Subgroups — C₂×H(5,𝔽₂)×Dic₁₇ #17940928

Order 4352.

12378 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	H(5,𝔽₂)×Dic₁₇	C₂	1	32	yes	maximal, direct product factor
2	C₂×#11763292	C₂	1	6	yes	maximal, complemented
3	C₂×#11763284	C₂	1	9	yes	maximal, complemented
4	C₂×#41×Dic₁₇	C₂	1	6	yes	maximal, complemented
5	C₂²×D₄×Dic₁₇	C₂	1	9	yes	maximal, complemented
6	C₂²×C₁₇×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₁₇	C₂²	1	6	yes	complemented
8	#11763292	C₂²	1	96	yes	complemented
9	#11763284	C₂²	1	144	yes	complemented
10	#41×Dic₁₇	C₂²	1	96	yes	complemented
11	C₂×Q₈×Dic₁₇	C₂²	1	2	yes	complemented
12	C₂×Q₈×Dic₁₇	C₂²	1	6	yes	complemented
13	C₂×D₄×Dic₁₇	C₂²	1	18	yes	complemented
14	C₂×D₄×Dic₁₇	C₂²	1	36	yes	complemented
15	C₂×D₄×Dic₁₇	C₂²	1	18	yes	complemented
16	C₂×D₄×Dic₁₇	C₂²	1	144	yes	complemented
17	C₂×#10587612	C₂²	1	18	yes	complemented
18	C₂²×C₄⋊Dic₁₇	C₂²	1	9	yes	complemented
19	C₂²×C₂²⋊Dic₁₇	C₂²	1	18	yes	complemented
20	C₂²×C₄×Dic₁₇	C₂²	1	9	yes	complemented
21	C₂×C₁₇×H(5,𝔽₂)	C₄	1	32	yes	complemented
22	C₂×C₁₇×H(5,𝔽₂)	C₂²	1	16	yes	—
23	C₂²×#41×C₁₇	C₂²	1	6	yes	—
24	C₂³×D₄×C₁₇	C₂²	1	9	yes	—
25	C₂⁵×C₁₇	C₂³	1	6	yes	—
26	C₁₇×H(5,𝔽₂)	C₂×C₄	1	256	yes	complemented
27	C₂×#41×C₁₇	C₂×C₄	1	96	yes	complemented
28	C₂×#41×C₁₇	C₂³	1	48	yes	—
29	C₂²×Q₈×C₁₇	C₂³	1	2	yes	—
30	C₂²×D₄×C₁₇	C₂×C₄	1	144	yes	complemented
31	C₂²×D₄×C₁₇	C₂³	1	18	yes	—
32	C₂²×D₄×C₁₇	C₂³	1	72	yes	—
33	C₂³×C₄×C₁₇	C₂³	1	9	yes	—
34	C₂³×Dic₁₇	—	4	24	no	—
35	C₂³×Dic₁₇	C₂³	1	9	yes	—
36	C₂³×Dic₁₇	C₂³	1	9	yes	—
37	C₂³×Dic₁₇	C₂³	1	18	yes	—
38	C₂³×Dic₁₇	C₂³	1	48	yes	complemented
39	C₂×C₂²⋊Dic₁₇	C₂³	1	36	yes	—
40	C₂×C₂²⋊Dic₁₇	C₂³	1	144	yes	complemented
41	Q₈×Dic₁₇	C₂³	1	16	yes	complemented
42	Q₈×Dic₁₇	C₂³	1	48	yes	complemented
43	D₄×Dic₁₇	C₂³	1	144	yes	complemented
44	D₄×Dic₁₇	C₂³	1	288	yes	complemented
45	D₄×Dic₁₇	C₂³	1	144	yes	complemented
46	#10587612	C₂³	1	144	yes	complemented
47	C₂×C₄⋊Dic₁₇	C₂³	1	18	yes	—
48	C₂×C₄⋊Dic₁₇	C₂³	1	6	yes	—
49	C₂×C₄⋊Dic₁₇	C₂³	1	72	yes	complemented
50	C₂×C₄×Dic₁₇	C₂³	1	6	yes	—
51	C₂×C₄×Dic₁₇	C₂³	1	18	yes	—
52	C₂×C₄×Dic₁₇	C₂³	1	72	yes	complemented
53	C₂⁴×C₁₇	—	4	12	no	—
54	C₂⁴×C₁₇	—	4	12	no	—
55	C₂⁴×C₁₇	C₂²×C₄	1	48	yes	complemented
56	C₂⁴×C₁₇	C₂⁴	1	9	yes	—
57	C₂⁴×C₁₇	C₂⁴	1	24	yes	—
58	#41×C₁₇	C₂²×C₄	1	384	yes	complemented
59	C₂×Q₈×C₁₇	C₂²×C₄	1	16	yes	complemented
60	C₂×Q₈×C₁₇	C₂⁴	1	8	yes	—
61	C₂×D₄×C₁₇	C₂²×C₄	1	144	yes	complemented
62	C₂×D₄×C₁₇	C₂²×C₄	1	576	yes	complemented
63	C₂×D₄×C₁₇	C₂⁴	1	72	yes	—
64	C₂²×C₄×C₁₇	C₂²×C₄	1	72	yes	complemented
65	C₂²×C₄×C₁₇	C₂⁴	1	6	yes	—
66	C₂²×C₄×C₁₇	C₂⁴	1	36	yes	—
67	C₂²×Dic₁₇	—	2	36	no	—
68	C₂²×Dic₁₇	—	4	96	no	—
69	C₂²×Dic₁₇	—	4	36	no	—
70	C₂²×Dic₁₇	—	4	18	no	—
71	C₂²×Dic₁₇	C₂⁴	1	9	yes	—
72	C₂²×Dic₁₇	C₂⁴	1	6	yes	—
73	C₂²×Dic₁₇	C₂⁴	1	36	yes	—
74	C₂²×Dic₁₇	C₂⁴	1	72	yes	—
75	C₂²×Dic₁₇	C₂⁴	1	36	yes	—
76	C₂²×Dic₁₇	C₂⁴	1	1	yes	characteristic
77	C₂²⋊Dic₁₇	C₂⁴	1	144	yes	—
78	C₄⋊Dic₁₇	C₂⁴	1	72	yes	—
79	C₄⋊Dic₁₇	C₂⁴	1	24	yes	—
80	C₄×Dic₁₇	C₂⁴	1	24	yes	—
81	C₄×Dic₁₇	C₂⁴	1	72	yes	—
82	C₂×C₄×H(5,𝔽₂)	—	17	1	no	maximal, Sylow
83	C₂³×C₁₇	—	2	18	no	—
84	C₂³×C₁₇	—	2	18	no	—
85	C₂³×C₁₇	—	4	24	no	—
86	C₂³×C₁₇	—	4	96	no	—
87	C₂³×C₁₇	—	4	24	no	—
88	C₂³×C₁₇	C₂³×C₄	1	36	yes	—
89	C₂³×C₁₇	C₂³×C₄	1	96	yes	complemented
90	C₂³×C₁₇	C₂⁵	1	18	yes	—
91	C₂³×C₁₇	C₂⁵	1	1	yes	characteristic
92	Q₈×C₁₇	C₂³×C₄	1	32	yes	complemented
93	D₄×C₁₇	C₂³×C₄	1	288	yes	complemented
94	C₂×C₄×C₁₇	C₂³×C₄	1	24	yes	—
95	C₂×C₄×C₁₇	C₂³×C₄	1	144	yes	complemented
96	C₂×C₄×C₁₇	C₂⁵	1	12	yes	—
97	C₂×Dic₁₇	—	2	18	no	—
98	C₂×Dic₁₇	—	2	12	no	—
99	C₂×Dic₁₇	—	2	72	no	—
100	C₂×Dic₁₇	—	4	72	no	—
101	C₂×Dic₁₇	—	4	36	no	—
102	C₂×Dic₁₇	H(5,𝔽₂)	1	4	yes	direct product factor
103	C₂×Dic₁₇	C₂⁵	1	12	yes	—
104	C₂×Dic₁₇	C₂⁵	1	18	yes	—
105	C₂×Dic₁₇	C₂⁵	1	2	yes	—
106	C₂²×H(5,𝔽₂)	D₁₇	1	1	yes	characteristic
107	C₄×H(5,𝔽₂)	—	17	32	no	—
108	C₂×#520	—	17	6	no	—
109	C₂×#518	—	17	9	no	—
110	C₂×C₄×#41	—	17	6	no	—
111	C₂²×C₄×D₄	—	17	9	no	—
112	C₂²×C₁₇	—	2	18	no	—
113	C₂²×C₁₇	—	2	18	no	—
114	C₂²×C₁₇	—	2	72	no	—
115	C₂²×C₁₇	—	4	96	no	—
116	C₂²×C₁₇	C₂⁴×C₄	1	36	yes	—
117	C₂²×C₁₇	C₂⁴×C₄	1	2	yes	—
118	C₂²×C₁₇	C₂×H(5,𝔽₂)	1	2	yes	—
119	C₂²×C₁₇	#585	1	2	yes	—
120	C₂²×C₁₇	C₂⁶	1	1	yes	characteristic
121	C₄×C₁₇	C₂⁴×C₄	1	24	yes	—
122	Dic₁₇	—	2	18	no	—
123	Dic₁₇	—	2	12	no	—
124	Dic₁₇	C₂×H(5,𝔽₂)	1	4	yes	direct product factor
125	C₂×H(5,𝔽₂)	Dic₁₇	1	32	yes	direct product factor
126	C₂×H(5,𝔽₂)	C₂×D₁₇	1	16	yes	—
127	C₂²×#41	C₂×D₁₇	1	6	yes	—
128	C₂³×D₄	C₂×D₁₇	1	9	yes	—
129	C₂⁴×C₄	—	17	6	no	—
130	#520	—	17	96	no	—
131	#518	—	17	144	no	—
132	C₄×#41	—	17	96	no	—
133	C₂×C₄×Q₈	—	17	2	no	—
134	C₂×C₄×Q₈	—	17	6	no	—
135	C₂×C₄×D₄	—	17	18	no	—
136	C₂×C₄×D₄	—	17	36	no	—
137	C₂×C₄×D₄	—	17	144	no	—
138	C₂×C₄×D₄	—	17	18	no	—
139	C₂×#117	—	17	18	no	—
140	C₂²×C₄⋊C₄	—	17	9	no	—
141	C₂²×C₂²⋊C₄	—	17	18	no	—
142	C₂²×C₄²	—	17	9	no	—
143	C₂×C₁₇	—	2	36	no	—
144	C₂×C₁₇	C₄×H(5,𝔽₂)	1	4	yes	complemented
145	C₂×C₁₇	C₂⁵×C₄	1	1	yes	characteristic
146	C₂×C₁₇	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
147	C₂×C₁₇	C₂×#585	1	1	yes	characteristic
148	C₂⁵	C₂²×D₁₇	1	6	yes	—
149	H(5,𝔽₂)	C₂×Dic₁₇	1	256	yes	direct product factor
150	C₂×#41	C₂×Dic₁₇	1	96	yes	complemented
151	C₂×#41	C₂²×D₁₇	1	48	yes	—
152	C₂²×Q₈	C₂²×D₁₇	1	2	yes	—
153	C₂²×D₄	C₂×Dic₁₇	1	144	yes	complemented
154	C₂²×D₄	C₂²×D₁₇	1	18	yes	—
155	C₂²×D₄	C₂²×D₁₇	1	72	yes	—
156	C₂³×C₄	—	17	9	no	—
157	C₂³×C₄	—	17	9	no	—
158	C₂³×C₄	—	17	18	no	—
159	C₂³×C₄	—	17	48	no	—
160	C₂³×C₄	—	68	24	no	—
161	C₂³×C₄	C₂²×D₁₇	1	9	yes	—
162	C₄×Q₈	—	17	16	no	—
163	C₄×Q₈	—	17	48	no	—
164	C₄×D₄	—	17	144	no	—
165	C₄×D₄	—	17	288	no	—
166	C₄×D₄	—	17	144	no	—
167	#117	—	17	144	no	—
168	C₂×C₄⋊C₄	—	17	18	no	—
169	C₂×C₄⋊C₄	—	17	72	no	—
170	C₂×C₄⋊C₄	—	17	6	no	—
171	C₂×C₂²⋊C₄	—	17	36	no	—
172	C₂×C₂²⋊C₄	—	17	144	no	—
173	C₂×C₄²	—	17	18	no	—
174	C₂×C₄²	—	17	6	no	—
175	C₂×C₄²	—	17	72	no	—
176	C₁₇	C₂×C₄×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
177	C₂⁴	—	4	12	no	—
178	C₂⁴	—	4	12	no	—
179	C₂⁴	C₂²×Dic₁₇	1	48	yes	complemented
180	C₂⁴	C₂³×D₁₇	1	9	yes	—
181	C₂⁴	C₂³×D₁₇	1	24	yes	—
182	#41	C₂²×Dic₁₇	1	384	yes	complemented
183	C₂×Q₈	C₂²×Dic₁₇	1	16	yes	complemented
184	C₂×Q₈	C₂³×D₁₇	1	8	yes	—
185	C₂×D₄	C₂²×Dic₁₇	1	144	yes	complemented
186	C₂×D₄	C₂²×Dic₁₇	1	576	yes	complemented
187	C₂×D₄	C₂³×D₁₇	1	72	yes	—
188	C₂²×C₄	—	17	1	no	—
189	C₂²×C₄	—	17	36	no	—
190	C₂²×C₄	—	17	6	no	—
191	C₂²×C₄	—	17	9	no	—
192	C₂²×C₄	—	17	36	no	—
193	C₂²×C₄	—	17	72	no	—
194	C₂²×C₄	—	34	36	no	—
195	C₂²×C₄	—	68	96	no	—
196	C₂²×C₄	—	68	18	no	—
197	C₂²×C₄	—	68	36	no	—
198	C₂²×C₄	C₂²×Dic₁₇	1	72	yes	complemented
199	C₂²×C₄	C₂³×D₁₇	1	6	yes	—
200	C₂²×C₄	C₂³×D₁₇	1	36	yes	—
201	C₄⋊C₄	—	17	72	no	—
202	C₄⋊C₄	—	17	24	no	—
203	C₂²⋊C₄	—	17	144	no	—
204	C₄²	—	17	24	no	—
205	C₄²	—	17	72	no	—
206	C₂³	—	2	18	no	—
207	C₂³	—	2	18	no	—
208	C₂³	—	4	24	no	—
209	C₂³	—	4	96	no	—
210	C₂³	—	4	24	no	—
211	C₂³	C₂³×Dic₁₇	1	36	yes	—
212	C₂³	C₂³×Dic₁₇	1	96	yes	complemented
213	C₂³	C₂⁴×D₁₇	1	18	yes	—
214	C₂³	C₂⁴×D₁₇	1	1	yes	characteristic
215	Q₈	C₂³×Dic₁₇	1	32	yes	complemented
216	D₄	C₂³×Dic₁₇	1	288	yes	complemented
217	C₂×C₄	—	17	4	no	—
218	C₂×C₄	—	17	2	no	—
219	C₂×C₄	—	17	12	no	—
220	C₂×C₄	—	17	18	no	—
221	C₂×C₄	—	34	72	no	—
222	C₂×C₄	—	34	18	no	—
223	C₂×C₄	—	34	12	no	—
224	C₂×C₄	—	68	72	no	—
225	C₂×C₄	—	68	36	no	—
226	C₂×C₄	C₂³×Dic₁₇	1	24	yes	—
227	C₂×C₄	C₂³×Dic₁₇	1	144	yes	complemented
228	C₂×C₄	C₂⁴×D₁₇	1	12	yes	—
229	C₂²	—	2	72	no	—
230	C₂²	—	2	18	no	—
231	C₂²	—	2	18	no	—
232	C₂²	—	4	96	no	—
233	C₂²	C₂⁴×Dic₁₇	1	36	yes	—
234	C₂²	C₂⁴×Dic₁₇	1	2	yes	—
235	C₂²	H(5,𝔽₂)×D₁₇	1	2	yes	—
236	C₂²	#11763644	1	2	yes	—
237	C₂²	C₂⁵×D₁₇	1	1	yes	characteristic
238	C₄	—	17	4	no	—
239	C₄	—	34	18	no	—
240	C₄	—	34	12	no	—
241	C₄	C₂⁴×Dic₁₇	1	24	yes	—
242	C₂	—	2	36	no	—
243	C₂	H(5,𝔽₂)×Dic₁₇	1	4	yes	direct product factor
244	C₂	C₂⁵×Dic₁₇	1	1	yes	characteristic
245	C₂	C₂×H(5,𝔽₂)×D₁₇	1	1	yes	characteristic
246	C₂	C₂×#11763644	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₁₇
commutator	C₂×C₁₇	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₁₇
Fitting, prime radical	C₂²×C₁₇×H(5,𝔽₂)	C₂
socle	C₂³×C₁₇	C₂⁵
cosocle	C₂²×C₁₇	C₂⁶
hypercenter	C₂²×H(5,𝔽₂)	D₁₇
hypocenter	C₁₇	C₂×C₄×H(5,𝔽₂)
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. H(5,𝔽₂)×Dic₁₇	C₂	direct (N × H)
2. C₂×#11763292	C₂	semidirect
3. C₂×#11763284	C₂	semidirect
4. C₂×#41×Dic₁₇	C₂	semidirect
5. C₂²×D₄×Dic₁₇	C₂	semidirect
7. C₂⁴×Dic₁₇	C₂²	semidirect
8. #11763292	C₂²	semidirect
9. #11763284	C₂²	semidirect
10. #41×Dic₁₇	C₂²	semidirect
11. C₂×Q₈×Dic₁₇	C₂²	semidirect
12. C₂×Q₈×Dic₁₇	C₂²	semidirect
13. C₂×D₄×Dic₁₇	C₂²	semidirect
14. C₂×D₄×Dic₁₇	C₂²	semidirect
15. C₂×D₄×Dic₁₇	C₂²	semidirect
16. C₂×D₄×Dic₁₇	C₂²	semidirect
17. C₂×#10587612	C₂²	semidirect
18. C₂²×C₄⋊Dic₁₇	C₂²	semidirect
19. C₂²×C₂²⋊Dic₁₇	C₂²	semidirect
20. C₂²×C₄×Dic₁₇	C₂²	semidirect
21. C₂×C₁₇×H(5,𝔽₂)	C₄	semidirect
26. C₁₇×H(5,𝔽₂)	C₂×C₄	semidirect
27. C₂×#41×C₁₇	C₂×C₄	semidirect
30. C₂²×D₄×C₁₇	C₂×C₄	semidirect
38. C₂³×Dic₁₇	C₂³	semidirect
40. C₂×C₂²⋊Dic₁₇	C₂³	semidirect
41. Q₈×Dic₁₇	C₂³	semidirect
42. Q₈×Dic₁₇	C₂³	semidirect
43. D₄×Dic₁₇	C₂³	semidirect
44. D₄×Dic₁₇	C₂³	semidirect
45. D₄×Dic₁₇	C₂³	semidirect
46. #10587612	C₂³	semidirect
49. C₂×C₄⋊Dic₁₇	C₂³	semidirect
52. C₂×C₄×Dic₁₇	C₂³	semidirect
55. C₂⁴×C₁₇	C₂²×C₄	semidirect
58. #41×C₁₇	C₂²×C₄	semidirect
59. C₂×Q₈×C₁₇	C₂²×C₄	semidirect
61. C₂×D₄×C₁₇	C₂²×C₄	semidirect
62. C₂×D₄×C₁₇	C₂²×C₄	semidirect
64. C₂²×C₄×C₁₇	C₂²×C₄	semidirect
89. C₂³×C₁₇	C₂³×C₄	semidirect
92. Q₈×C₁₇	C₂³×C₄	semidirect
93. D₄×C₁₇	C₂³×C₄	semidirect
95. C₂×C₄×C₁₇	C₂³×C₄	semidirect
102. C₂×Dic₁₇	H(5,𝔽₂)	direct (N × H)
124. Dic₁₇	C₂×H(5,𝔽₂)	direct (N × H)
125. C₂×H(5,𝔽₂)	Dic₁₇	direct (N × H)
144. C₂×C₁₇	C₄×H(5,𝔽₂)	semidirect
149. H(5,𝔽₂)	C₂×Dic₁₇	direct (N × H)
150. C₂×#41	C₂×Dic₁₇	semidirect
153. C₂²×D₄	C₂×Dic₁₇	semidirect
176. C₁₇	C₂×C₄×H(5,𝔽₂)	semidirect
179. C₂⁴	C₂²×Dic₁₇	semidirect
182. #41	C₂²×Dic₁₇	semidirect
183. C₂×Q₈	C₂²×Dic₁₇	semidirect
185. C₂×D₄	C₂²×Dic₁₇	semidirect
186. C₂×D₄	C₂²×Dic₁₇	semidirect
198. C₂²×C₄	C₂²×Dic₁₇	semidirect
212. C₂³	C₂³×Dic₁₇	semidirect
215. Q₈	C₂³×Dic₁₇	semidirect
216. D₄	C₂³×Dic₁₇	semidirect
227. C₂×C₄	C₂³×Dic₁₇	semidirect
243. C₂	H(5,𝔽₂)×Dic₁₇	direct (N × H)

Maximal subgroups

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

subgroup	index
H(5,𝔽₂)×Dic₁₇	2
C₂×#11763292	2
C₂×#11763284	2
C₂×#41×Dic₁₇	2
C₂²×D₄×Dic₁₇	2
C₂²×C₁₇×H(5,𝔽₂)	2
C₂×C₄×H(5,𝔽₂)	17

Subgroups — C2×H(5,𝔽2)×Dic17 #17940928

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×H(5,𝔽₂)×Dic₁₇ #17940928