Subgroups — C₂×#11719434 #13128233

Order 1664.

1080 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88200	C₂	1	2	yes	maximal
3	C₂×#88198	C₂	1	2	yes	maximal, complemented
4	#11719434	C₂	1	16	yes	maximal, direct product factor
5	C₂×#88150	C₂	1	2	yes	maximal
6	C₂×#88146	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊C₄×D₁₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#88144	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#88143	C₂	1	2	yes	maximal, complemented
10	C₂×#88142	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#88126	C₂	1	1	yes	maximal, characteristic
12	C₂×#88124	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
14	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
16	C₂²×Dic₂₆	C₂²	1	2	yes	—
17	C₁₃×C₄⋊Q₈	C₂²	1	8	yes	complemented
18	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
22	#88200	C₂²	1	16	yes	—
23	#88198	C₂²	1	16	yes	complemented
24	C₂×#6103	C₂²	1	1	yes	characteristic
25	C₂×#6103	C₂²	1	1	yes	characteristic
26	C₂×#6103	C₂²	1	2	yes	—
27	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
30	C₂×#6101	C₂²	1	2	yes	—
31	C₂×#6101	C₂²	1	2	yes	—
32	C₂×#6101	C₂²	1	1	yes	characteristic
33	C₂×#6101	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
38	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
39	#88150	C₂²	1	16	yes	—
40	#88146	C₂²	1	8	yes	—
41	C₄⋊C₄×D₁₃	C₂²	1	8	yes	—
42	#88144	C₂²	1	8	yes	complemented
43	#88143	C₂²	1	16	yes	complemented
44	#88142	C₂²	1	8	yes	complemented
45	#88126	C₂²	1	8	yes	—
46	#88124	C₂²	1	8	yes	complemented
47	C₂³×D₁₃	C₂³	1	1	yes	characteristic
48	C₂×Q₈×C₁₃	—	4	4	no	—
49	C₂×Q₈×C₁₃	C₂³	1	8	yes	—
50	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
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	—	4	4	no	—
61	C₂×C₄×D₁₃	Q₈	1	8	yes	complemented
62	C₂×C₄×D₁₃	C₂³	1	8	yes	—
63	C₂×C₄×D₁₃	C₂³	1	4	yes	—
64	C₂×Dic₂₆	—	4	4	no	—
65	C₂×Dic₂₆	C₂³	1	8	yes	—
66	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
67	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
68	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
69	C₄²×C₁₃	C₂³	1	4	yes	—
70	#6103	C₂³	1	4	yes	—
71	#6103	C₂³	1	4	yes	—
72	#6103	C₂³	1	8	yes	—
73	C₄⋊Dic₁₃	C₂³	1	4	yes	—
74	C₄⋊Dic₁₃	C₂³	1	4	yes	—
75	C₄⋊Dic₁₃	C₂³	1	8	yes	—
76	#6101	C₂³	1	8	yes	—
77	#6101	C₂³	1	8	yes	—
78	#6101	C₂³	1	4	yes	—
79	#6101	C₂³	1	4	yes	—
80	#6101	C₂³	1	8	yes	—
81	#6101	C₂³	1	8	yes	—
82	#6101	C₂³	1	8	yes	—
83	C₄×Dic₁₃	C₂³	1	4	yes	—
84	C₄×Dic₁₃	C₂³	1	8	yes	—
85	C₂×#544	—	13	1	no	maximal, Sylow
86	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₂²×D₁₃	—	2	2	no	—
88	C₂²×D₁₃	—	2	2	no	—
89	C₂²×D₁₃	C₂×Q₈	1	4	yes	—
90	C₂²×D₁₃	C₂⁴	1	2	yes	—
91	Q₈×C₁₃	—	4	8	no	—
92	C₂×C₄×C₁₃	—	2	4	no	—
93	C₂×C₄×C₁₃	—	2	4	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	C₂×Q₈	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×C₄×C₁₃	C₂⁴	1	4	yes	—
101	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
102	C₂×Dic₁₃	—	2	2	no	—
103	C₂×Dic₁₃	—	2	4	no	—
104	C₂×Dic₁₃	—	2	2	no	—
105	C₂×Dic₁₃	—	2	4	no	—
106	C₂×Dic₁₃	C₂×Q₈	1	4	yes	—
107	C₂×Dic₁₃	C₂⁴	1	2	yes	—
108	C₂×Dic₁₃	C₂⁴	1	4	yes	—
109	C₂×Dic₁₃	C₂⁴	1	2	yes	—
110	C₂×Dic₁₃	C₂⁴	1	4	yes	—
111	C₂×Dic₁₃	C₂⁴	1	2	yes	—
112	C₄×D₁₃	—	4	8	no	—
113	C₄×D₁₃	C₂×Q₈	1	16	yes	complemented
114	Dic₂₆	—	4	8	no	—
115	#544	—	13	16	no	—
116	C₂×C₄⋊Q₈	—	13	2	no	—
117	C₂×C₄⋊Q₈	—	13	1	no	—
118	C₂×C₄⋊Q₈	D₁₃	1	1	yes	characteristic, complemented
119	C₂×#125	—	13	2	no	—
120	C₂×#125	—	13	1	no	—
121	C₂×#125	—	13	1	no	—
122	C₂×C₂²⋊Q₈	—	13	2	no	—
123	C₂×C₂²⋊Q₈	—	13	2	no	—
124	C₂×#117	—	13	1	no	—
125	C₂×#117	—	13	1	no	—
126	C₂²×C₄⋊C₄	—	13	1	no	—
127	C₂²×C₁₃	C₂²×Q₈	1	2	yes	—
128	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
129	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
130	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
131	C₂×D₁₃	—	2	8	no	—
132	C₂×D₁₃	—	2	2	no	—
133	C₂×D₁₃	—	2	2	no	—
134	C₂×D₁₃	C₂²×Q₈	1	4	yes	—
135	C₄×C₁₃	—	2	4	no	—
136	C₄×C₁₃	—	2	4	no	—
137	C₄×C₁₃	—	2	2	no	—
138	C₄×C₁₃	—	2	2	no	—
139	C₄×C₁₃	C₂²×Q₈	1	4	yes	—
140	Dic₁₃	—	2	2	no	—
141	Dic₁₃	—	2	4	no	—
142	Dic₁₃	—	2	2	no	—
143	Dic₁₃	—	2	4	no	—
144	Dic₁₃	C₂²×Q₈	1	4	yes	—
145	C₂²×Q₈	—	13	2	no	—
146	C₂²×Q₈	C₂×D₁₃	1	2	yes	—
147	C₂³×C₄	—	13	1	no	—
148	C₂³×C₄	—	13	2	no	—
149	C₄⋊Q₈	—	13	16	no	—
150	C₄⋊Q₈	—	13	8	no	—
151	C₄⋊Q₈	C₂×D₁₃	1	8	yes	complemented
152	#125	—	13	16	no	—
153	#125	—	13	8	no	—
154	#125	—	13	8	no	—
155	C₂²⋊Q₈	—	13	16	no	—
156	C₂²⋊Q₈	—	13	16	no	—
157	#117	—	13	8	no	—
158	#117	—	13	8	no	—
159	C₂×C₄⋊C₄	—	13	2	no	—
160	C₂×C₄⋊C₄	—	13	1	no	—
161	C₂×C₄⋊C₄	—	13	1	no	—
162	C₂×C₄⋊C₄	—	13	1	no	—
163	C₂×C₄⋊C₄	—	13	1	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	—	13	2	no	—
168	C₂×C₄⋊C₄	—	13	2	no	—
169	C₂×C₄⋊C₄	—	13	8	no	—
170	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
171	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
172	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
173	C₂×C₂²⋊C₄	—	13	2	no	—
174	C₂×C₂²⋊C₄	—	13	1	no	—
175	C₂×C₂²⋊C₄	—	13	1	no	—
176	C₂×C₄²	—	13	2	no	—
177	C₂×C₄²	—	13	1	no	—
178	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
179	C₂×C₁₃	#544	1	4	yes	complemented
180	C₂×C₁₃	C₂³×Q₈	1	1	yes	characteristic
181	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
182	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
183	D₁₃	—	2	4	no	—
184	C₂⁴	—	13	1	no	—
185	C₂×Q₈	—	4	4	no	—
186	C₂×Q₈	—	13	8	no	—
187	C₂×Q₈	—	52	4	no	—
188	C₂×Q₈	C₂²×D₁₃	1	8	yes	—
189	C₂²×C₄	—	13	1	no	—
190	C₂²×C₄	—	13	8	no	—
191	C₂²×C₄	—	13	4	no	—
192	C₂²×C₄	—	13	1	no	—
193	C₂²×C₄	—	13	2	no	—
194	C₂²×C₄	—	13	1	no	—
195	C₂²×C₄	—	13	2	no	—
196	C₂²×C₄	—	13	8	no	—
197	C₂²×C₄	—	52	4	no	—
198	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
199	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
202	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
203	C₄⋊C₄	—	13	8	no	—
204	C₄⋊C₄	—	13	4	no	—
205	C₄⋊C₄	—	13	4	no	—
206	C₄⋊C₄	—	13	4	no	—
207	C₄⋊C₄	—	13	4	no	—
208	C₄⋊C₄	—	13	8	no	—
209	C₄⋊C₄	—	13	8	no	—
210	C₄⋊C₄	—	13	8	no	—
211	C₄⋊C₄	—	13	8	no	—
212	C₄⋊C₄	—	13	8	no	—
213	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
214	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
215	C₄⋊C₄	C₂²×D₁₃	1	8	yes	—
216	C₂²⋊C₄	—	13	8	no	—
217	C₂²⋊C₄	—	13	4	no	—
218	C₂²⋊C₄	—	13	4	no	—
219	C₄²	—	13	8	no	—
220	C₄²	—	13	4	no	—
221	C₄²	C₂²×D₁₃	1	4	yes	—
222	C₁₃	C₂×#544	1	1	yes	characteristic, complemented, Sylow
223	C₂³	—	13	4	no	—
224	C₂³	—	13	2	no	—
225	C₂³	—	26	2	no	—
226	C₂³	—	26	2	no	—
227	C₂³	C₂³×D₁₃	1	1	yes	characteristic
228	Q₈	—	4	8	no	—
229	Q₈	—	52	8	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	13	4	no	—
235	C₂×C₄	—	13	2	no	—
236	C₂×C₄	—	13	16	no	—
237	C₂×C₄	—	13	2	no	—
238	C₂×C₄	—	13	4	no	—
239	C₂×C₄	—	13	2	no	—
240	C₂×C₄	—	13	4	no	—
241	C₂×C₄	—	26	4	no	—
242	C₂×C₄	—	26	2	no	—
243	C₂×C₄	—	26	2	no	—
244	C₂×C₄	—	26	4	no	—
245	C₂×C₄	—	52	8	no	—
246	C₂×C₄	Q₈×D₁₃	1	4	yes	—
247	C₂×C₄	C₂³×D₁₃	1	4	yes	—
248	C₂×C₄	C₂³×D₁₃	1	2	yes	—
249	C₂×C₄	C₂³×D₁₃	1	2	yes	—
250	C₂×C₄	C₂³×D₁₃	1	4	yes	—
251	C₂×C₄	C₂³×D₁₃	1	2	yes	—
252	C₂²	—	13	4	no	—
253	C₂²	—	26	8	no	—
254	C₂²	—	26	2	no	—
255	C₂²	—	26	2	no	—
256	C₂²	#88251	1	2	yes	—
257	C₂²	C₂×Q₈×D₁₃	1	2	yes	—
258	C₂²	#88254	1	2	yes	—
259	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	4	no	—
263	C₄	—	2	2	no	—
264	C₄	—	13	4	no	—
265	C₄	—	26	4	no	—
266	C₄	—	26	2	no	—
267	C₄	—	26	2	no	—
268	C₄	—	26	4	no	—
269	C₄	C₂×Q₈×D₁₃	1	4	yes	—
270	C₂	—	26	4	no	—
271	C₂	#11719434	1	4	yes	direct product factor
272	C₂	C₂×#88251	1	1	yes	characteristic
273	C₂	C₂²×Q₈×D₁₃	1	1	yes	characteristic
274	C₂	C₂×#88254	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃
Fitting	C₂×C₁₃×C₄⋊Q₈	C₂
socle	C₂³×C₁₃	C₂⁴
prime radical	C₂³×D₁₃	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	C₂×#544
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×C₁₃×C₄⋊Q₈	C₂	semidirect
3. C₂×#88198	C₂	semidirect
4. #11719434	C₂	direct (N × H)
8. C₂×#88144	C₂	semidirect
9. C₂×#88143	C₂	semidirect
10. C₂×#88142	C₂	semidirect
12. C₂×#88124	C₂	semidirect
17. C₁₃×C₄⋊Q₈	C₂²	semidirect
23. #88198	C₂²	semidirect
42. #88144	C₂²	semidirect
43. #88143	C₂²	semidirect
44. #88142	C₂²	semidirect
46. #88124	C₂²	semidirect
61. C₂×C₄×D₁₃	Q₈	semidirect
113. C₄×D₁₃	C₂×Q₈	semidirect
118. C₂×C₄⋊Q₈	D₁₃	semidirect
151. C₄⋊Q₈	C₂×D₁₃	semidirect
179. C₂×C₁₃	#544	semidirect
222. C₁₃	C₂×#544	semidirect
271. C₂	#11719434	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×C₄⋊Q₈	2
C₂×#88200	2
C₂×#88198	2
#11719434	2
C₂×#88150	2
C₂×#88146	2
C₂×C₄⋊C₄×D₁₃	2
C₂×#88144	2
C₂×#88143	2
C₂×#88142	2
C₂×#88126	2
C₂×#88124	2
C₂×#544	13

Subgroups — C2×#11719434 #13128233

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11719434 #13128233