Subgroups — C₂×#14541925 #17940836

Order 4352.

2126 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763614	C₂	1	2	yes	maximal, complemented
2	#14541925	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11763247	C₂	1	6	yes	maximal, complemented
4	C₂×#11763241	C₂	1	3	yes	maximal, complemented
5	C₂²×D₂₇₂	C₂	1	3	yes	maximal, complemented
6	C₂×C₁₇×#504	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#10587696	—	2	2	no	—
8	#11763614	—	2	8	no	—
9	#11763614	C₂²	1	16	yes	complemented
10	C₂×#10587594	—	2	6	no	—
11	C₂×#10587590	C₂²	1	6	yes	complemented
12	C₂²×D₁₃₆	C₂²	1	6	yes	complemented
13	#11763247	C₂²	1	48	yes	complemented
14	#11763241	C₂²	1	24	yes	complemented
15	C₂×Dic₁₃₆	C₂²	1	1	yes	characteristic, complemented
16	C₂×#10587471	C₂²	1	6	yes	complemented
17	C₂×D₂₇₂	C₂²	1	3	yes	complemented
18	C₂×D₂₇₂	C₂²	1	6	yes	complemented
19	C₂×D₂₇₂	C₂²	1	24	yes	complemented
20	C₂×C₁₇×#131	C₂²	1	1	yes	characteristic
21	C₁₇×#504	C₂²	1	8	yes	complemented
22	C₂×C₁₇×M₃₂	C₂²	1	3	yes	complemented
23	C₂²×C₁₆×C₁₇	C₂²	1	3	yes	complemented
24	C₂×#41×C₁₇	D₄	1	1	yes	characteristic
25	#10587696	—	2	16	no	—
26	#10587696	—	4	8	no	—
27	C₂×#63256	—	4	2	no	—
28	C₂×D₄×D₁₇	—	4	6	no	—
29	C₂×#63252	—	2	6	no	—
30	C₂²×D₆₈	—	2	6	no	—
31	C₁₇×#131	D₄	1	4	yes	—
32	C₁₇×#131	C₂³	1	4	yes	—
33	C₂×M₁₆×C₁₇	D₄	1	3	yes	—
34	C₂²×C₈×C₁₇	C₂³	1	3	yes	—
35	#10587594	—	2	48	no	—
36	C₂×Dic₆₈	C₂³	1	2	yes	—
37	#10587590	—	2	12	no	—
38	#10587590	C₂³	1	24	yes	complemented
39	C₂×D₁₃₆	—	2	6	no	—
40	C₂×D₁₃₆	—	2	12	no	—
41	C₂×D₁₃₆	C₂³	1	6	yes	—
42	C₂×D₁₃₆	C₂³	1	24	yes	complemented
43	C₂×#63219	—	2	6	no	—
44	C₁₇×M₃₂	C₂³	1	12	yes	complemented
45	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₁₇	C₂³	1	3	yes	—
47	C₂×C₁₆×C₁₇	C₂³	1	12	yes	complemented
48	Dic₁₃₆	C₂³	1	4	yes	complemented
49	#10587471	C₂³	1	24	yes	complemented
50	D₂₇₂	C₂³	1	12	yes	complemented
51	D₂₇₂	C₂³	1	24	yes	complemented
52	C₂³×D₁₇	—	4	6	no	—
53	#41×C₁₇	D₈	1	4	yes	—
54	#41×C₁₇	C₂×D₄	1	4	yes	—
55	C₂×Q₈×C₁₇	D₈	1	1	yes	characteristic
56	C₂×D₄×C₁₇	D₈	1	3	yes	—
57	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
58	C₂×C₂²⋊D₁₇	—	4	6	no	—
59	#63256	—	4	16	no	—
60	D₄×D₁₇	—	4	48	no	—
61	#63252	—	2	24	no	—
62	#63252	—	4	12	no	—
63	C₂×D₆₈	—	2	6	no	—
64	C₂×D₆₈	—	2	24	no	—
65	C₂×D₆₈	—	4	6	no	—
66	C₂×D₆₈	—	4	12	no	—
67	C₂×C₄×D₁₇	—	4	6	no	—
68	C₂×Dic₃₄	—	2	2	no	—
69	M₁₆×C₁₇	C₂×D₄	1	12	yes	—
70	C₂×C₈×C₁₇	C₂×D₄	1	3	yes	—
71	C₂×C₈×C₁₇	C₂×D₄	1	6	yes	—
72	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
73	C₂×C₈×C₁₇	C₂⁴	1	6	yes	—
74	Dic₆₈	—	2	2	no	—
75	Dic₆₈	C₂⁴	1	4	yes	—
76	D₁₃₆	—	2	6	no	—
77	D₁₃₆	—	2	24	no	—
78	D₁₃₆	C₂⁴	1	12	yes	—
79	#63219	—	2	24	no	—
80	C₁₆×C₁₇	C₂⁴	1	2	yes	—
81	C₁₆×C₁₇	C₂⁴	1	6	yes	—
82	C₂×#3415	—	17	1	no	maximal, Sylow
83	C₂³×C₁₇	C₂×D₈	1	3	yes	—
84	C₂²×D₁₇	—	4	6	no	—
85	C₂²×D₁₇	—	4	24	no	—
86	C₂²×D₁₇	—	16	12	no	—
87	Q₈×C₁₇	C₂×D₈	1	4	yes	—
88	D₄×C₁₇	C₂×D₈	1	12	yes	—
89	C₂×C₄×C₁₇	C₂×D₈	1	3	yes	—
90	C₂×C₄×C₁₇	C₂×D₈	1	6	yes	—
91	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₇	C₂²×D₄	1	6	yes	—
93	C₂²⋊D₁₇	—	4	24	no	—
94	C₂×Dic₁₇	—	4	2	no	—
95	D₆₈	—	2	12	no	—
96	D₆₈	—	4	6	no	—
97	D₆₈	—	4	24	no	—
98	C₄×D₁₇	—	4	24	no	—
99	Dic₃₄	—	2	4	no	—
100	Dic₃₄	—	4	2	no	—
101	C₈×C₁₇	C₂²×D₄	1	1	yes	characteristic
102	C₈×C₁₇	C₂²×D₄	1	6	yes	—
103	C₈×C₁₇	C₂⁵	1	1	yes	characteristic
104	C₂×#576	—	17	2	no	—
105	#3415	—	17	16	no	—
106	C₂×#509	—	17	6	no	—
107	C₂×#508	—	17	3	no	—
108	C₂²×D₁₆	—	17	3	no	—
109	C₂×#504	D₁₇	1	1	yes	characteristic, complemented
110	C₂²×C₁₇	—	2	6	no	—
111	C₂²×C₁₇	C₂²×D₈	1	6	yes	—
112	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
113	C₂×D₁₇	—	4	12	no	—
114	C₂×D₁₇	—	8	12	no	—
115	C₂×D₁₇	—	16	24	no	—
116	C₄×C₁₇	C₂²×D₈	1	1	yes	characteristic
117	C₄×C₁₇	C₂²×D₈	1	6	yes	—
118	C₄×C₁₇	C₂³×D₄	1	1	yes	characteristic
119	Dic₁₇	—	4	4	no	—
120	C₂×H(5,𝔽₂)	—	34	2	no	—
121	#576	—	17	16	no	—
122	#576	—	34	8	no	—
123	C₂×AGL(1,ℤ/8)	—	34	6	no	—
124	C₂×#135	—	17	6	no	—
125	C₂²×D₈	—	17	6	no	—
126	C₂×#131	C₂×D₁₇	1	1	yes	characteristic
127	#509	—	17	48	no	—
128	#508	—	17	24	no	—
129	C₂×Q₃₂	—	17	1	no	—
130	C₂×SD₃₂	—	17	6	no	—
131	C₂×D₁₆	—	17	3	no	—
132	C₂×D₁₆	—	17	6	no	—
133	C₂×D₁₆	—	17	24	no	—
134	#504	C₂×D₁₇	1	8	yes	complemented
135	C₂×M₃₂	C₂×D₁₇	1	3	yes	complemented
136	C₂²×C₁₆	C₂×D₁₇	1	3	yes	complemented
137	C₂×C₁₇	—	2	6	no	—
138	C₂×C₁₇	#3415	1	2	yes	complemented
139	C₂×C₁₇	C₂³×D₈	1	1	yes	characteristic
140	D₁₇	—	8	12	no	—
141	H(5,𝔽₂)	—	34	16	no	—
142	H(5,𝔽₂)	—	68	8	no	—
143	C₂×#41	—	34	6	no	—
144	C₂×#41	—	68	2	no	—
145	C₂×#41	D₆₈	1	1	yes	characteristic
146	C₂²×D₄	—	34	6	no	—
147	C₂²×D₄	—	68	6	no	—
148	AGL(1,ℤ/8)	—	34	48	no	—
149	#135	—	17	24	no	—
150	#135	—	34	12	no	—
151	C₂×Q₁₆	—	17	2	no	—
152	C₂×SD₁₆	—	34	6	no	—
153	C₂×D₈	—	17	6	no	—
154	C₂×D₈	—	17	24	no	—
155	C₂×D₈	—	34	6	no	—
156	C₂×D₈	—	34	12	no	—
157	#131	D₆₈	1	4	yes	—
158	#131	C₂²×D₁₇	1	4	yes	—
159	C₂×M₁₆	D₆₈	1	3	yes	—
160	C₂²×C₈	C₂²×D₁₇	1	3	yes	—
161	Q₃₂	—	17	4	no	—
162	SD₃₂	—	17	24	no	—
163	D₁₆	—	17	12	no	—
164	D₁₆	—	17	24	no	—
165	M₃₂	C₂²×D₁₇	1	12	yes	complemented
166	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
167	C₂×C₁₆	C₂²×D₁₇	1	3	yes	—
168	C₂×C₁₆	C₂²×D₁₇	1	12	yes	complemented
169	C₁₇	C₂×#3415	1	1	yes	characteristic, complemented, Sylow
170	C₂⁴	—	68	6	no	—
171	#41	—	34	24	no	—
172	#41	—	68	16	no	—
173	#41	—	68	12	no	—
174	#41	D₁₃₆	1	4	yes	—
175	#41	C₂×D₆₈	1	4	yes	—
176	C₂×Q₈	—	34	2	no	—
177	C₂×Q₈	D₁₃₆	1	1	yes	characteristic
178	C₂×D₄	—	34	6	no	—
179	C₂×D₄	—	34	24	no	—
180	C₂×D₄	—	68	6	no	—
181	C₂×D₄	—	68	48	no	—
182	C₂×D₄	—	68	6	no	—
183	C₂×D₄	—	68	12	no	—
184	C₂×D₄	D₁₃₆	1	3	yes	—
185	C₂²×C₄	—	68	6	no	—
186	C₂²×C₄	C₂×D₆₈	1	3	yes	—
187	Q₁₆	—	17	4	no	—
188	Q₁₆	—	34	2	no	—
189	SD₁₆	—	34	24	no	—
190	D₈	—	17	12	no	—
191	D₈	—	34	6	no	—
192	D₈	—	34	24	no	—
193	M₁₆	C₂×D₆₈	1	12	yes	—
194	C₂×C₈	C₂×D₆₈	1	3	yes	—
195	C₂×C₈	C₂×D₆₈	1	6	yes	—
196	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
197	C₂×C₈	C₂³×D₁₇	1	6	yes	—
198	C₁₆	C₂³×D₁₇	1	2	yes	—
199	C₁₆	C₂³×D₁₇	1	6	yes	—
200	C₂³	—	272	12	no	—
201	C₂³	—	68	24	no	—
202	C₂³	—	68	6	no	—
203	C₂³	C₂×D₁₃₆	1	3	yes	—
204	Q₈	—	34	4	no	—
205	Q₈	—	68	2	no	—
206	Q₈	C₂×D₁₃₆	1	4	yes	—
207	D₄	—	34	12	no	—
208	D₄	—	68	24	no	—
209	D₄	—	68	24	no	—
210	D₄	—	68	6	no	—
211	D₄	C₂×D₁₃₆	1	12	yes	—
212	C₂×C₄	—	68	2	no	—
213	C₂×C₄	—	68	24	no	—
214	C₂×C₄	C₂×D₁₃₆	1	6	yes	—
215	C₂×C₄	C₂×D₁₃₆	1	3	yes	—
216	C₂×C₄	C₂²×D₆₈	1	6	yes	—
217	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
218	C₈	C₂²×D₆₈	1	1	yes	characteristic
219	C₈	C₂²×D₆₈	1	6	yes	—
220	C₈	C₂⁴×D₁₇	1	1	yes	characteristic
221	C₂²	—	2	6	no	—
222	C₂²	—	272	24	no	—
223	C₂²	—	68	12	no	—
224	C₂²	—	136	12	no	—
225	C₂²	C₂²×D₁₃₆	1	6	yes	—
226	C₂²	C₂²×D₁₃₆	1	1	yes	characteristic
227	C₄	—	68	4	no	—
228	C₄	C₂²×D₁₃₆	1	6	yes	—
229	C₄	C₂²×D₁₃₆	1	1	yes	characteristic
230	C₄	C₂³×D₆₈	1	1	yes	characteristic
231	C₂	—	2	6	no	—
232	C₂	—	136	12	no	—
233	C₂	#14541925	1	2	yes	direct product factor
234	C₂	C₂³×D₁₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₃₆
commutator, cosocle	C₈×C₁₇	C₂⁵
Frattini	C₈	C₂⁴×D₁₇
Fitting	C₂×C₁₇×#504	C₂
socle	C₂²×C₁₇	C₂²×D₈
hypercenter	C₂×#504	D₁₇
hypocenter	C₁₇	C₂×#3415
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×#11763614	C₂	semidirect
2. #14541925	C₂	direct (N × H)
3. C₂×#11763247	C₂	semidirect
4. C₂×#11763241	C₂	semidirect
5. C₂²×D₂₇₂	C₂	semidirect
6. C₂×C₁₇×#504	C₂	semidirect
9. #11763614	C₂²	semidirect
11. C₂×#10587590	C₂²	semidirect
12. C₂²×D₁₃₆	C₂²	semidirect
13. #11763247	C₂²	semidirect
14. #11763241	C₂²	semidirect
15. C₂×Dic₁₃₆	C₂²	semidirect
16. C₂×#10587471	C₂²	semidirect
17. C₂×D₂₇₂	C₂²	semidirect
18. C₂×D₂₇₂	C₂²	semidirect
19. C₂×D₂₇₂	C₂²	semidirect
21. C₁₇×#504	C₂²	semidirect
22. C₂×C₁₇×M₃₂	C₂²	semidirect
23. C₂²×C₁₆×C₁₇	C₂²	semidirect
38. #10587590	C₂³	semidirect
42. C₂×D₁₃₆	C₂³	semidirect
44. C₁₇×M₃₂	C₂³	semidirect
47. C₂×C₁₆×C₁₇	C₂³	semidirect
48. Dic₁₃₆	C₂³	semidirect
49. #10587471	C₂³	semidirect
50. D₂₇₂	C₂³	semidirect
51. D₂₇₂	C₂³	semidirect
109. C₂×#504	D₁₇	semidirect
134. #504	C₂×D₁₇	semidirect
135. C₂×M₃₂	C₂×D₁₇	semidirect
136. C₂²×C₁₆	C₂×D₁₇	semidirect
138. C₂×C₁₇	#3415	semidirect
165. M₃₂	C₂²×D₁₇	semidirect
168. C₂×C₁₆	C₂²×D₁₇	semidirect
169. C₁₇	C₂×#3415	semidirect
233. C₂	#14541925	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×#11763614	2
#14541925	2
C₂×#11763247	2
C₂×#11763241	2
C₂²×D₂₇₂	2
C₂×C₁₇×#504	2
C₂×#3415	17

Subgroups — C2×#14541925 #17940836

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14541925 #17940836