Subgroups — C₂×#13123925 #16727868

Order 3328.

2022 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	#13123925	C₂	1	16	yes	maximal, direct product factor
2	C₂×#11718685	C₂	1	6	yes	maximal, complemented
3	C₂×Q₈×#1055	C₂	1	2	yes	maximal, complemented
4	C₂×#11718587	C₂	1	3	yes	maximal
5	C₂×#11718586	C₂	1	3	yes	maximal
6	C₂²×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₁₃	C₄	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₁₃	C₄	1	8	yes	—
9	C₂×C₄×Q₈×C₁₃	C₄	1	3	yes	—
10	C₂×C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₁₃	C₂²	1	8	yes	—
12	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₁₃	C₂²	1	3	yes	—
14	C₂²×C₁₃×C₄⋊C₄	C₄	1	3	yes	—
15	C₂²×C₄²×C₁₃	C₂²	1	3	yes	—
16	C₂×C₂²⋊#1055	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊#1055	C₂²	1	3	yes	—
18	#11718685	C₂²	1	48	yes	complemented
19	Q₈×#1055	C₂²	1	16	yes	complemented
20	#11718587	C₂²	1	24	yes	—
21	#11718586	C₂²	1	24	yes	—
22	C₂×C₄⋊#1055	C₂²	1	6	yes	—
23	C₂×C₄⋊#1055	C₂²	1	6	yes	—
24	C₂×#88043	C₂²	1	6	yes	—
25	C₂×C₄×#1055	C₂²	1	6	yes	—
26	C₂²×Q₈×C₁₃	—	2	8	no	—
27	C₂²×Q₈×C₁₃	C₂×C₄	1	8	yes	—
28	C₂²×Q₈×C₁₃	C₂×C₄	1	3	yes	—
29	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₁₃	C₂×C₄	1	3	yes	—
31	C₂³×C₄×C₁₃	C₂×C₄	1	3	yes	—
32	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₄×Q₈×C₁₃	C₂×C₄	1	24	yes	—
34	C₄×Q₈×C₁₃	C₂×C₄	1	12	yes	—
35	C₄×Q₈×C₁₃	C₂×C₄	1	8	yes	—
36	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
37	C₄×Q₈×C₁₃	C₂³	1	4	yes	—
38	C₄×Q₈×C₁₃	C₂³	1	12	yes	—
39	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	24	yes	—
40	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	6	yes	—
41	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
42	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
43	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
44	C₂×C₄²×C₁₃	C₂×C₄	1	12	yes	—
45	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
47	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
48	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
49	C₂²×#1055	C₂³	1	2	yes	—
50	C₂²×#1055	C₂³	1	6	yes	—
51	C₂²⋊#1055	C₂³	1	4	yes	—
52	C₂²⋊#1055	C₂³	1	12	yes	—
53	C₄⋊#1055	C₂³	1	24	yes	—
54	C₄⋊#1055	C₂³	1	24	yes	—
55	#88043	C₂³	1	24	yes	—
56	C₄×#1055	C₂³	1	24	yes	—
57	C₂×#2984	—	13	1	no	maximal, Sylow
58	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×Q₈×C₁₃	—	2	32	no	—
60	C₂×Q₈×C₁₃	—	2	12	no	—
61	C₂×Q₈×C₁₃	M₁₆	1	8	yes	—
62	C₂×Q₈×C₁₃	C₂²×C₄	1	12	yes	—
63	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₁₃	—	2	12	no	—
65	C₂²×C₄×C₁₃	—	2	12	no	—
66	C₂²×C₄×C₁₃	—	2	4	no	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
71	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
72	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
73	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
74	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
75	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
76	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
77	C₁₃×C₄⋊C₄	C₂²×C₄	1	24	yes	—
78	C₁₃×C₄⋊C₄	C₂²×C₄	1	12	yes	—
79	C₁₃×C₄⋊C₄	C₂²×C₄	1	12	yes	—
80	C₄²×C₁₃	C₂²×C₄	1	12	yes	—
81	C₄²×C₁₃	C₂²×C₄	1	12	yes	—
82	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
83	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
84	C₄²×C₁₃	C₂⁴	1	6	yes	—
85	C₄²×C₁₃	C₂⁴	1	6	yes	—
86	C₂×#1055	—	2	4	no	—
87	C₂×#1055	—	2	12	no	—
88	C₂×#1055	C₂⁴	1	4	yes	—
89	C₂×#1055	C₂⁴	1	12	yes	—
90	C₂²×C₄×Q₈	D₁₃	1	1	yes	characteristic
91	#2984	—	13	16	no	—
92	C₂×#446	—	13	6	no	—
93	C₂×C₈×Q₈	—	13	2	no	—
94	C₂×#432	—	13	3	no	—
95	C₂×#431	—	13	3	no	—
96	C₂³×C₁₃	—	2	2	no	—
97	C₂³×C₁₃	—	2	2	no	—
98	C₂³×C₁₃	#131	1	4	yes	—
99	C₂³×C₁₃	C₂³×C₄	1	2	yes	—
100	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
101	Q₈×C₁₃	—	2	24	no	—
102	Q₈×C₁₃	C₂×M₁₆	1	16	yes	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	6	no	—
105	C₂×C₄×C₁₃	—	2	24	no	—
106	C₂×C₄×C₁₃	—	2	6	no	—
107	C₂×C₄×C₁₃	—	2	6	no	—
108	C₂×C₄×C₁₃	—	2	4	no	—
109	C₂×C₄×C₁₃	—	2	24	no	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	C₂×M₁₆	1	12	yes	—
112	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
113	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
114	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
115	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
116	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
118	C₂×C₄×C₁₃	2₋¹⁺⁴	1	4	yes	—
119	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
120	#1055	—	2	4	no	—
121	#1055	—	2	12	no	—
122	C₂³×Q₈	Dic₁₃	1	1	yes	characteristic
123	C₂×C₄×Q₈	Dic₁₃	1	8	yes	—
124	C₂×C₄×Q₈	Dic₁₃	1	3	yes	—
125	C₂×C₄×Q₈	Dic₁₃	1	1	yes	characteristic
126	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
127	C₂×C₄×Q₈	C₂×D₁₃	1	3	yes	—
128	C₂×C₄×Q₈	C₂×D₁₃	1	8	yes	—
129	C₂²×C₄⋊C₄	Dic₁₃	1	3	yes	—
130	C₂²×C₄²	C₂×D₁₃	1	3	yes	—
131	#446	—	13	48	no	—
132	C₈×Q₈	—	13	16	no	—
133	#432	—	13	24	no	—
134	#431	—	13	24	no	—
135	C₂×C₄⋊C₈	—	13	6	no	—
136	C₂×C₄⋊C₈	—	13	6	no	—
137	C₂×C₂²⋊C₈	—	13	1	no	—
138	C₂×C₂²⋊C₈	—	13	3	no	—
139	C₂×#97	—	13	6	no	—
140	C₂×C₄×C₈	—	13	6	no	—
141	C₂²×C₁₃	—	2	2	no	—
142	C₂²×C₁₃	—	2	8	no	—
143	C₂²×C₁₃	—	2	2	no	—
144	C₂²×C₁₃	#519	1	2	yes	—
145	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
146	C₂²×C₁₃	C₂×#131	1	4	yes	—
147	C₂²×C₁₃	C₂×#131	1	2	yes	—
148	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
149	C₄×C₁₃	—	2	2	no	—
150	C₄×C₁₃	—	2	6	no	—
151	C₄×C₁₃	—	2	6	no	—
152	C₄×C₁₃	—	2	6	no	—
153	C₄×C₁₃	#519	1	2	yes	—
154	C₄×C₁₃	C₂²×M₁₆	1	12	yes	—
155	C₄×C₁₃	C₂×2₋¹⁺⁴	1	2	yes	—
156	C₂²×Q₈	—	2	8	no	—
157	C₂²×Q₈	C₂×Dic₁₃	1	8	yes	—
158	C₂²×Q₈	C₂×Dic₁₃	1	3	yes	—
159	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
160	C₂³×C₄	C₂×Dic₁₃	1	3	yes	—
161	C₂³×C₄	C₂×Dic₁₃	1	3	yes	—
162	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
163	C₂²×C₈	—	13	2	no	—
164	C₂²×C₈	—	13	6	no	—
165	C₄×Q₈	C₂×Dic₁₃	1	12	yes	—
166	C₄×Q₈	C₂×Dic₁₃	1	24	yes	—
167	C₄×Q₈	C₂×Dic₁₃	1	4	yes	—
168	C₄×Q₈	C₂×Dic₁₃	1	8	yes	—
169	C₄×Q₈	C₂²×D₁₃	1	4	yes	—
170	C₄×Q₈	C₂²×D₁₃	1	12	yes	—
171	C₂×C₄⋊C₄	C₂×Dic₁₃	1	6	yes	—
172	C₂×C₄⋊C₄	C₂×Dic₁₃	1	24	yes	—
173	C₂×C₄⋊C₄	C₂×Dic₁₃	1	3	yes	—
174	C₂×C₄⋊C₄	C₂×Dic₁₃	1	3	yes	—
175	C₂×C₄²	C₂×Dic₁₃	1	3	yes	—
176	C₂×C₄²	C₂×Dic₁₃	1	12	yes	—
177	C₂×C₄²	C₂×Dic₁₃	1	3	yes	—
178	C₂×C₄²	C₂²×D₁₃	1	12	yes	—
179	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
180	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
181	C₄⋊C₈	—	13	24	no	—
182	C₄⋊C₈	—	13	24	no	—
183	C₂²⋊C₈	—	13	4	no	—
184	C₂²⋊C₈	—	13	12	no	—
185	#97	—	13	24	no	—
186	C₄×C₈	—	13	24	no	—
187	C₂×C₁₃	—	2	4	no	—
188	C₂×C₁₃	#2984	1	4	yes	complemented
189	C₂×C₁₃	C₂×#519	1	1	yes	characteristic
190	C₂×C₁₃	C₂³×M₁₆	1	1	yes	characteristic
191	C₂×C₁₃	C₂²×#131	1	1	yes	characteristic
192	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
193	C₂×Q₈	—	2	32	no	—
194	C₂×Q₈	—	2	12	no	—
195	C₂×Q₈	#6099	1	8	yes	—
196	C₂×Q₈	C₂²×Dic₁₃	1	12	yes	—
197	C₂×Q₈	C₂²×Dic₁₃	1	4	yes	—
198	C₂²×C₄	—	2	12	no	—
199	C₂²×C₄	—	2	12	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
203	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
204	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
205	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
206	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
207	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
208	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
209	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
210	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
211	C₂×C₈	—	13	4	no	—
212	C₂×C₈	—	13	12	no	—
213	C₂×C₈	—	26	4	no	—
214	C₂×C₈	—	26	12	no	—
215	C₄⋊C₄	C₂²×Dic₁₃	1	24	yes	—
216	C₄⋊C₄	C₂²×Dic₁₃	1	12	yes	—
217	C₄⋊C₄	C₂²×Dic₁₃	1	12	yes	—
218	C₄²	C₂²×Dic₁₃	1	12	yes	—
219	C₄²	C₂²×Dic₁₃	1	12	yes	—
220	C₄²	C₂²×Dic₁₃	1	6	yes	—
221	C₄²	C₂²×Dic₁₃	1	6	yes	—
222	C₄²	C₂³×D₁₃	1	6	yes	—
223	C₄²	C₂³×D₁₃	1	6	yes	—
224	C₁₃	C₂×#2984	1	1	yes	characteristic, complemented, Sylow
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	#88202	1	4	yes	—
228	C₂³	C₂³×Dic₁₃	1	2	yes	—
229	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
230	Q₈	—	2	24	no	—
231	Q₈	C₂×#6099	1	16	yes	—
232	C₂×C₄	—	2	6	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	24	no	—
235	C₂×C₄	—	2	6	no	—
236	C₂×C₄	—	2	6	no	—
237	C₂×C₄	—	2	24	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	C₂×#6099	1	12	yes	—
241	C₂×C₄	#88254	1	4	yes	—
242	C₂×C₄	C₂³×Dic₁₃	1	2	yes	—
243	C₂×C₄	C₂³×Dic₁₃	1	6	yes	—
244	C₂×C₄	C₂³×Dic₁₃	1	6	yes	—
245	C₂×C₄	C₂³×Dic₁₃	1	6	yes	—
246	C₂×C₄	C₂³×Dic₁₃	1	6	yes	—
247	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
248	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
249	C₈	—	26	4	no	—
250	C₈	—	26	12	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	8	no	—
253	C₂²	—	2	2	no	—
254	C₂²	C₂²×#6099	1	2	yes	—
255	C₂²	#11719517	1	2	yes	—
256	C₂²	C₂×#88202	1	4	yes	—
257	C₂²	C₂×#88202	1	2	yes	—
258	C₂²	C₂⁴×Dic₁₃	1	1	yes	characteristic
259	C₄	—	2	6	no	—
260	C₄	—	2	6	no	—
261	C₄	—	2	6	no	—
262	C₄	—	2	2	no	—
263	C₄	C₂²×#6099	1	12	yes	—
264	C₄	#11719517	1	2	yes	—
265	C₄	C₂×#88254	1	2	yes	—
266	C₂	—	2	4	no	—
267	C₂	#13123925	1	4	yes	direct product factor
268	C₂	C₂×#11719517	1	1	yes	characteristic
269	C₂	C₂³×#6099	1	1	yes	characteristic
270	C₂	C₂²×#88202	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₄×Q₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₁₃
hypocenter	C₁₃	C₂×#2984
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #13123925	C₂	direct (N × H)
2. C₂×#11718685	C₂	semidirect
3. C₂×Q₈×#1055	C₂	semidirect
18. #11718685	C₂²	semidirect
19. Q₈×#1055	C₂²	semidirect
188. C₂×C₁₃	#2984	semidirect
224. C₁₃	C₂×#2984	semidirect
267. C₂	#13123925	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13123925	2
C₂×#11718685	2
C₂×Q₈×#1055	2
C₂×#11718587	2
C₂×#11718586	2
C₂²×C₄×Q₈×C₁₃	2
C₂×#2984	13

Subgroups — C2×#13123925 #16727868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13123925 #16727868