Subgroups — C₂×#13125350 #16732524

Order 3328.

1942 conjugacy classes of subgroups.

The subgroup lattice has 248 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	2	yes	maximal
2	#13125350	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11718343	C₂	1	2	yes	maximal, complemented
4	C₂×#11718344	C₂	1	4	yes	maximal, complemented
5	C₂×#11718341	C₂	1	2	yes	maximal, complemented
6	C₂×#11718767	C₂	1	4	yes	maximal
7	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88180	C₂²	1	4	yes	—
11	C₂×#88178	—	2	4	no	—
12	#11719175	—	2	8	no	—
13	#11719175	C₂²	1	16	yes	—
14	C₂×#88124	C₂²	1	4	yes	—
15	C₂×#88123	C₂²	1	4	yes	—
16	C₂×C₄×Dic₂₆	—	2	4	no	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₁₃	C₂²	1	8	yes	—
21	#11718767	C₂²	1	32	yes	—
22	C₂×#88058	C₂²	1	8	yes	—
23	C₂×#88057	C₂²	1	8	yes	—
24	#11718344	C₂²	1	32	yes	complemented
25	#11718343	C₂²	1	16	yes	complemented
26	#11718341	C₂²	1	16	yes	complemented
27	C₂³×C₄×C₁₃	D₄	1	2	yes	—
28	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₂₆	—	2	4	no	—
30	C₂²×C₈×C₁₃	Q₈	1	16	yes	complemented
31	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	8	yes	—
34	C₂×C₄²×C₁₃	D₄	1	2	yes	—
35	C₂×C₄²×C₁₃	D₄	1	4	yes	—
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
39	C₂×C₂²⋊Dic₁₃	—	2	4	no	—
40	#88180	—	2	8	no	—
41	#88180	C₂³	1	16	yes	—
42	C₂×C₄⋊Dic₁₃	—	2	4	no	—
43	C₂×C₄⋊Dic₁₃	C₂³	1	8	yes	—
44	#88178	—	2	32	no	—
45	C₂×#6101	—	2	8	no	—
46	C₂×C₄×Dic₁₃	—	2	4	no	—
47	#88124	—	2	8	no	—
48	#88124	C₂³	1	16	yes	—
49	#88123	—	2	8	no	—
50	#88123	C₂³	1	16	yes	—
51	C₄×Dic₂₆	—	2	32	no	—
52	C₄×C₈×C₁₃	C₂³	1	4	yes	—
53	C₄×C₈×C₁₃	C₂³	1	4	yes	—
54	C₄×C₈×C₁₃	C₂³	1	8	yes	—
55	#88058	C₂³	1	32	yes	—
56	#88057	C₂³	1	32	yes	—
57	C₂×#3160	—	13	1	no	maximal, Sylow
58	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	—	2	8	no	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
66	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
68	C₂²×Dic₁₃	—	2	8	no	—
69	C₂×Dic₂₆	—	2	16	no	—
70	C₂×Dic₂₆	—	4	16	no	—
71	C₂×C₈×C₁₃	C₂×Q₈	1	8	yes	—
72	C₂×C₈×C₁₃	C₂×Q₈	1	8	yes	—
73	C₂×C₈×C₁₃	C₂×Q₈	1	32	yes	complemented
74	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
75	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
76	C₄²×C₁₃	C₂×D₄	1	8	yes	—
77	C₄²×C₁₃	C₂×D₄	1	2	yes	—
78	C₄²×C₁₃	C₂×D₄	1	2	yes	—
79	C₄²×C₁₃	C₂⁴	1	2	yes	—
80	C₄²×C₁₃	C₂⁴	1	2	yes	—
81	C₂²⋊Dic₁₃	—	2	16	no	—
82	C₄⋊Dic₁₃	—	2	16	no	—
83	C₄⋊Dic₁₃	—	2	8	no	—
84	C₄⋊Dic₁₃	C₂⁴	1	16	yes	—
85	#6101	—	2	32	no	—
86	C₄×Dic₁₃	—	2	16	no	—
87	C₂×#533	—	13	2	no	—
88	#3160	—	13	16	no	—
89	C₂×#500	—	13	2	no	—
90	C₂×#499	—	13	4	no	—
91	C₂×#498	—	13	2	no	—
92	C₂×#427	—	13	4	no	—
93	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
94	C₂³×C₁₃	—	2	4	no	—
95	C₂³×C₁₃	C₄⋊Q₈	1	4	yes	—
96	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
97	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	—	2	4	no	—
99	C₂×C₄×C₁₃	—	2	16	no	—
100	C₂×C₄×C₁₃	C₄⋊Q₈	1	8	yes	—
101	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
102	C₂×C₄×C₁₃	#135	1	8	yes	—
103	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
104	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
105	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
107	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
108	C₂×C₄×C₁₃	C₂²×Q₈	1	8	yes	—
109	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
110	C₂×Dic₁₃	—	2	16	no	—
111	C₂×Dic₁₃	—	4	16	no	—
112	Dic₂₆	—	4	32	no	—
113	C₈×C₁₃	C₂²×Q₈	1	8	yes	—
114	C₈×C₁₃	C₂²×Q₈	1	8	yes	—
115	C₂³×C₈	C₂×D₁₃	1	2	yes	—
116	#533	—	13	16	no	—
117	#533	—	26	8	no	—
118	C₂×C₄⋊Q₈	—	13	4	no	—
119	C₂×#125	—	13	4	no	—
120	C₂×C₂²⋊Q₈	—	26	4	no	—
121	C₂×C₄×Q₈	—	26	4	no	—
122	C₂×#117	—	13	4	no	—
123	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
124	#500	—	13	16	no	—
125	#499	—	13	32	no	—
126	#498	—	13	16	no	—
127	#427	—	13	32	no	—
128	C₂×#107	—	13	8	no	—
129	C₂×#106	—	13	8	no	—
130	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
131	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
132	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
133	C₂×C₄×C₈	C₂×D₁₃	1	8	yes	—
134	C₂²×C₁₃	—	2	8	no	—
135	C₂²×C₁₃	—	2	4	no	—
136	C₂²×C₁₃	C₂×C₄⋊Q₈	1	4	yes	—
137	C₂²×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
138	C₂²×C₁₃	C₂×#135	1	4	yes	—
139	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
140	C₄×C₁₃	—	2	4	no	—
141	C₄×C₁₃	C₂×C₄⋊Q₈	1	4	yes	—
142	C₄×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
143	C₄×C₁₃	C₂×#135	1	8	yes	—
144	C₄×C₁₃	C₂³×Q₈	1	2	yes	—
145	Dic₁₃	—	4	16	no	—
146	C₂²×Q₈	—	26	4	no	—
147	C₂³×C₄	D₅₂	1	2	yes	—
148	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
149	C₂²×C₈	Dic₂₆	1	16	yes	complemented
150	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
151	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
152	C₂²×C₈	C₂²×D₁₃	1	8	yes	—
153	C₄⋊Q₈	—	13	16	no	—
154	C₄⋊Q₈	—	26	8	no	—
155	#125	—	13	16	no	—
156	#125	—	26	8	no	—
157	C₂²⋊Q₈	—	26	32	no	—
158	C₄×Q₈	—	26	32	no	—
159	#117	—	13	16	no	—
160	#117	—	26	8	no	—
161	C₂×C₄⋊C₄	—	13	8	no	—
162	C₂×C₄⋊C₄	—	26	8	no	—
163	C₂×C₄⋊C₄	—	26	4	no	—
164	C₂×C₂²⋊C₄	—	26	4	no	—
165	C₂×C₄²	—	26	4	no	—
166	C₂×C₄²	D₅₂	1	4	yes	—
167	C₂×C₄²	D₅₂	1	2	yes	—
168	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
169	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
170	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
171	#107	—	13	32	no	—
172	#106	—	13	32	no	—
173	C₄×C₈	C₂²×D₁₃	1	4	yes	—
174	C₄×C₈	C₂²×D₁₃	1	8	yes	—
175	C₄×C₈	C₂²×D₁₃	1	4	yes	—
176	C₂×C₁₃	—	2	4	no	—
177	C₂×C₁₃	#3160	1	4	yes	complemented
178	C₂×C₁₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
179	C₂×C₁₃	C₂²×#135	1	2	yes	—
180	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
181	C₂×Q₈	—	26	16	no	—
182	C₂×Q₈	—	52	16	no	—
183	C₂²×C₄	—	2	8	no	—
184	C₂²×C₄	—	26	8	no	—
185	C₂²×C₄	C₂×Dic₂₆	1	8	yes	—
186	C₂²×C₄	C₂×D₅₂	1	2	yes	—
187	C₂²×C₄	C₂×D₅₂	1	8	yes	—
188	C₂²×C₄	C₂×D₅₂	1	2	yes	—
189	C₂²×C₄	C₂×D₅₂	1	2	yes	—
190	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
191	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
192	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
193	C₂×C₈	C₂×Dic₂₆	1	8	yes	—
194	C₂×C₈	C₂×Dic₂₆	1	8	yes	—
195	C₂×C₈	C₂×Dic₂₆	1	32	yes	complemented
196	C₂×C₈	C₂³×D₁₃	1	4	yes	—
197	C₂×C₈	C₂³×D₁₃	1	4	yes	—
198	C₄⋊C₄	—	13	16	no	—
199	C₄⋊C₄	—	26	8	no	—
200	C₄⋊C₄	—	26	32	no	—
201	C₄⋊C₄	—	26	16	no	—
202	C₂²⋊C₄	—	26	16	no	—
203	C₄²	—	26	16	no	—
204	C₄²	C₂×D₅₂	1	8	yes	—
205	C₄²	C₂×D₅₂	1	2	yes	—
206	C₄²	C₂×D₅₂	1	2	yes	—
207	C₄²	C₂³×D₁₃	1	2	yes	—
208	C₄²	C₂³×D₁₃	1	2	yes	—
209	C₁₃	C₂×#3160	1	1	yes	characteristic, complemented, Sylow
210	C₂³	—	2	4	no	—
211	C₂³	#88123	1	4	yes	—
212	C₂³	C₂²×D₅₂	1	2	yes	—
213	C₂³	C₂²×D₅₂	1	1	yes	characteristic
214	Q₈	—	52	32	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	16	no	—
217	C₂×C₄	—	26	16	no	—
218	C₂×C₄	—	52	16	no	—
219	C₂×C₄	#88123	1	8	yes	—
220	C₂×C₄	#88123	1	4	yes	—
221	C₂×C₄	#88158	1	8	yes	—
222	C₂×C₄	C₂²×Dic₂₆	1	4	yes	—
223	C₂×C₄	C₂²×Dic₂₆	1	8	yes	—
224	C₂×C₄	C₂²×D₅₂	1	4	yes	—
225	C₂×C₄	C₂²×D₅₂	1	2	yes	—
226	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
227	C₂×C₄	C₂²×D₅₂	1	4	yes	—
228	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
229	C₈	C₂²×Dic₂₆	1	8	yes	—
230	C₈	C₂²×Dic₂₆	1	8	yes	—
231	C₂²	—	2	4	no	—
232	C₂²	—	2	8	no	—
233	C₂²	C₂×#88123	1	4	yes	—
234	C₂²	C₂×#88123	1	2	yes	—
235	C₂²	C₂×#88158	1	4	yes	—
236	C₂²	C₂³×D₅₂	1	1	yes	characteristic
237	C₄	—	2	4	no	—
238	C₄	—	52	16	no	—
239	C₄	C₂×#88123	1	2	yes	—
240	C₄	C₂×#88123	1	4	yes	—
241	C₄	C₂×#88158	1	8	yes	—
242	C₄	C₂³×Dic₂₆	1	2	yes	—
243	C₂	—	2	4	no	—
244	C₂	#13125350	1	4	yes	direct product factor
245	C₂	C₂²×#88123	1	1	yes	characteristic
246	C₂	C₂²×#88158	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂⁴×C₁₃	C₂×D₄
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	C₂×#3160
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13125350	C₂	direct (N × H)
3. C₂×#11718343	C₂	semidirect
4. C₂×#11718344	C₂	semidirect
5. C₂×#11718341	C₂	semidirect
24. #11718344	C₂²	semidirect
25. #11718343	C₂²	semidirect
26. #11718341	C₂²	semidirect
30. C₂²×C₈×C₁₃	Q₈	semidirect
73. C₂×C₈×C₁₃	C₂×Q₈	semidirect
149. C₂²×C₈	Dic₂₆	semidirect
177. C₂×C₁₃	#3160	semidirect
195. C₂×C₈	C₂×Dic₂₆	semidirect
209. C₁₃	C₂×#3160	semidirect
244. C₂	#13125350	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
#13125350	2
C₂×#11718343	2
C₂×#11718344	2
C₂×#11718341	2
C₂×#11718767	2
C₂²×C₄×C₈×C₁₃	2
C₂×#3160	13

Subgroups — C2×#13125350 #16732524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13125350 #16732524