Subgroups — C₂×#13123776 #16727291

Order 3328.

1894 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×#11719175	C₂	1	1	yes	maximal, characteristic
2	#13123776	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11718362	C₂	1	8	yes	maximal, complemented
4	C₂×#11718763	C₂	1	2	yes	maximal
5	C₂×#11718585	C₂	1	1	yes	maximal, characteristic
6	C₂×#11718762	C₂	1	2	yes	maximal
7	C₂²×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88180	C₂²	1	2	yes	—
12	C₂×#88178	C₄	1	4	yes	—
13	C₂²×#6099	C₂²	1	2	yes	—
14	#11719175	C₄	1	8	yes	—
15	#11719175	C₂²	1	8	yes	—
16	C₂×#88124	C₄	1	2	yes	—
17	C₂×#88123	C₄	1	2	yes	—
18	C₂×C₄×Dic₂₆	C₂²	1	4	yes	—
19	C₂×C₁₃×#97	C₂²	1	4	yes	—
20	C₂×C₁₃×#97	C₂²	1	8	yes	—
21	#11718763	C₂²	1	16	yes	—
22	#11718762	C₂²	1	16	yes	—
23	C₂×#88055	C₂²	1	4	yes	—
24	C₂×#88054	C₂²	1	8	yes	—
25	C₂×C₈×Dic₁₃	C₂²	1	4	yes	—
26	#11718585	C₂²	1	8	yes	—
27	C₂×C₄⋊#1055	C₂²	1	4	yes	—
28	#11718362	C₂²	1	64	yes	complemented
29	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×Dic₂₆	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₁₃	—	2	4	no	—
33	C₂²×C₈×C₁₃	Q₈	1	8	yes	complemented
34	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
38	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
42	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
43	C₂×C₂²⋊Dic₁₃	C₂×C₄	1	4	yes	—
44	#88180	C₂×C₄	1	8	yes	—
45	#88180	C₂³	1	8	yes	—
46	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
47	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
48	#88178	C₂×C₄	1	32	yes	—
49	C₂×#6101	C₂×C₄	1	8	yes	—
50	C₂×C₄×Dic₁₃	C₂³	1	4	yes	—
51	C₂×#6099	—	2	8	no	—
52	C₂×#6099	C₂³	1	8	yes	—
53	C₂²×#1055	C₂³	1	4	yes	—
54	#88124	C₂×C₄	1	16	yes	—
55	#88123	C₂×C₄	1	16	yes	—
56	C₄×Dic₂₆	C₂×C₄	1	16	yes	—
57	C₄×Dic₂₆	C₂³	1	16	yes	—
58	C₁₃×#97	C₂³	1	16	yes	—
59	#88055	C₂³	1	16	yes	—
60	#88054	C₂³	1	32	yes	—
61	C₈×Dic₁₃	C₂³	1	16	yes	—
62	C₄⋊#1055	C₂³	1	16	yes	—
63	C₂×#2961	—	13	1	no	maximal, Sylow
64	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	—	4	4	no	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
72	C₂²×C₄×C₁₃	#41	1	4	yes	—
73	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₂²×Dic₁₃	C₂²×C₄	1	8	yes	—
76	C₂×Dic₂₆	—	4	8	no	—
77	C₂×Dic₂₆	C₂²×C₄	1	16	yes	—
78	C₂×C₈×C₁₃	—	2	4	no	—
79	C₂×C₈×C₁₃	—	2	8	no	—
80	C₂×C₈×C₁₃	C₂×Q₈	1	8	yes	—
81	C₂×C₈×C₁₃	C₂×Q₈	1	16	yes	complemented
82	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
83	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
84	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
85	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
86	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
87	C₄²×C₁₃	C₂⁴	1	4	yes	—
88	C₂²⋊Dic₁₃	C₂²×C₄	1	16	yes	—
89	C₄⋊Dic₁₃	C₂²×C₄	1	16	yes	—
90	C₄⋊Dic₁₃	C₂²×C₄	1	16	yes	—
91	#6101	C₂²×C₄	1	32	yes	—
92	C₄×Dic₁₃	C₂²×C₄	1	8	yes	—
93	C₄×Dic₁₃	C₂⁴	1	8	yes	—
94	#6099	—	2	16	no	—
95	C₂×#1055	—	2	8	no	—
96	C₂×#1055	C₂⁴	1	8	yes	—
97	C₂×#533	—	13	1	no	—
98	#2961	—	13	16	no	—
99	C₂×#446	—	13	8	no	—
100	C₂×#424	—	13	2	no	—
101	C₂×#423	—	13	1	no	—
102	C₂×#405	—	13	2	no	—
103	C₂²×#97	D₁₃	1	1	yes	characteristic
104	C₂³×C₁₃	—	2	4	no	—
105	C₂³×C₁₃	C₄×Q₈	1	4	yes	—
106	C₂³×C₁₃	C₂³×C₄	1	2	yes	—
107	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	—	2	4	no	—
109	C₂×C₄×C₁₃	—	2	4	no	—
110	C₂×C₄×C₁₃	—	4	8	no	—
111	C₂×C₄×C₁₃	C₄×Q₈	1	8	yes	—
112	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
113	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
114	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
115	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
117	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
118	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
119	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
120	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
121	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
122	C₂×Dic₁₃	—	2	16	no	—
123	C₂×Dic₁₃	C₂³×C₄	1	16	yes	—
124	Dic₂₆	—	4	16	no	—
125	C₈×C₁₃	—	2	4	no	—
126	C₈×C₁₃	C₂²×Q₈	1	8	yes	—
127	#1055	—	2	8	no	—
128	C₂²×M₁₆	—	13	2	no	—
129	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
130	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
131	#533	—	13	8	no	—
132	#533	—	13	8	no	—
133	C₂×C₄⋊Q₈	—	13	2	no	—
134	C₂×#125	—	13	2	no	—
135	C₂×C₂²⋊Q₈	—	13	4	no	—
136	C₂×C₄×Q₈	—	13	4	no	—
137	C₂×#117	—	13	2	no	—
138	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
139	#446	—	13	64	no	—
140	#424	—	13	16	no	—
141	#423	—	13	8	no	—
142	C₂×C₄⋊C₈	—	13	8	no	—
143	C₂×C₄⋊C₈	—	13	4	no	—
144	#405	—	13	16	no	—
145	C₂×#97	—	13	4	no	—
146	C₂×#97	C₂×D₁₃	1	4	yes	—
147	C₂×#97	C₂×D₁₃	1	8	yes	—
148	C₂×C₄×C₈	—	13	4	no	—
149	C₂²×C₁₃	—	2	8	no	—
150	C₂²×C₁₃	—	2	4	no	—
151	C₂²×C₁₃	C₂×C₄×Q₈	1	4	yes	—
152	C₂²×C₁₃	C₂×C₄×Q₈	1	2	yes	—
153	C₂²×C₁₃	#568	1	4	yes	—
154	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
155	C₄×C₁₃	—	2	4	no	—
156	C₄×C₁₃	—	2	4	no	—
157	C₄×C₁₃	C₂×C₄×Q₈	1	2	yes	—
158	C₄×C₁₃	C₂×C₄×Q₈	1	4	yes	—
159	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
160	C₄×C₁₃	C₂²×#41	1	1	yes	characteristic
161	Dic₁₃	—	2	16	no	—
162	C₂²×Q₈	—	13	4	no	—
163	C₂³×C₄	C₄×D₁₃	1	2	yes	—
164	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
165	C₂×M₁₆	—	13	8	no	—
166	C₂×M₁₆	—	26	8	no	—
167	C₂²×C₈	—	2	4	no	—
168	C₂²×C₈	—	13	4	no	—
169	C₂²×C₈	Dic₂₆	1	8	yes	complemented
170	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
171	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
172	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
173	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
174	C₄⋊Q₈	—	13	16	no	—
175	#125	—	13	16	no	—
176	C₂²⋊Q₈	—	13	32	no	—
177	C₄×Q₈	—	13	16	no	—
178	C₄×Q₈	—	13	16	no	—
179	#117	—	13	8	no	—
180	#117	—	13	8	no	—
181	C₂×C₄⋊C₄	—	13	4	no	—
182	C₂×C₄⋊C₄	—	13	8	no	—
183	C₂×C₄⋊C₄	—	13	4	no	—
184	C₂×C₂²⋊C₄	—	13	4	no	—
185	C₂×C₄²	—	13	4	no	—
186	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
187	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
188	C₂×C₄²	C₄×D₁₃	1	4	yes	—
189	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
190	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
191	C₄⋊C₈	—	13	32	no	—
192	C₄⋊C₈	—	13	16	no	—
193	#97	—	13	16	no	—
194	#97	C₂²×D₁₃	1	16	yes	—
195	C₄×C₈	—	13	16	no	—
196	C₂×C₁₃	—	2	4	no	—
197	C₂×C₁₃	#2961	1	4	yes	complemented
198	C₂×C₁₃	C₂²×C₄×Q₈	1	1	yes	characteristic
199	C₂×C₁₃	C₂×#568	1	2	yes	—
200	C₂⁴	C₂×C₄×D₁₃	1	1	yes	characteristic
201	C₂×Q₈	—	13	16	no	—
202	C₂×Q₈	—	52	8	no	—
203	C₂²×C₄	—	4	4	no	—
204	C₂²×C₄	—	13	8	no	—
205	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
206	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
207	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
208	C₂²×C₄	C₂×C₄×D₁₃	1	8	yes	—
209	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
211	C₂²×C₄	#6127	1	4	yes	—
212	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
213	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
214	M₁₆	—	26	16	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	8	no	—
217	C₂×C₈	—	13	8	no	—
218	C₂×C₈	—	26	8	no	—
219	C₂×C₈	C₂×Dic₂₆	1	8	yes	—
220	C₂×C₈	C₂×Dic₂₆	1	16	yes	complemented
221	C₂×C₈	C₂³×D₁₃	1	4	yes	—
222	C₂×C₈	C₂³×D₁₃	1	4	yes	—
223	C₄⋊C₄	—	13	16	no	—
224	C₄⋊C₄	—	13	32	no	—
225	C₄⋊C₄	—	13	16	no	—
226	C₂²⋊C₄	—	13	16	no	—
227	C₄²	—	13	8	no	—
228	C₄²	—	13	8	no	—
229	C₄²	C₂×C₄×D₁₃	1	4	yes	—
230	C₄²	C₂×C₄×D₁₃	1	4	yes	—
231	C₄²	C₂×C₄×D₁₃	1	4	yes	—
232	C₄²	C₂³×D₁₃	1	4	yes	—
233	C₁₃	C₂×#2961	1	1	yes	characteristic, complemented, Sylow
234	C₂³	—	2	4	no	—
235	C₂³	C₄×Dic₂₆	1	4	yes	—
236	C₂³	C₂²×C₄×D₁₃	1	2	yes	—
237	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
238	Q₈	—	52	16	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	8	no	—
242	C₂×C₄	—	13	16	no	—
243	C₂×C₄	—	26	16	no	—
244	C₂×C₄	C₄×Dic₂₆	1	8	yes	—
245	C₂×C₄	C₄×Dic₂₆	1	4	yes	—
246	C₂×C₄	C₂²×Dic₂₆	1	2	yes	—
247	C₂×C₄	C₂²×Dic₂₆	1	4	yes	—
248	C₂×C₄	C₂²×C₄×D₁₃	1	4	yes	—
249	C₂×C₄	C₂²×C₄×D₁₃	1	4	yes	—
250	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
251	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
252	C₂×C₄	C₂×#6127	1	2	yes	—
253	C₂×C₄	C₂×#6127	1	4	yes	—
254	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
255	C₈	—	2	4	no	—
256	C₈	—	26	8	no	—
257	C₈	C₂²×Dic₂₆	1	8	yes	—
258	C₂²	—	2	4	no	—
259	C₂²	—	2	8	no	—
260	C₂²	C₂×C₄×Dic₂₆	1	4	yes	—
261	C₂²	C₂×C₄×Dic₂₆	1	2	yes	—
262	C₂²	#11719450	1	4	yes	—
263	C₂²	C₂³×C₄×D₁₃	1	1	yes	characteristic
264	C₄	—	2	4	no	—
265	C₄	—	2	4	no	—
266	C₄	—	26	16	no	—
267	C₄	C₂×C₄×Dic₂₆	1	2	yes	—
268	C₄	C₂×C₄×Dic₂₆	1	4	yes	—
269	C₄	C₂³×Dic₂₆	1	1	yes	characteristic
270	C₄	C₂²×#6127	1	1	yes	characteristic
271	C₂	—	2	4	no	—
272	C₂	#13123776	1	4	yes	direct product factor
273	C₂	C₂²×C₄×Dic₂₆	1	1	yes	characteristic
274	C₂	C₂×#11719450	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₁₃×#97	C₂
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂²×#97	D₁₃
hypocenter	C₁₃	C₂×#2961
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. #13123776	C₂	direct (N × H)
3. C₂×#11718362	C₂	semidirect
28. #11718362	C₂²	semidirect
33. C₂²×C₈×C₁₃	Q₈	semidirect
81. C₂×C₈×C₁₃	C₂×Q₈	semidirect
169. C₂²×C₈	Dic₂₆	semidirect
197. C₂×C₁₃	#2961	semidirect
220. C₂×C₈	C₂×Dic₂₆	semidirect
233. C₁₃	C₂×#2961	semidirect
272. C₂	#13123776	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×#11719175	2
#13123776	2
C₂×#11718362	2
C₂×#11718763	2
C₂×#11718585	2
C₂×#11718762	2
C₂²×C₁₃×#97	2
C₂×#2961	13

Subgroups — C2×#13123776 #16727291

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13123776 #16727291