Subgroups — C₂×#11718484 #13123729

Order 1664.

464 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88204	C₂	1	1	yes	maximal, characteristic
2	C₂×#88202	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88161	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×#104	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#88062	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#88045	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#88043	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718484	C₂	1	8	yes	maximal, direct product factor
9	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
10	C₂×#6127	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	#88204	C₄	1	4	yes	complemented
14	#88204	C₂²	1	4	yes	—
15	#88202	—	2	2	no	—
16	#88202	C₂²	1	4	yes	complemented
17	C₂×#6107	C₄	1	1	yes	characteristic, complemented
18	C₂×#6106	C₄	1	1	yes	characteristic, complemented
19	C₂×#6105	C₄	1	1	yes	characteristic, complemented
20	C₂×#6104	C₄	1	1	yes	characteristic, complemented
21	C₂×#6099	—	2	1	no	—
22	C₂×#6099	C₂²	1	1	yes	characteristic
23	C₂²×#1055	—	2	1	no	—
24	C₂²×#1055	C₂²	1	1	yes	characteristic
25	C₂²×#1055	C₂²	1	1	yes	characteristic
26	#88161	—	2	2	no	—
27	#88161	C₂²	1	4	yes	complemented
28	C₂×#6094	—	2	1	no	—
29	C₂×C₈×D₁₃	—	2	1	no	—
30	C₁₃×#104	C₂²	1	4	yes	complemented
31	#88062	C₂²	1	4	yes	complemented
32	#88045	C₂²	1	4	yes	complemented
33	#88043	C₂²	1	4	yes	complemented
34	C₁₃×#41	—	2	1	no	—
35	C₁₃×#41	—	2	1	no	—
36	C₁₃×#41	C₂×C₄	1	2	yes	—
37	C₁₃×#41	C₂³	1	2	yes	—
38	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×D₄×C₁₃	—	2	1	no	—
40	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	—	2	1	no	—
42	C₂²×C₄×C₁₃	—	2	1	no	—
43	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₁₃	—	2	1	no	—
45	#6127	—	2	1	no	—
46	#6127	—	2	1	no	—
47	#6127	C₂×C₄	1	2	yes	—
48	#6127	C₂³	1	2	yes	—
49	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₃	—	2	1	no	—
51	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
52	C₁₃×M₁₆	—	2	1	no	—
53	C₁₃×M₁₆	C₂³	1	2	yes	—
54	C₂×C₈×C₁₃	—	2	1	no	—
55	C₄²×C₁₃	—	2	1	no	—
56	C₄²×C₁₃	C₂³	1	2	yes	—
57	#6107	C₂×C₄	1	4	yes	complemented
58	#6106	C₂×C₄	1	4	yes	complemented
59	#6105	C₂×C₄	1	4	yes	complemented
60	#6104	C₂×C₄	1	4	yes	complemented
61	#6099	—	2	1	no	—
62	#6099	—	2	4	no	—
63	#6099	C₂³	1	2	yes	—
64	C₂×#1055	—	2	1	no	—
65	C₂×#1055	—	2	4	no	—
66	C₂×#1055	C₂×C₄	1	1	yes	characteristic
67	C₂×#1055	C₂×C₄	1	1	yes	characteristic
68	C₂×#1055	C₂×C₄	1	2	yes	—
69	C₂×#1055	D₄	1	1	yes	characteristic
70	C₂×#1055	D₄	1	1	yes	characteristic
71	C₂×#1055	D₄	1	2	yes	—
72	C₂×#1055	C₂³	1	2	yes	—
73	C₂×#1055	C₂³	1	2	yes	—
74	#6094	—	2	4	no	—
75	C₈×D₁₃	—	2	4	no	—
76	C₂×#444	—	13	1	no	maximal, Sylow
77	C₂³×C₁₃	—	2	1	no	—
78	C₂³×C₁₃	#41	1	1	yes	characteristic
79	C₂²×D₁₃	—	2	1	no	—
80	Q₈×C₁₃	—	2	1	no	—
81	Q₈×C₁₃	C₂²×C₄	1	2	yes	—
82	D₄×C₁₃	—	2	2	no	—
83	D₄×C₁₃	—	2	2	no	—
84	D₄×C₁₃	—	2	1	no	—
85	D₄×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	4	2	no	—
91	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₁₃	—	2	2	no	—
98	C₂²⋊D₁₃	—	2	2	no	—
99	C₂×Dic₁₃	—	2	1	no	—
100	D₅₂	—	2	1	no	—
101	D₅₂	C₂²×C₄	1	2	yes	—
102	C₄×D₁₃	—	2	2	no	—
103	C₄×D₁₃	—	2	2	no	—
104	Dic₂₆	—	2	1	no	—
105	Dic₂₆	C₂²×C₄	1	2	yes	—
106	C₈×C₁₃	—	2	2	no	—
107	#1055	—	2	2	no	—
108	#1055	C₂²×C₄	1	2	yes	—
109	#1055	C₂²×C₄	1	2	yes	—
110	#1055	C₂×D₄	1	2	yes	—
111	#1055	C₂×D₄	1	2	yes	—
112	C₂×#135	—	13	1	no	—
113	C₂×#131	—	13	1	no	—
114	C₂×#131	—	13	1	no	—
115	#444	—	13	8	no	—
116	C₂×#108	—	13	1	no	—
117	C₂×#104	—	13	1	no	—
118	C₂×#104	D₁₃	1	1	yes	characteristic, complemented
119	C₂×#97	—	13	1	no	—
120	C₂²×C₁₃	—	2	2	no	—
121	C₂²×C₁₃	—	2	2	no	—
122	C₂²×C₁₃	—	4	2	no	—
123	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
124	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
125	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
126	C₂×D₁₃	—	2	2	no	—
127	C₂×D₁₃	—	4	2	no	—
128	C₄×C₁₃	—	2	2	no	—
129	C₄×C₁₃	—	4	2	no	—
130	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
131	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
132	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
133	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
134	Dic₁₃	—	2	2	no	—
135	C₂×#41	—	13	1	no	—
136	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
137	#135	—	13	4	no	—
138	#135	—	13	4	no	—
139	C₂×Q₁₆	—	13	1	no	—
140	C₂×SD₁₆	—	13	1	no	—
141	C₂×SD₁₆	—	13	1	no	—
142	C₂×D₈	—	13	1	no	—
143	#131	—	13	4	no	—
144	#131	—	13	4	no	—
145	#131	—	26	2	no	—
146	#131	—	26	2	no	—
147	C₂×M₁₆	—	13	1	no	—
148	C₂×M₁₆	—	26	1	no	—
149	C₂×M₁₆	—	26	1	no	—
150	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
151	C₂²×C₈	—	13	1	no	—
152	C₂²×C₈	—	13	1	no	—
153	C₂²×C₈	—	26	1	no	—
154	C₂²×C₈	—	26	1	no	—
155	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
156	#108	—	13	4	no	—
157	#104	—	13	4	no	—
158	#104	C₂×D₁₃	1	4	yes	complemented
159	#97	—	13	4	no	—
160	C₂×C₁₃	—	2	1	no	—
161	C₂×C₁₃	—	2	1	no	—
162	C₂×C₁₃	—	4	2	no	—
163	C₂×C₁₃	#444	1	2	yes	complemented
164	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
165	D₁₃	—	4	2	no	—
166	#41	—	2	1	no	—
167	#41	—	2	1	no	—
168	#41	—	13	2	no	—
169	#41	—	13	2	no	—
170	#41	—	26	1	no	—
171	#41	—	26	1	no	—
172	#41	C₄×D₁₃	1	2	yes	—
173	#41	C₂²×D₁₃	1	2	yes	—
174	C₂×Q₈	—	13	1	no	—
175	C₂×Q₈	C₄×D₁₃	1	1	yes	characteristic
176	C₂×D₄	—	2	1	no	—
177	C₂×D₄	—	13	1	no	—
178	C₂×D₄	—	26	1	no	—
179	C₂×D₄	C₄×D₁₃	1	1	yes	characteristic
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	26	1	no	—
183	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
184	Q₁₆	—	13	4	no	—
185	SD₁₆	—	13	4	no	—
186	SD₁₆	—	13	4	no	—
187	D₈	—	13	4	no	—
188	M₁₆	—	2	1	no	—
189	M₁₆	—	13	2	no	—
190	M₁₆	—	26	4	no	—
191	M₁₆	—	26	4	no	—
192	M₁₆	—	26	1	no	—
193	M₁₆	C₂²×D₁₃	1	2	yes	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	13	1	no	—
196	C₂×C₈	—	13	1	no	—
197	C₂×C₈	—	13	2	no	—
198	C₂×C₈	—	13	2	no	—
199	C₂×C₈	—	13	1	no	—
200	C₂×C₈	—	13	1	no	—
201	C₂×C₈	—	13	2	no	—
202	C₂×C₈	—	13	2	no	—
203	C₂×C₈	—	26	4	no	—
204	C₂×C₈	—	26	4	no	—
205	C₂×C₈	—	26	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	C₂²×D₁₃	1	2	yes	—
208	C₁₃	C₂×#444	1	1	yes	characteristic, complemented, Sylow
209	C₂³	—	2	1	no	—
210	C₂³	—	26	1	no	—
211	C₂³	#6129	1	1	yes	characteristic
212	Q₈	—	2	1	no	—
213	Q₈	—	13	2	no	—
214	Q₈	—	26	1	no	—
215	Q₈	C₂×C₄×D₁₃	1	2	yes	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	1	no	—
219	D₄	—	13	2	no	—
220	D₄	—	26	2	no	—
221	D₄	—	26	2	no	—
222	D₄	—	26	1	no	—
223	D₄	C₂×C₄×D₁₃	1	2	yes	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	26	1	no	—
230	C₂×C₄	—	26	2	no	—
231	C₂×C₄	—	26	2	no	—
232	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
234	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
235	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
236	C₂×C₄	#6129	1	1	yes	characteristic
237	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	13	2	no	—
240	C₈	—	13	2	no	—
241	C₈	—	13	2	no	—
242	C₈	—	13	2	no	—
243	C₈	—	26	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	26	2	no	—
248	C₂²	—	52	2	no	—
249	C₂²	#88135	1	1	yes	characteristic
250	C₂²	#88135	1	1	yes	characteristic
251	C₂²	C₂×#6129	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	4	2	no	—
254	C₄	—	26	2	no	—
255	C₄	#88135	1	1	yes	characteristic
256	C₄	#88135	1	1	yes	characteristic
257	C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
258	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	4	2	no	—
262	C₂	—	52	2	no	—
263	C₂	#11718484	1	2	yes	direct product factor
264	C₂	C₂×#88135	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₃
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#104	C₂
socle	C₂²×C₁₃	C₄×D₄
prime radical	C₂×#88204	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₂×#104	D₁₃
hypocenter	C₁₃	C₂×#444
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#104

Complements

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

N	H ≅ G/N	type
2. C₂×#88202	C₂	semidirect
3. C₂×#88161	C₂	semidirect
4. C₂×C₁₃×#104	C₂	semidirect
5. C₂×#88062	C₂	semidirect
6. C₂×#88045	C₂	semidirect
7. C₂×#88043	C₂	semidirect
8. #11718484	C₂	direct (N × H)
13. #88204	C₄	semidirect
16. #88202	C₂²	semidirect
17. C₂×#6107	C₄	semidirect
18. C₂×#6106	C₄	semidirect
19. C₂×#6105	C₄	semidirect
20. C₂×#6104	C₄	semidirect
27. #88161	C₂²	semidirect
30. C₁₃×#104	C₂²	semidirect
31. #88062	C₂²	semidirect
32. #88045	C₂²	semidirect
33. #88043	C₂²	semidirect
57. #6107	C₂×C₄	semidirect
58. #6106	C₂×C₄	semidirect
59. #6105	C₂×C₄	semidirect
60. #6104	C₂×C₄	semidirect
118. C₂×#104	D₁₃	semidirect
158. #104	C₂×D₁₃	semidirect
163. C₂×C₁₃	#444	semidirect
208. C₁₃	C₂×#444	semidirect
263. C₂	#11718484	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×#88204	2
C₂×#88202	2
C₂×#88161	2
C₂×C₁₃×#104	2
C₂×#88062	2
C₂×#88045	2
C₂×#88043	2
#11718484	2
C₂×#444	13

Subgroups — C2×#11718484 #13123729

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11718484 #13123729