C₄×C₁₃	—	2	4	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	4	no	—
95	C₂×C₄×C₁₃	#41	1	4	yes	—
96	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
97	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
98	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
99	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
100	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
101	C₂×Dic₁₃	—	2	4	no	—
102	C₂×Dic₁₃	—	2	2	no	—
103	C₂×Dic₁₃	—	2	4	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	4	yes	—
108	C₂×Dic₁₃	C₂⁴	1	2	yes	—
109	C₂×Dic₁₃	C₂⁴	1	4	yes	—
110	C₂×Dic₁₃	C₂⁴	1	2	yes	—
111	C₄×D₁₃	—	2	8	no	—
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	—	13	1	no	—
122	C₂×C₄×Q₈	—	13	1	no	—
123	C₂×C₄×Q₈	D₁₃	1	1	yes	characteristic, complemented
124	C₂×#117	—	13	1	no	—
125	C₂²×C₁₃	C₂×#41	1	2	yes	—
126	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
127	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
128	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
129	C₂×D₁₃	—	2	2	no	—
130	C₂×D₁₃	—	2	2	no	—
131	C₂×D₁₃	—	2	2	no	—
132	C₂×D₁₃	—	4	4	no	—
133	C₄×C₁₃	—	2	4	no	—
134	C₄×C₁₃	—	2	2	no	—
135	C₄×C₁₃	—	2	2	no	—
136	C₄×C₁₃	—	2	4	no	—
137	C₄×C₁₃	C₂×#41	1	4	yes	—
138	Dic₁₃	—	2	4	no	—
139	Dic₁₃	—	2	2	no	—
140	Dic₁₃	—	2	4	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	—	13	1	no	—
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	—	13	8	no	—
154	C₄×Q₈	—	13	8	no	—
155	C₄×Q₈	C₂×D₁₃	1	8	yes	complemented
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	—	13	2	no	—
173	C₂×C₄²	—	13	1	no	—
174	C₂×C₄²	—	13	2	no	—
175	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
176	C₂×C₄²	C₂×D₁₃	1	2	yes	—
177	C₂×C₁₃	#541	1	4	yes	complemented
178	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
179	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
180	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
181	D₁₃	—	4	2	no	—
182	C₂⁴	—	13	1	no	—
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	—	13	1	no	—
188	C₂²×C₄	—	13	2	no	—
189	C₂²×C₄	—	13	1	no	—
190	C₂²×C₄	—	13	2	no	—
191	C₂²×C₄	—	13	1	no	—
192	C₂²×C₄	—	13	4	no	—
193	C₂²×C₄	—	26	4	no	—
194	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
195	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
196	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
197	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
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	—	13	8	no	—
215	C₄²	—	13	4	no	—
216	C₄²	—	13	8	no	—
217	C₄²	C₂²×D₁₃	1	4	yes	—
218	C₄²	C₂²×D₁₃	1	8	yes	—
219	C₁₃	C₂×#541	1	1	yes	characteristic, complemented, Sylow
220	C₂³	—	13	2	no	—
221	C₂³	—	26	2	no	—
222	C₂³	—	26	2	no	—
223	C₂³	—	26	2	no	—
224	C₂³	C₂³×D₁₃	1	1	yes	characteristic
225	Q₈	—	2	8	no	—
226	Q₈	—	26	8	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	13	2	no	—
232	C₂×C₄	—	13	4	no	—
233	C₂×C₄	—	13	2	no	—
234	C₂×C₄	—	13	4	no	—
235	C₂×C₄	—	13	2	no	—
236	C₂×C₄	—	26	2	no	—
237	C₂×C₄	—	26	4	no	—
238	C₂×C₄	—	26	2	no	—
239	C₂×C₄	—	26	4	no	—
240	C₂×C₄	—	26	2	no	—
241	C₂×C₄	—	26	8	no	—
242	C₂×C₄	#6127	1	4	yes	—
243	C₂×C₄	C₂³×D₁₃	1	2	yes	—
244	C₂×C₄	C₂³×D₁₃	1	2	yes	—
245	C₂×C₄	C₂³×D₁₃	1	4	yes	—
246	C₂×C₄	C₂³×D₁₃	1	4	yes	—
247	C₂×C₄	C₂³×D₁₃	1	2	yes	—
248	C₂²	—	26	2	no	—
249	C₂²	—	26	2	no	—
250	C₂²	—	26	2	no	—
251	C₂²	—	52	4	no	—
252	C₂²	C₂×#6127	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	4	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	4	no	—
260	C₄	—	26	2	no	—
261	C₄	—	26	4	no	—
262	C₄	—	26	2	no	—
263	C₄	—	26	4	no	—
264	C₄	—	26	2	no	—
265	C₄	C₂×#6127	1	4	yes	—
266	C₂	—	52	2	no	—
267	C₂	#11719288	1	4	yes	direct product factor
268	C₂	C₂²×#6127	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×C₄×Q₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂⁴
prime radical	C₂³×D₁₃	C₂³
hypercenter	C₂×C₄×Q₈	D₁₃
hypocenter	C₁₃	C₂×#541
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×Q₈×C₁₃	C₂	semidirect
3. C₂×#88198	C₂	semidirect
4. #11719288	C₂	direct (N × H)
7. C₂×#88144	C₂	semidirect
8. C₂×#88143	C₂	semidirect
11. C₂×#88124	C₂	semidirect
12. C₂×C₄×Dic₂₆	C₂	semidirect
16. C₄×Q₈×C₁₃	C₂²	semidirect
22. #88198	C₂²	semidirect
41. #88144	C₂²	semidirect
42. #88143	C₂²	semidirect
45. #88124	C₂²	semidirect
46. C₄×Dic₂₆	C₂²	semidirect
123. C₂×C₄×Q₈	D₁₃	semidirect
155. C₄×Q₈	C₂×D₁₃	semidirect
177. C₂×C₁₃	#541	semidirect
219. C₁₃	C₂×#541	semidirect
267. C₂	#11719288	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×C₄×Q₈×C₁₃	2
C₂×#88200	2
C₂×#88198	2
#11719288	2
C₂×#88152	2
C₂×#88151	2
C₂×#88144	2
C₂×#88143	2
C₂×#88130	2
C₂×#88126	2
C₂×#88124	2
C₂×C₄×Dic₂₆	2
C₂×#541	13

Subgroups — C2×#11719288 #13128200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11719288 #13128200