Subgroups — C₂×#13128238 #16769760

Order 3328.

7022 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	#13128238	C₂	1	32	yes	maximal, direct product factor
2	C₂×#11719328	C₂	1	12	yes	maximal, complemented
3	C₂×#11719188	C₂	1	3	yes	maximal
4	C₂×#11719327	C₂	1	12	yes	maximal, complemented
5	C₂²×#88131	C₂	1	3	yes	maximal
6	C₂×C₁₃×#543	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₁₃	C₂²	1	1	yes	characteristic
8	C₁₃×#543	C₂²	1	16	yes	—
9	C₂×C₁₃×C₂²⋊Q₈	C₂²	1	12	yes	—
10	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	3	yes	—
11	C₂²×C₂²⋊Dic₁₃	C₂²	1	3	yes	—
12	C₂×Q₈×Dic₁₃	C₂²	1	4	yes	complemented
13	C₂×#88198	C₂²	1	12	yes	complemented
14	C₂×#88178	C₂²	1	12	yes	—
15	C₂²×#6101	C₂²	1	6	yes	—
16	C₂²×C₄×Dic₁₃	C₂²	1	6	yes	—
17	#11719328	C₂²	1	192	yes	complemented
18	#11719327	C₂²	1	192	yes	complemented
19	C₂×#88143	C₂²	1	24	yes	complemented
20	C₂×#88142	C₂²	1	12	yes	complemented
21	C₂×#88141	C₂²	1	12	yes	complemented
22	#11719188	C₂²	1	48	yes	—
23	C₂×#88132	C₂²	1	24	yes	—
24	C₂×#88131	C₂²	1	12	yes	—
25	C₂×#88131	C₂²	1	48	yes	—
26	C₂×#88131	C₂²	1	12	yes	—
27	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
28	C₂²×Q₈×C₁₃	C₂³	1	4	yes	—
29	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
30	C₂³×Dic₁₃	Q₈	1	16	yes	complemented
31	C₂³×Dic₁₃	C₂³	1	6	yes	—
32	C₂³×Dic₁₃	C₂³	1	3	yes	—
33	C₂³×Dic₁₃	C₂³	1	3	yes	—
34	C₂³×Dic₁₃	C₂³	1	8	yes	—
35	C₂²×Dic₂₆	C₂³	1	12	yes	—
36	C₁₃×C₂²⋊Q₈	C₂³	1	96	yes	—
37	C₂×C₁₃×C₄⋊C₄	C₂³	1	12	yes	—
38	C₂×C₁₃×C₂²⋊C₄	C₂³	1	12	yes	—
39	C₂×C₁₃×C₂²⋊C₄	C₂³	1	24	yes	—
40	C₂×C₂²⋊Dic₁₃	C₂³	1	12	yes	—
41	C₂×C₂²⋊Dic₁₃	C₂³	1	24	yes	—
42	Q₈×Dic₁₃	C₂³	1	32	yes	complemented
43	#88198	C₂³	1	96	yes	complemented
44	C₂×C₄⋊Dic₁₃	C₂³	1	12	yes	—
45	#88178	C₂³	1	96	yes	—
46	C₂×#6101	C₂³	1	24	yes	—
47	C₂×#6101	C₂³	1	48	yes	—
48	C₂×#6101	C₂³	1	24	yes	—
49	C₂×C₄×Dic₁₃	C₂³	1	12	yes	—
50	C₂×C₄×Dic₁₃	C₂³	1	12	yes	—
51	C₂×C₄×Dic₁₃	C₂³	1	48	yes	—
52	#88143	C₂³	1	192	yes	complemented
53	#88142	C₂³	1	96	yes	complemented
54	#88141	C₂³	1	96	yes	complemented
55	#88132	C₂³	1	192	yes	—
56	#88131	C₂³	1	96	yes	—
57	#88131	C₂³	1	96	yes	—
58	C₂×#3458	—	13	1	no	maximal, Sylow
59	C₂⁴×C₁₃	—	4	4	no	—
60	C₂⁴×C₁₃	C₂×Q₈	1	8	yes	—
61	C₂⁴×C₁₃	C₂⁴	1	3	yes	—
62	C₂⁴×C₁₃	C₂⁴	1	4	yes	—
63	C₂×Q₈×C₁₃	—	4	8	no	—
64	C₂×Q₈×C₁₃	C₂⁴	1	16	yes	—
65	C₂²×C₄×C₁₃	—	4	12	no	—
66	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
67	C₂²×C₄×C₁₃	C₂⁴	1	24	yes	—
68	C₂²×Dic₁₃	—	4	12	no	—
69	C₂²×Dic₁₃	C₂×Q₈	1	24	yes	—
70	C₂²×Dic₁₃	C₂×Q₈	1	24	yes	—
71	C₂²×Dic₁₃	C₂×Q₈	1	64	yes	complemented
72	C₂²×Dic₁₃	C₂⁴	1	12	yes	—
73	C₂²×Dic₁₃	C₂⁴	1	24	yes	—
74	C₂²×Dic₁₃	C₂⁴	1	12	yes	—
75	C₂²×Dic₁₃	C₂⁴	1	12	yes	—
76	C₂²×Dic₁₃	C₂⁴	1	1	yes	characteristic
77	C₂²×Dic₁₃	C₂⁴	1	3	yes	—
78	C₂×Dic₂₆	—	4	24	no	—
79	C₂×Dic₂₆	C₂⁴	1	48	yes	—
80	C₁₃×C₄⋊C₄	C₂⁴	1	48	yes	—
81	C₁₃×C₂²⋊C₄	C₂⁴	1	48	yes	—
82	C₂²⋊Dic₁₃	C₂⁴	1	48	yes	—
83	C₄⋊Dic₁₃	C₂⁴	1	48	yes	—
84	#6101	C₂⁴	1	96	yes	—
85	#6101	C₂⁴	1	96	yes	—
86	C₄×Dic₁₃	C₂⁴	1	48	yes	—
87	C₄×Dic₁₃	C₂⁴	1	48	yes	—
88	#3458	—	13	32	no	—
89	C₂×#544	—	13	12	no	—
90	C₂×#543	—	13	3	no	—
91	C₂×#543	D₁₃	1	1	yes	characteristic
92	C₂×#533	—	13	12	no	—
93	C₂²×#117	—	13	3	no	—
94	C₂³×C₁₃	—	2	12	no	—
95	C₂³×C₁₃	—	4	16	no	—
96	C₂³×C₁₃	—	4	8	no	—
97	C₂³×C₁₃	C₂²×Q₈	1	12	yes	—
98	C₂³×C₁₃	C₂²×Q₈	1	16	yes	—
99	C₂³×C₁₃	C₂⁵	1	6	yes	—
100	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
101	Q₈×C₁₃	—	4	16	no	—
102	C₂×C₄×C₁₃	—	2	24	no	—
103	C₂×C₄×C₁₃	—	4	24	no	—
104	C₂×C₄×C₁₃	C₂⁵	1	24	yes	—
105	C₂×Dic₁₃	—	2	24	no	—
106	C₂×Dic₁₃	—	4	24	no	—
107	C₂×Dic₁₃	C₂²×Q₈	1	4	yes	—
108	C₂×Dic₁₃	C₂²×Q₈	1	12	yes	—
109	C₂×Dic₁₃	C₂²×Q₈	1	48	yes	—
110	C₂×Dic₁₃	C₂²×Q₈	1	48	yes	—
111	C₂×Dic₁₃	C₂⁵	1	24	yes	—
112	C₂×Dic₁₃	C₂⁵	1	2	yes	—
113	C₂×Dic₁₃	C₂⁵	1	6	yes	—
114	Dic₂₆	—	4	48	no	—
115	C₂⁴×C₄	—	13	1	no	—
116	#544	—	13	192	no	—
117	#543	—	13	48	no	—
118	#543	C₂×D₁₃	1	16	yes	—
119	#533	—	13	192	no	—
120	C₂×C₄⋊Q₈	—	13	12	no	—
121	C₂×C₄⋊Q₈	—	13	12	no	—
122	C₂×#125	—	13	24	no	—
123	C₂×C₂²⋊Q₈	—	13	12	no	—
124	C₂×C₂²⋊Q₈	—	13	24	no	—
125	C₂×C₂²⋊Q₈	C₂×D₁₃	1	12	yes	—
126	C₂×C₄×Q₈	—	13	4	no	—
127	C₂×C₄×Q₈	—	13	12	no	—
128	C₂×#117	—	13	12	no	—
129	C₂×#117	—	13	12	no	—
130	C₂×#117	—	13	48	no	—
131	C₂²×C₄⋊C₄	—	13	6	no	—
132	C₂²×C₂²⋊C₄	—	13	3	no	—
133	C₂²×C₂²⋊C₄	C₂×D₁₃	1	3	yes	—
134	C₂²×C₄²	—	13	6	no	—
135	C₂²×C₁₃	—	2	12	no	—
136	C₂²×C₁₃	—	2	24	no	—
137	C₂²×C₁₃	—	4	16	no	—
138	C₂²×C₁₃	C₂³×Q₈	1	12	yes	—
139	C₂²×C₁₃	C₂³×Q₈	1	2	yes	—
140	C₂²×C₁₃	#585	1	4	yes	—
141	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
142	C₄×C₁₃	—	2	24	no	—
143	Dic₁₃	—	2	24	no	—
144	Dic₁₃	C₂³×Q₈	1	4	yes	—
145	Dic₁₃	C₂³×Q₈	1	12	yes	—
146	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
147	C₂²×Q₈	—	13	12	no	—
148	C₂²×Q₈	C₂²×D₁₃	1	4	yes	—
149	C₂³×C₄	—	13	16	no	—
150	C₂³×C₄	—	13	3	no	—
151	C₂³×C₄	—	13	8	no	—
152	C₂³×C₄	—	13	3	no	—
153	C₂³×C₄	—	13	6	no	—
154	C₂³×C₄	C₂²×D₁₃	1	6	yes	—
155	C₄⋊Q₈	—	13	96	no	—
156	C₄⋊Q₈	—	13	96	no	—
157	#125	—	13	192	no	—
158	C₂²⋊Q₈	—	13	96	no	—
159	C₂²⋊Q₈	—	13	192	no	—
160	C₂²⋊Q₈	C₂²×D₁₃	1	96	yes	—
161	C₄×Q₈	—	13	32	no	—
162	C₄×Q₈	—	13	96	no	—
163	#117	—	13	96	no	—
164	#117	—	13	96	no	—
165	C₂×C₄⋊C₄	—	13	24	no	—
166	C₂×C₄⋊C₄	—	13	48	no	—
167	C₂×C₄⋊C₄	—	13	24	no	—
168	C₂×C₄⋊C₄	—	13	12	no	—
169	C₂×C₄⋊C₄	C₂²×D₁₃	1	12	yes	—
170	C₂×C₂²⋊C₄	—	13	12	no	—
171	C₂×C₂²⋊C₄	—	13	24	no	—
172	C₂×C₂²⋊C₄	C₂²×D₁₃	1	12	yes	—
173	C₂×C₂²⋊C₄	C₂²×D₁₃	1	24	yes	—
174	C₂×C₄²	—	13	12	no	—
175	C₂×C₄²	—	13	12	no	—
176	C₂×C₄²	—	13	48	no	—
177	C₂×C₁₃	—	2	12	no	—
178	C₂×C₁₃	#3458	1	4	yes	complemented
179	C₂×C₁₃	C₂⁴×Q₈	1	1	yes	characteristic
180	C₂×C₁₃	C₂×#585	1	2	yes	—
181	C₂⁴	—	4	4	no	—
182	C₂⁴	Q₈×D₁₃	1	8	yes	—
183	C₂⁴	C₂³×D₁₃	1	3	yes	—
184	C₂⁴	C₂³×D₁₃	1	4	yes	—
185	C₂×Q₈	—	4	8	no	—
186	C₂×Q₈	—	13	48	no	—
187	C₂×Q₈	—	52	24	no	—
188	C₂×Q₈	C₂³×D₁₃	1	16	yes	—
189	C₂²×C₄	—	4	12	no	—
190	C₂²×C₄	—	13	24	no	—
191	C₂²×C₄	—	13	24	no	—
192	C₂²×C₄	—	13	64	no	—
193	C₂²×C₄	—	13	12	no	—
194	C₂²×C₄	—	13	12	no	—
195	C₂²×C₄	—	13	1	no	—
196	C₂²×C₄	—	13	3	no	—
197	C₂²×C₄	—	13	12	no	—
198	C₂²×C₄	—	13	24	no	—
199	C₂²×C₄	—	52	12	no	—
200	C₂²×C₄	C₂³×D₁₃	1	12	yes	—
201	C₂²×C₄	C₂³×D₁₃	1	24	yes	—
202	C₄⋊C₄	—	13	96	no	—
203	C₄⋊C₄	—	13	96	no	—
204	C₄⋊C₄	—	13	48	no	—
205	C₄⋊C₄	C₂³×D₁₃	1	48	yes	—
206	C₂²⋊C₄	—	13	48	no	—
207	C₂²⋊C₄	C₂³×D₁₃	1	48	yes	—
208	C₄²	—	13	48	no	—
209	C₄²	—	13	48	no	—
210	C₁₃	C₂×#3458	1	1	yes	characteristic, complemented, Sylow
211	C₂³	—	2	12	no	—
212	C₂³	—	4	16	no	—
213	C₂³	—	4	8	no	—
214	C₂³	C₂×Q₈×D₁₃	1	12	yes	—
215	C₂³	C₂×Q₈×D₁₃	1	16	yes	—
216	C₂³	C₂⁴×D₁₃	1	6	yes	—
217	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
218	Q₈	—	4	16	no	—
219	Q₈	—	52	48	no	—
220	C₂×C₄	—	2	24	no	—
221	C₂×C₄	—	4	24	no	—
222	C₂×C₄	—	13	48	no	—
223	C₂×C₄	—	13	48	no	—
224	C₂×C₄	—	13	4	no	—
225	C₂×C₄	—	13	12	no	—
226	C₂×C₄	—	13	2	no	—
227	C₂×C₄	—	13	6	no	—
228	C₂×C₄	—	13	24	no	—
229	C₂×C₄	—	26	24	no	—
230	C₂×C₄	—	52	24	no	—
231	C₂×C₄	C₂⁴×D₁₃	1	24	yes	—
232	C₂²	—	2	24	no	—
233	C₂²	—	2	12	no	—
234	C₂²	—	4	16	no	—
235	C₂²	C₂²×Q₈×D₁₃	1	12	yes	—
236	C₂²	C₂²×Q₈×D₁₃	1	2	yes	—
237	C₂²	#11719650	1	4	yes	—
238	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
239	C₄	—	2	24	no	—
240	C₄	—	13	4	no	—
241	C₄	—	13	12	no	—
242	C₄	—	26	24	no	—
243	C₂	—	2	12	no	—
244	C₂	#13128238	1	4	yes	direct product factor
245	C₂	C₂³×Q₈×D₁₃	1	1	yes	characteristic
246	C₂	C₂×#11719650	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂×C₁₃×#543	C₂
socle	C₂³×C₁₃	C₂⁵
prime radical	C₂⁵×C₁₃	C₂³
hypercenter	C₂×#543	D₁₃
hypocenter	C₁₃	C₂×#3458
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#543

Complements

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

N	H ≅ G/N	type
1. #13128238	C₂	direct (N × H)
2. C₂×#11719328	C₂	semidirect
4. C₂×#11719327	C₂	semidirect
12. C₂×Q₈×Dic₁₃	C₂²	semidirect
13. C₂×#88198	C₂²	semidirect
17. #11719328	C₂²	semidirect
18. #11719327	C₂²	semidirect
19. C₂×#88143	C₂²	semidirect
20. C₂×#88142	C₂²	semidirect
21. C₂×#88141	C₂²	semidirect
30. C₂³×Dic₁₃	Q₈	semidirect
42. Q₈×Dic₁₃	C₂³	semidirect
43. #88198	C₂³	semidirect
52. #88143	C₂³	semidirect
53. #88142	C₂³	semidirect
54. #88141	C₂³	semidirect
71. C₂²×Dic₁₃	C₂×Q₈	semidirect
178. C₂×C₁₃	#3458	semidirect
210. C₁₃	C₂×#3458	semidirect
244. C₂	#13128238	direct (N × H)

Maximal subgroups

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

subgroup	index
#13128238	2
C₂×#11719328	2
C₂×#11719188	2
C₂×#11719327	2
C₂²×#88131	2
C₂×C₁₃×#543	2
C₂×#3458	13

Subgroups — C2×#13128238 #16769760

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13128238 #16769760