Subgroups — C₂³×#10587610 #17940967

Order 4352.

14078 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₃₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#10587610	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×#63225	C₂	1	2	yes	maximal, complemented
5	C₂³×C₂²⋊Dic₁₇	C₂	1	2	yes	maximal
6	C₂⁵×C₄×C₁₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₃₄	—	2	16	no	—
8	C₂³×Dic₃₄	C₂²	1	28	yes	complemented
9	C₂⁴×Dic₁₇	C₂²	1	2	yes	—
10	C₂⁴×Dic₁₇	C₂²	1	2	yes	—
11	C₂×#10587610	C₂²	1	448	yes	direct product factor
12	C₂²×C₄⋊Dic₁₇	C₂²	1	28	yes	complemented
13	C₂²×#63225	C₂²	1	56	yes	complemented
14	C₂²×C₂²⋊Dic₁₇	C₂²	1	56	yes	—
15	C₂⁶×C₁₇	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₁₇	—	2	16	no	—
17	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄×C₁₇	C₂²	1	28	yes	—
19	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
20	C₂⁵×C₁₇	—	2	8	no	—
21	C₂⁵×C₁₇	—	2	8	no	—
22	C₂⁵×C₁₇	Q₈	1	16	yes	complemented
23	C₂⁵×C₁₇	C₂³	1	14	yes	—
24	C₂⁵×C₁₇	C₂³	1	1	yes	characteristic
25	C₂³×C₄×C₁₇	—	2	8	no	—
26	C₂³×C₄×C₁₇	—	2	224	no	—
27	C₂³×C₄×C₁₇	D₄	1	16	yes	—
28	C₂³×C₄×C₁₇	C₂³	1	14	yes	—
29	C₂³×C₄×C₁₇	C₂³	1	112	yes	—
30	C₂³×C₄×C₁₇	C₂³	1	14	yes	—
31	C₂³×Dic₁₇	—	2	16	no	—
32	C₂³×Dic₁₇	—	2	16	no	—
33	C₂³×Dic₁₇	C₂³	1	28	yes	—
34	C₂³×Dic₁₇	C₂³	1	28	yes	—
35	C₂²×Dic₃₄	—	2	224	no	—
36	C₂²×Dic₃₄	C₂³	1	112	yes	complemented
37	C₂×C₂²⋊Dic₁₇	C₂³	1	224	yes	—
38	C₂×C₄⋊Dic₁₇	C₂³	1	112	yes	complemented
39	#10587610	C₂³	1	512	yes	direct product factor
40	C₂×#63225	C₂³	1	224	yes	complemented
41	C₂⁴×C₁₇	—	2	128	no	—
42	C₂⁴×C₁₇	—	2	56	no	—
43	C₂⁴×C₁₇	—	2	56	no	—
44	C₂⁴×C₁₇	C₂×D₄	1	8	yes	—
45	C₂⁴×C₁₇	C₂×Q₈	1	8	yes	—
46	C₂⁴×C₁₇	C₂×Q₈	1	112	yes	complemented
47	C₂⁴×C₁₇	#41	1	8	yes	—
48	C₂⁴×C₁₇	C₂⁴	1	28	yes	—
49	C₂⁴×C₁₇	C₂⁴	1	7	yes	—
50	C₂²×C₄×C₁₇	—	2	56	no	—
51	C₂²×C₄×C₁₇	—	2	448	no	—
52	C₂²×C₄×C₁₇	C₂×D₄	1	112	yes	—
53	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	—
54	C₂²×C₄×C₁₇	C₂⁴	1	64	yes	—
55	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	—
56	C₂²×Dic₁₇	—	2	112	no	—
57	C₂²×Dic₁₇	—	2	112	no	—
58	C₂²×Dic₁₇	C₂⁴	1	56	yes	—
59	C₂²×Dic₁₇	C₂⁴	1	56	yes	—
60	C₂×Dic₃₄	—	2	448	no	—
61	C₂×Dic₃₄	C₂⁴	1	64	yes	complemented
62	C₂²⋊Dic₁₇	C₂⁴	1	128	yes	—
63	C₄⋊Dic₁₇	C₂⁴	1	64	yes	complemented
64	#63225	C₂⁴	1	128	yes	complemented
65	C₂³×C₂²⋊Q₈	—	17	1	no	maximal, Sylow
66	C₂³×C₁₇	—	2	448	no	—
67	C₂³×C₁₇	—	2	56	no	—
68	C₂³×C₁₇	—	2	56	no	—
69	C₂³×C₁₇	C₂²⋊Q₈	1	64	yes	complemented
70	C₂³×C₁₇	C₂²×D₄	1	28	yes	—
71	C₂³×C₁₇	C₂²×Q₈	1	28	yes	—
72	C₂³×C₁₇	C₂²×Q₈	1	112	yes	complemented
73	C₂³×C₁₇	C₂×#41	1	28	yes	—
74	C₂³×C₁₇	C₂⁵	1	8	yes	—
75	C₂³×C₁₇	C₂⁵	1	7	yes	—
76	C₂×C₄×C₁₇	—	2	56	no	—
77	C₂×C₄×C₁₇	—	2	128	no	—
78	C₂×C₄×C₁₇	C₂²×D₄	1	112	yes	—
79	C₂×C₄×C₁₇	C₂⁵	1	8	yes	—
80	C₂×C₄×C₁₇	C₂⁵	1	8	yes	—
81	C₂×Dic₁₇	—	2	112	no	—
82	C₂×Dic₁₇	—	2	112	no	—
83	C₂×Dic₁₇	C₂⁵	1	16	yes	—
84	C₂×Dic₁₇	C₂⁵	1	16	yes	—
85	Dic₃₄	—	2	128	no	—
86	C₂⁴×Q₈	—	17	1	no	—
87	C₂⁵×C₄	D₁₇	1	1	yes	characteristic
88	C₂²×C₂²⋊Q₈	—	17	56	no	—
89	C₂³×C₄⋊C₄	—	17	2	no	—
90	C₂³×C₄⋊C₄	—	17	1	no	—
91	C₂³×C₂²⋊C₄	—	17	2	no	—
92	C₂²×C₁₇	—	2	8	no	—
93	C₂²×C₁₇	—	2	8	no	—
94	C₂²×C₁₇	—	2	224	no	—
95	C₂²×C₁₇	C₂×C₂²⋊Q₈	1	112	yes	complemented
96	C₂²×C₁₇	C₂³×D₄	1	14	yes	—
97	C₂²×C₁₇	C₂³×Q₈	1	14	yes	—
98	C₂²×C₁₇	C₂³×Q₈	1	16	yes	complemented
99	C₂²×C₁₇	C₂²×#41	1	14	yes	—
100	C₂²×C₁₇	C₂⁶	1	1	yes	characteristic
101	C₄×C₁₇	—	2	8	no	—
102	C₄×C₁₇	C₂³×D₄	1	16	yes	—
103	Dic₁₇	—	2	16	no	—
104	Dic₁₇	—	2	16	no	—
105	C₂⁶	C₂×D₁₇	1	1	yes	characteristic
106	C₂³×Q₈	—	17	28	no	—
107	C₂³×Q₈	—	34	16	no	—
108	C₂⁴×C₄	—	2	16	no	—
109	C₂⁴×C₄	—	17	2	no	—
110	C₂⁴×C₄	—	17	2	no	—
111	C₂⁴×C₄	C₂×D₁₇	1	28	yes	—
112	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
113	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
114	C₂×C₂²⋊Q₈	—	17	448	no	—
115	C₂²×C₄⋊C₄	—	17	56	no	—
116	C₂²×C₄⋊C₄	—	17	28	no	—
117	C₂²×C₂²⋊C₄	—	17	56	no	—
118	C₂×C₁₇	—	2	16	no	—
119	C₂×C₁₇	C₂²×C₂²⋊Q₈	1	28	yes	complemented
120	C₂×C₁₇	C₂⁴×D₄	1	1	yes	characteristic
121	C₂×C₁₇	C₂⁴×Q₈	1	1	yes	characteristic
122	C₂×C₁₇	C₂³×#41	1	1	yes	characteristic
123	C₂⁵	—	2	8	no	—
124	C₂⁵	—	2	8	no	—
125	C₂⁵	Dic₃₄	1	16	yes	complemented
126	C₂⁵	C₂²×D₁₇	1	14	yes	—
127	C₂⁵	C₂²×D₁₇	1	1	yes	characteristic
128	C₂²×Q₈	—	17	112	no	—
129	C₂²×Q₈	—	34	224	no	—
130	C₂³×C₄	—	2	8	no	—
131	C₂³×C₄	—	2	224	no	—
132	C₂³×C₄	—	17	28	no	—
133	C₂³×C₄	—	17	28	no	—
134	C₂³×C₄	—	34	16	no	—
135	C₂³×C₄	—	34	16	no	—
136	C₂³×C₄	C₂²⋊D₁₇	1	16	yes	—
137	C₂³×C₄	C₂²×D₁₇	1	14	yes	—
138	C₂³×C₄	C₂²×D₁₇	1	112	yes	—
139	C₂³×C₄	C₂²×D₁₇	1	14	yes	—
140	C₂²⋊Q₈	—	17	512	no	—
141	C₂×C₄⋊C₄	—	17	224	no	—
142	C₂×C₄⋊C₄	—	17	112	no	—
143	C₂×C₂²⋊C₄	—	17	224	no	—
144	C₁₇	C₂³×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
145	C₂⁴	—	2	56	no	—
146	C₂⁴	—	2	56	no	—
147	C₂⁴	—	2	128	no	—
148	C₂⁴	C₂×Dic₃₄	1	8	yes	—
149	C₂⁴	C₂×Dic₃₄	1	112	yes	complemented
150	C₂⁴	#63252	1	8	yes	—
151	C₂⁴	C₂×C₂²⋊D₁₇	1	8	yes	—
152	C₂⁴	C₂³×D₁₇	1	28	yes	—
153	C₂⁴	C₂³×D₁₇	1	7	yes	—
154	C₂×Q₈	—	17	64	no	—
155	C₂×Q₈	—	34	448	no	—
156	C₂²×C₄	—	2	56	no	—
157	C₂²×C₄	—	2	448	no	—
158	C₂²×C₄	—	17	56	no	—
159	C₂²×C₄	—	17	56	no	—
160	C₂²×C₄	—	34	112	no	—
161	C₂²×C₄	—	34	112	no	—
162	C₂²×C₄	C₂×C₂²⋊D₁₇	1	112	yes	—
163	C₂²×C₄	C₂³×D₁₇	1	28	yes	—
164	C₂²×C₄	C₂³×D₁₇	1	64	yes	—
165	C₂²×C₄	C₂³×D₁₇	1	28	yes	—
166	C₄⋊C₄	—	17	128	no	—
167	C₄⋊C₄	—	17	64	no	—
168	C₂²⋊C₄	—	17	128	no	—
169	C₂³	—	2	448	no	—
170	C₂³	—	2	56	no	—
171	C₂³	—	2	56	no	—
172	C₂³	#10587610	1	64	yes	direct product factor
173	C₂³	C₂²×Dic₃₄	1	28	yes	—
174	C₂³	C₂²×Dic₃₄	1	112	yes	complemented
175	C₂³	C₂×#63252	1	28	yes	—
176	C₂³	C₂²×C₂²⋊D₁₇	1	28	yes	—
177	C₂³	C₂⁴×D₁₇	1	8	yes	—
178	C₂³	C₂⁴×D₁₇	1	7	yes	—
179	Q₈	—	34	128	no	—
180	C₂×C₄	—	2	56	no	—
181	C₂×C₄	—	2	128	no	—
182	C₂×C₄	—	17	16	no	—
183	C₂×C₄	—	17	16	no	—
184	C₂×C₄	—	34	112	no	—
185	C₂×C₄	—	34	112	no	—
186	C₂×C₄	C₂²×C₂²⋊D₁₇	1	112	yes	—
187	C₂×C₄	C₂⁴×D₁₇	1	8	yes	—
188	C₂×C₄	C₂⁴×D₁₇	1	8	yes	—
189	C₂²	—	2	224	no	—
190	C₂²	—	2	8	no	—
191	C₂²	—	2	8	no	—
192	C₂²	C₂×#10587610	1	112	yes	direct product factor
193	C₂²	C₂³×C₂²⋊D₁₇	1	14	yes	—
194	C₂²	C₂³×Dic₃₄	1	14	yes	—
195	C₂²	C₂³×Dic₃₄	1	16	yes	complemented
196	C₂²	C₂²×#63252	1	14	yes	—
197	C₂²	C₂⁵×D₁₇	1	1	yes	characteristic
198	C₄	—	2	8	no	—
199	C₄	—	34	16	no	—
200	C₄	—	34	16	no	—
201	C₄	C₂³×C₂²⋊D₁₇	1	16	yes	—
202	C₂	—	2	16	no	—
203	C₂	C₂²×#10587610	1	28	yes	direct product factor
204	C₂	C₂⁴×C₂²⋊D₁₇	1	1	yes	characteristic
205	C₂	C₂⁴×Dic₃₄	1	1	yes	characteristic
206	C₂	C₂³×#63252	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₁₇
commutator, cosocle	C₂²×C₁₇	C₂⁶
Frattini	C₂²	C₂⁵×D₁₇
Fitting	C₂⁵×C₄×C₁₇	C₂
socle	C₂⁵×C₁₇	C₂³
prime radical	C₂⁶×C₁₇	C₂²
hypercenter	C₂⁵×C₄	D₁₇
hypocenter	C₁₇	C₂³×C₂²⋊Q₈
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂⁵ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×Dic₃₄	C₂	semidirect
2. C₂²×#10587610	C₂	direct (N × H)
3. C₂³×C₄⋊Dic₁₇	C₂	semidirect
4. C₂³×#63225	C₂	semidirect
8. C₂³×Dic₃₄	C₂²	semidirect
11. C₂×#10587610	C₂²	direct (N × H)
12. C₂²×C₄⋊Dic₁₇	C₂²	semidirect
13. C₂²×#63225	C₂²	semidirect
22. C₂⁵×C₁₇	Q₈	semidirect
36. C₂²×Dic₃₄	C₂³	semidirect
38. C₂×C₄⋊Dic₁₇	C₂³	semidirect
39. #10587610	C₂³	direct (N × H)
40. C₂×#63225	C₂³	semidirect
46. C₂⁴×C₁₇	C₂×Q₈	semidirect
61. C₂×Dic₃₄	C₂⁴	semidirect
63. C₄⋊Dic₁₇	C₂⁴	semidirect
64. #63225	C₂⁴	semidirect
69. C₂³×C₁₇	C₂²⋊Q₈	semidirect
72. C₂³×C₁₇	C₂²×Q₈	semidirect
95. C₂²×C₁₇	C₂×C₂²⋊Q₈	semidirect
98. C₂²×C₁₇	C₂³×Q₈	semidirect
119. C₂×C₁₇	C₂²×C₂²⋊Q₈	semidirect
125. C₂⁵	Dic₃₄	semidirect
144. C₁₇	C₂³×C₂²⋊Q₈	semidirect
149. C₂⁴	C₂×Dic₃₄	semidirect
172. C₂³	#10587610	direct (N × H)
174. C₂³	C₂²×Dic₃₄	semidirect
192. C₂²	C₂×#10587610	direct (N × H)
195. C₂²	C₂³×Dic₃₄	semidirect
203. C₂	C₂²×#10587610	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Dic₃₄	2
C₂²×#10587610	2
C₂³×C₄⋊Dic₁₇	2
C₂³×#63225	2
C₂³×C₂²⋊Dic₁₇	2
C₂⁵×C₄×C₁₇	2
C₂³×C₂²⋊Q₈	17

Subgroups — C23×#10587610 #17940967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#10587610 #17940967