Subgroups — #17981043

Order 4352.

11674 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(5,𝔽₂)×#595	C₂	1	32	yes	maximal, complemented
2	#14545764	C₂	1	12	yes	maximal, complemented
3	C₂×#11763876	C₂	1	18	yes	maximal, complemented
4	C₂×H(5,𝔽₂)×D₁₇	C₂	1	1	yes	maximal, characteristic
5	#11763878	C₂²	1	96	yes	complemented
6	#11763877	C₂²	1	8	yes	complemented
7	#11763876	C₂²	1	72	yes	complemented
8	#11763876	C₂²	1	144	yes	complemented
9	#41×#595	C₂²	1	96	yes	complemented
10	C₂×D₄×#595	C₂²	1	144	yes	complemented
11	C₂×#10587677	C₂²	1	36	yes	complemented
12	C₂²×C₂²⋊#595	C₂²	1	24	yes	complemented
13	C₂×#10587696	C₄	1	6	yes	complemented
14	C₂×#10587691	C₄	1	9	yes	complemented
15	H(5,𝔽₂)×D₁₇	C₄	1	16	yes	complemented
16	H(5,𝔽₂)×D₁₇	C₂²	1	16	yes	—
17	C₂×#41×D₁₇	C₂²	1	6	yes	—
18	C₂²×D₄×D₁₇	C₂²	1	9	yes	—
19	C₂×C₁₇×H(5,𝔽₂)	C₄	1	1	yes	characteristic, complemented
20	C₂⁴×D₁₇	C₂³	1	6	yes	—
21	C₂³×#595	C₂³	1	48	yes	complemented
22	C₁₇×H(5,𝔽₂)	C₂×C₄	1	16	yes	complemented
23	C₂×#41×C₁₇	C₂×C₄	1	6	yes	complemented
24	C₂²×D₄×C₁₇	C₂×C₄	1	9	yes	complemented
25	C₂²×C₂²⋊D₁₇	C₂×C₄	1	18	yes	complemented
26	#10587696	C₂×C₄	1	96	yes	complemented
27	#41×D₁₇	C₂×C₄	1	48	yes	complemented
28	#41×D₁₇	C₂³	1	48	yes	—
29	C₂×#63256	C₂×C₄	1	6	yes	complemented
30	C₂×Q₈×D₁₇	C₂³	1	2	yes	—
31	#10587691	C₂×C₄	1	144	yes	complemented
32	C₂×#63254	C₂×C₄	1	18	yes	complemented
33	C₂×D₄×D₁₇	C₂×C₄	1	72	yes	complemented
34	C₂×D₄×D₁₇	C₂×C₄	1	36	yes	complemented
35	C₂×D₄×D₁₇	C₂³	1	72	yes	—
36	C₂×D₄×D₁₇	C₂³	1	18	yes	—
37	C₂×#63252	C₂×C₄	1	18	yes	complemented
38	C₂²×D₆₈	C₂×C₄	1	9	yes	complemented
39	C₂²×C₄×D₁₇	C₂³	1	9	yes	—
40	C₂×C₂²⋊#595	C₂³	1	72	yes	—
41	C₂×C₂²⋊#595	C₂³	1	144	yes	complemented
42	Q₈×#595	C₂³	1	64	yes	complemented
43	D₄×#595	C₂³	1	576	yes	complemented
44	#10587677	C₂³	1	48	yes	—
45	#10587677	C₂³	1	144	yes	complemented
46	C₂×C₄⋊#595	C₂³	1	72	yes	complemented
47	C₂×C₄×#595	C₂³	1	72	yes	complemented
48	C₂⁴×C₁₇	C₂²×C₄	1	6	yes	complemented
49	C₂³×D₁₇	—	8	12	no	—
50	C₂³×D₁₇	C₂²×C₄	1	18	yes	complemented
51	C₂³×D₁₇	C₂²×C₄	1	24	yes	complemented
52	C₂³×D₁₇	C₂⁴	1	9	yes	—
53	C₂³×D₁₇	C₂⁴	1	24	yes	—
54	C₂²×#595	—	8	48	no	—
55	C₂²×#595	C₂⁴	1	144	yes	—
56	#41×C₁₇	C₂²×C₄	1	48	yes	complemented
57	C₂×Q₈×C₁₇	C₂²×C₄	1	2	yes	complemented
58	C₂×D₄×C₁₇	C₂²×C₄	1	18	yes	complemented
59	C₂×D₄×C₁₇	C₂²×C₄	1	72	yes	complemented
60	C₂²×C₄×C₁₇	C₂²×C₄	1	9	yes	complemented
61	C₂×C₂²⋊D₁₇	C₂²×C₄	1	36	yes	complemented
62	C₂×C₂²⋊D₁₇	C₂²×C₄	1	144	yes	complemented
63	C₂²×Dic₁₇	C₂²×C₄	1	9	yes	complemented
64	#63256	C₂²×C₄	1	48	yes	complemented
65	Q₈×D₁₇	C₂²×C₄	1	8	yes	complemented
66	Q₈×D₁₇	C₂⁴	1	8	yes	—
67	#63254	C₂²×C₄	1	144	yes	complemented
68	D₄×D₁₇	C₂²×C₄	1	72	yes	complemented
69	D₄×D₁₇	C₂²×C₄	1	288	yes	complemented
70	D₄×D₁₇	C₂⁴	1	72	yes	—
71	#63252	C₂²×C₄	1	144	yes	complemented
72	C₂×D₆₈	C₂²×C₄	1	18	yes	complemented
73	C₂×D₆₈	C₂²×C₄	1	72	yes	complemented
74	C₂×C₄×D₁₇	C₂²×C₄	1	18	yes	complemented
75	C₂×C₄×D₁₇	C₂²×C₄	1	36	yes	complemented
76	C₂×C₄×D₁₇	C₂⁴	1	36	yes	—
77	C₂×C₄×D₁₇	C₂⁴	1	6	yes	—
78	C₂×Dic₃₄	C₂²×C₄	1	6	yes	complemented
79	C₂²⋊#595	C₂⁴	1	16	yes	—
80	C₂²⋊#595	C₂⁴	1	144	yes	—
81	C₄⋊#595	C₂⁴	1	96	yes	—
82	C₄×#595	C₂⁴	1	96	yes	—
83	#62652	—	17	1	no	maximal, Sylow
84	C₂³×C₁₇	—	8	6	no	—
85	C₂³×C₁₇	C₂³×C₄	1	9	yes	—
86	C₂³×C₁₇	C₂³×C₄	1	24	yes	—
87	C₂²×D₁₇	—	4	48	no	—
88	C₂²×D₁₇	—	4	18	no	—
89	C₂²×D₁₇	—	8	24	no	—
90	C₂²×D₁₇	—	8	18	no	—
91	C₂²×D₁₇	C₂³×C₄	1	9	yes	—
92	C₂²×D₁₇	C₂³×C₄	1	18	yes	—
93	C₂²×D₁₇	C₂³×C₄	1	72	yes	—
94	C₂²×D₁₇	C₂⁵	1	1	yes	characteristic
95	C₂²×D₁₇	C₂⁵	1	18	yes	—
96	C₂×#595	—	4	144	no	—
97	C₂×#595	C₂⁵	1	32	yes	—
98	Q₈×C₁₇	C₂³×C₄	1	8	yes	—
99	D₄×C₁₇	C₂³×C₄	1	72	yes	—
100	C₂×C₄×C₁₇	C₂³×C₄	1	6	yes	—
101	C₂×C₄×C₁₇	C₂³×C₄	1	36	yes	—
102	C₂²⋊D₁₇	C₂³×C₄	1	144	yes	—
103	C₂×Dic₁₇	C₂³×C₄	1	6	yes	—
104	C₂×Dic₁₇	C₂³×C₄	1	36	yes	—
105	D₆₈	C₂³×C₄	1	72	yes	—
106	C₄×D₁₇	C₂³×C₄	1	12	yes	—
107	C₄×D₁₇	C₂³×C₄	1	72	yes	—
108	C₄×D₁₇	C₂⁵	1	12	yes	—
109	Dic₃₄	C₂³×C₄	1	24	yes	—
110	C₂²×H(5,𝔽₂)	—	17	1	no	—
111	C₄×H(5,𝔽₂)	—	17	32	no	—
112	#3428	—	17	12	no	—
113	C₂×#518	—	17	18	no	—
114	C₂²×C₁₇	—	4	9	no	—
115	C₂²×C₁₇	—	4	24	no	—
116	C₂²×C₁₇	C₂⁴×C₄	1	1	yes	characteristic
117	C₂²×C₁₇	C₂⁴×C₄	1	18	yes	—
118	C₂×D₁₇	—	2	36	no	—
119	C₂×D₁₇	—	2	2	no	—
120	C₂×D₁₇	—	4	18	no	—
121	C₂×D₁₇	—	4	9	no	—
122	C₂×D₁₇	—	4	72	no	—
123	C₂×D₁₇	C₂⁴×C₄	1	1	yes	characteristic
124	C₂×D₁₇	C₂⁴×C₄	1	18	yes	—
125	C₂×D₁₇	C₂⁶	1	1	yes	characteristic
126	#595	—	2	32	no	—
127	C₄×C₁₇	C₂⁴×C₄	1	12	yes	—
128	Dic₁₇	C₂⁴×C₄	1	12	yes	—
129	C₂×H(5,𝔽₂)	—	17	16	no	—
130	C₂×H(5,𝔽₂)	—	17	16	no	—
131	C₂×H(5,𝔽₂)	—	17	6	no	—
132	C₂×H(5,𝔽₂)	—	17	9	no	—
133	C₂×H(5,𝔽₂)	#595	1	1	yes	characteristic, complemented
134	C₂²×#41	—	17	6	no	—
135	C₂³×D₄	—	17	9	no	—
136	#520	—	17	96	no	—
137	#519	—	17	8	no	—
138	#518	—	17	72	no	—
139	#518	—	17	144	no	—
140	C₄×#41	—	17	96	no	—
141	C₂×C₄×D₄	—	17	144	no	—
142	C₂×#117	—	17	36	no	—
143	C₂²×C₂²⋊C₄	—	17	24	no	—
144	C₂×C₁₇	—	2	1	no	—
145	C₂×C₁₇	—	2	18	no	—
146	C₂×C₁₇	C₂⁵×C₄	1	1	yes	characteristic
147	D₁₇	—	2	1	no	—
148	D₁₇	—	2	18	no	—
149	D₁₇	H(7,𝔽₂)	1	2	yes	—
150	C₂⁵	—	17	6	no	—
151	H(5,𝔽₂)	—	17	96	no	—
152	H(5,𝔽₂)	—	17	144	no	—
153	H(5,𝔽₂)	C₂×#595	1	16	yes	complemented
154	C₂×#41	—	17	48	no	—
155	C₂×#41	—	17	48	no	—
156	C₂×#41	—	17	6	no	—
157	C₂×#41	—	17	18	no	—
158	C₂×#41	—	17	18	no	—
159	C₂×#41	C₂×#595	1	6	yes	complemented
160	C₂²×Q₈	—	17	2	no	—
161	C₂²×D₄	—	17	18	no	—
162	C₂²×D₄	—	17	72	no	—
163	C₂²×D₄	—	17	18	no	—
164	C₂²×D₄	—	17	36	no	—
165	C₂²×D₄	—	17	72	no	—
166	C₂²×D₄	—	17	9	no	—
167	C₂²×D₄	C₂×#595	1	9	yes	complemented
168	C₂³×C₄	—	17	9	no	—
169	C₂³×C₄	—	17	48	no	—
170	C₄×Q₈	—	17	64	no	—
171	C₄×D₄	—	17	576	no	—
172	#117	—	17	48	no	—
173	#117	—	17	144	no	—
174	C₂×C₄⋊C₄	—	17	72	no	—
175	C₂×C₂²⋊C₄	—	17	72	no	—
176	C₂×C₂²⋊C₄	—	17	144	no	—
177	C₂×C₄²	—	17	72	no	—
178	C₁₇	#62652	1	1	yes	characteristic, complemented, Sylow
179	C₂⁴	—	17	9	no	—
180	C₂⁴	—	17	24	no	—
181	C₂⁴	—	17	24	no	—
182	C₂⁴	—	17	18	no	—
183	C₂⁴	—	136	12	no	—
184	C₂⁴	C₂²×#595	1	6	yes	complemented
185	#41	—	17	48	no	—
186	#41	—	17	144	no	—
187	#41	—	17	144	no	—
188	#41	C₂²×#595	1	48	yes	complemented
189	C₂×Q₈	—	17	8	no	—
190	C₂×Q₈	—	17	8	no	—
191	C₂×Q₈	—	17	6	no	—
192	C₂×Q₈	C₂²×#595	1	2	yes	complemented
193	C₂×D₄	—	17	36	no	—
194	C₂×D₄	—	17	144	no	—
195	C₂×D₄	—	17	72	no	—
196	C₂×D₄	—	17	288	no	—
197	C₂×D₄	—	17	72	no	—
198	C₂×D₄	—	17	18	no	—
199	C₂×D₄	—	17	72	no	—
200	C₂×D₄	C₂²×#595	1	18	yes	complemented
201	C₂×D₄	C₂²×#595	1	72	yes	complemented
202	C₂²×C₄	—	17	9	no	—
203	C₂²×C₄	—	17	18	no	—
204	C₂²×C₄	—	17	36	no	—
205	C₂²×C₄	—	17	36	no	—
206	C₂²×C₄	—	17	6	no	—
207	C₂²×C₄	—	17	144	no	—
208	C₂²×C₄	—	136	48	no	—
209	C₂²×C₄	C₂²×#595	1	9	yes	complemented
210	C₄⋊C₄	—	17	96	no	—
211	C₂²⋊C₄	—	17	16	no	—
212	C₂²⋊C₄	—	17	144	no	—
213	C₄²	—	17	96	no	—
214	C₂³	—	8	6	no	—
215	C₂³	—	17	1	no	—
216	C₂³	—	17	18	no	—
217	C₂³	—	17	18	no	—
218	C₂³	—	17	9	no	—
219	C₂³	—	17	72	no	—
220	C₂³	—	68	48	no	—
221	C₂³	—	68	18	no	—
222	C₂³	—	136	18	no	—
223	C₂³	—	136	24	no	—
224	C₂³	C₂³×#595	1	9	yes	—
225	C₂³	C₂³×#595	1	24	yes	—
226	Q₈	—	17	24	no	—
227	Q₈	C₂³×#595	1	8	yes	—
228	D₄	—	17	144	no	—
229	D₄	—	17	72	no	—
230	D₄	C₂³×#595	1	72	yes	—
231	C₂×C₄	—	17	6	no	—
232	C₂×C₄	—	17	36	no	—
233	C₂×C₄	—	17	12	no	—
234	C₂×C₄	—	17	12	no	—
235	C₂×C₄	—	17	72	no	—
236	C₂×C₄	—	17	32	no	—
237	C₂×C₄	—	68	144	no	—
238	C₂×C₄	C₂³×#595	1	6	yes	—
239	C₂×C₄	C₂³×#595	1	36	yes	—
240	C₂²	—	4	9	no	—
241	C₂²	—	4	24	no	—
242	C₂²	—	17	1	no	—
243	C₂²	—	17	1	no	—
244	C₂²	—	17	18	no	—
245	C₂²	—	34	36	no	—
246	C₂²	—	34	2	no	—
247	C₂²	—	68	9	no	—
248	C₂²	—	68	18	no	—
249	C₂²	—	68	72	no	—
250	C₂²	C₂⁴×#595	1	1	yes	characteristic
251	C₂²	C₂⁴×#595	1	18	yes	—
252	C₄	—	17	12	no	—
253	C₄	—	34	32	no	—
254	C₄	C₂⁴×#595	1	12	yes	—
255	C₂	—	2	1	no	—
256	C₂	—	2	18	no	—
257	C₂	—	17	2	no	—
258	C₂	—	34	1	no	—
259	C₂	—	34	18	no	—
260	C₂	C₂⁵×#595	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁵×#595
commutator, socle	C₂×C₁₇	C₂⁵×C₄
Fitting	C₂×C₁₇×H(5,𝔽₂)	C₄
prime radical	C₂×H(5,𝔽₂)×D₁₇	C₂
cosocle	C₂×D₁₇	C₂⁶
hypercenter	C₂×H(5,𝔽₂)	#595
hypocenter	C₁₇	#62652
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. H(5,𝔽₂)×#595	C₂	semidirect
2. #14545764	C₂	semidirect
3. C₂×#11763876	C₂	semidirect
5. #11763878	C₂²	semidirect
6. #11763877	C₂²	semidirect
7. #11763876	C₂²	semidirect
8. #11763876	C₂²	semidirect
9. #41×#595	C₂²	semidirect
10. C₂×D₄×#595	C₂²	semidirect
11. C₂×#10587677	C₂²	semidirect
12. C₂²×C₂²⋊#595	C₂²	semidirect
13. C₂×#10587696	C₄	semidirect
14. C₂×#10587691	C₄	semidirect
15. H(5,𝔽₂)×D₁₇	C₄	semidirect
19. C₂×C₁₇×H(5,𝔽₂)	C₄	semidirect
21. C₂³×#595	C₂³	semidirect
22. C₁₇×H(5,𝔽₂)	C₂×C₄	semidirect
23. C₂×#41×C₁₇	C₂×C₄	semidirect
24. C₂²×D₄×C₁₇	C₂×C₄	semidirect
25. C₂²×C₂²⋊D₁₇	C₂×C₄	semidirect
26. #10587696	C₂×C₄	semidirect
27. #41×D₁₇	C₂×C₄	semidirect
29. C₂×#63256	C₂×C₄	semidirect
31. #10587691	C₂×C₄	semidirect
32. C₂×#63254	C₂×C₄	semidirect
33. C₂×D₄×D₁₇	C₂×C₄	semidirect
34. C₂×D₄×D₁₇	C₂×C₄	semidirect
37. C₂×#63252	C₂×C₄	semidirect
38. C₂²×D₆₈	C₂×C₄	semidirect
41. C₂×C₂²⋊#595	C₂³	semidirect
42. Q₈×#595	C₂³	semidirect
43. D₄×#595	C₂³	semidirect
45. #10587677	C₂³	semidirect
46. C₂×C₄⋊#595	C₂³	semidirect
47. C₂×C₄×#595	C₂³	semidirect
48. C₂⁴×C₁₇	C₂²×C₄	semidirect
50. C₂³×D₁₇	C₂²×C₄	semidirect
51. C₂³×D₁₇	C₂²×C₄	semidirect
56. #41×C₁₇	C₂²×C₄	semidirect
57. C₂×Q₈×C₁₇	C₂²×C₄	semidirect
58. C₂×D₄×C₁₇	C₂²×C₄	semidirect
59. C₂×D₄×C₁₇	C₂²×C₄	semidirect
60. C₂²×C₄×C₁₇	C₂²×C₄	semidirect
61. C₂×C₂²⋊D₁₇	C₂²×C₄	semidirect
62. C₂×C₂²⋊D₁₇	C₂²×C₄	semidirect
63. C₂²×Dic₁₇	C₂²×C₄	semidirect
64. #63256	C₂²×C₄	semidirect
65. Q₈×D₁₇	C₂²×C₄	semidirect
67. #63254	C₂²×C₄	semidirect
68. D₄×D₁₇	C₂²×C₄	semidirect
69. D₄×D₁₇	C₂²×C₄	semidirect
71. #63252	C₂²×C₄	semidirect
72. C₂×D₆₈	C₂²×C₄	semidirect
73. C₂×D₆₈	C₂²×C₄	semidirect
74. C₂×C₄×D₁₇	C₂²×C₄	semidirect
75. C₂×C₄×D₁₇	C₂²×C₄	semidirect
78. C₂×Dic₃₄	C₂²×C₄	semidirect
133. C₂×H(5,𝔽₂)	#595	semidirect
153. H(5,𝔽₂)	C₂×#595	semidirect
159. C₂×#41	C₂×#595	semidirect
167. C₂²×D₄	C₂×#595	semidirect
178. C₁₇	#62652	semidirect
184. C₂⁴	C₂²×#595	semidirect
188. #41	C₂²×#595	semidirect
192. C₂×Q₈	C₂²×#595	semidirect
200. C₂×D₄	C₂²×#595	semidirect
201. C₂×D₄	C₂²×#595	semidirect
209. C₂²×C₄	C₂²×#595	semidirect

Maximal subgroups

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

subgroup	index
H(5,𝔽₂)×#595	2
#14545764	2
C₂×#11763876	2
C₂×H(5,𝔽₂)×D₁₇	2
#62652	17