Subgroups — C₂³×#10587582 #17940966

Order 4352.

14078 conjugacy classes of subgroups.

The subgroup lattice has 256 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	C₂⁴×C₄×D₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#10587582	C₂	1	56	yes	maximal, direct product factor
4	C₂³×#63225	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×#63225	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×#63227	C₂	1	1	yes	maximal, characteristic
7	C₂³×#63227	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄⋊C₄×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×D₁₇	C₂²	1	1	yes	characteristic
10	C₂³×Dic₃₄	—	2	16	no	—
11	C₂³×Dic₃₄	C₂²	1	28	yes	complemented
12	C₂⁴×Dic₁₇	C₂²	1	1	yes	characteristic
13	C₂⁴×Dic₁₇	C₂²	1	1	yes	characteristic
14	C₂⁴×Dic₁₇	C₂²	1	1	yes	characteristic
15	C₂³×C₄×D₁₇	—	2	16	no	—
16	C₂³×C₄×D₁₇	C₂²	1	28	yes	—
17	C₂×#10587582	C₂²	1	448	yes	direct product factor
18	C₂²×#63225	C₂²	1	28	yes	complemented
19	C₂²×#63225	C₂²	1	28	yes	complemented
20	C₂²×#63227	C₂²	1	28	yes	—
21	C₂²×#63227	C₂²	1	28	yes	—
22	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
23	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
24	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₁₇	C₂²	1	28	yes	complemented
26	C₂⁵×C₁₇	C₂³	1	1	yes	characteristic
27	C₂⁴×D₁₇	—	2	8	no	—
28	C₂⁴×D₁₇	—	2	8	no	—
29	C₂⁴×D₁₇	Q₈	1	16	yes	complemented
30	C₂⁴×D₁₇	C₂³	1	14	yes	—
31	C₂³×C₄×C₁₇	—	2	8	no	—
32	C₂³×C₄×C₁₇	—	2	8	no	—
33	C₂³×C₄×C₁₇	—	2	8	no	—
34	C₂³×C₄×C₁₇	C₂³	1	14	yes	—
35	C₂³×C₄×C₁₇	C₂³	1	14	yes	—
36	C₂³×C₄×C₁₇	C₂³	1	14	yes	—
37	C₂³×Dic₁₇	—	2	8	no	—
38	C₂³×Dic₁₇	—	2	8	no	—
39	C₂³×Dic₁₇	D₄	1	16	yes	—
40	C₂³×Dic₁₇	C₂³	1	14	yes	—
41	C₂³×Dic₁₇	C₂³	1	14	yes	—
42	C₂³×Dic₁₇	C₂³	1	14	yes	—
43	C₂²×C₄×D₁₇	—	2	224	no	—
44	C₂²×C₄×D₁₇	C₂³	1	112	yes	—
45	C₂²×Dic₃₄	—	2	224	no	—
46	C₂²×Dic₃₄	C₂³	1	112	yes	complemented
47	C₂×C₄⋊C₄×C₁₇	C₂³	1	112	yes	complemented
48	C₂×#63227	C₂³	1	112	yes	—
49	C₂×#63227	C₂³	1	112	yes	—
50	C₂×#63225	C₂³	1	112	yes	complemented
51	C₂×#63225	C₂³	1	112	yes	complemented
52	#10587582	C₂³	1	512	yes	direct product factor
53	C₂⁴×C₁₇	C₂×D₄	1	8	yes	—
54	C₂⁴×C₁₇	C₂×Q₈	1	8	yes	—
55	C₂⁴×C₁₇	#41	1	8	yes	—
56	C₂⁴×C₁₇	C₂⁴	1	7	yes	—
57	C₂³×D₁₇	—	2	128	no	—
58	C₂³×D₁₇	—	2	56	no	—
59	C₂³×D₁₇	—	2	56	no	—
60	C₂³×D₁₇	C₂×Q₈	1	112	yes	complemented
61	C₂³×D₁₇	C₂⁴	1	28	yes	—
62	C₂²×C₄×C₁₇	—	2	56	no	—
63	C₂²×C₄×C₁₇	—	2	56	no	—
64	C₂²×C₄×C₁₇	—	2	56	no	—
65	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	—
66	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	—
67	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	—
68	C₂²×Dic₁₇	—	2	56	no	—
69	C₂²×Dic₁₇	—	2	56	no	—
70	C₂²×Dic₁₇	C₂×D₄	1	112	yes	—
71	C₂²×Dic₁₇	C₂⁴	1	28	yes	—
72	C₂²×Dic₁₇	C₂⁴	1	28	yes	—
73	C₂²×Dic₁₇	C₂⁴	1	28	yes	—
74	C₂×C₄×D₁₇	—	2	448	no	—
75	C₂×C₄×D₁₇	C₂⁴	1	64	yes	—
76	C₂×Dic₃₄	—	2	448	no	—
77	C₂×Dic₃₄	C₂⁴	1	64	yes	complemented
78	C₄⋊C₄×C₁₇	C₂⁴	1	64	yes	complemented
79	#63227	C₂⁴	1	64	yes	—
80	#63227	C₂⁴	1	64	yes	—
81	#63225	C₂⁴	1	64	yes	complemented
82	#63225	C₂⁴	1	64	yes	complemented
83	C₂³×C₂²⋊Q₈	—	17	1	no	maximal, Sylow
84	C₂³×C₁₇	C₂²⋊Q₈	1	64	yes	complemented
85	C₂³×C₁₇	C₂²×D₄	1	28	yes	—
86	C₂³×C₁₇	C₂²×Q₈	1	28	yes	—
87	C₂³×C₁₇	C₂×#41	1	28	yes	—
88	C₂³×C₁₇	C₂⁵	1	7	yes	—
89	C₂²×D₁₇	—	2	448	no	—
90	C₂²×D₁₇	—	2	56	no	—
91	C₂²×D₁₇	—	2	56	no	—
92	C₂²×D₁₇	C₂²×Q₈	1	112	yes	complemented
93	C₂²×D₁₇	C₂⁵	1	8	yes	—
94	C₂×C₄×C₁₇	—	2	56	no	—
95	C₂×C₄×C₁₇	—	2	56	no	—
96	C₂×C₄×C₁₇	—	2	56	no	—
97	C₂×C₄×C₁₇	C₂⁵	1	8	yes	—
98	C₂×C₄×C₁₇	C₂⁵	1	8	yes	—
99	C₂×C₄×C₁₇	C₂⁵	1	8	yes	—
100	C₂×Dic₁₇	—	2	56	no	—
101	C₂×Dic₁₇	—	2	56	no	—
102	C₂×Dic₁₇	C₂²×D₄	1	112	yes	—
103	C₂×Dic₁₇	C₂⁵	1	8	yes	—
104	C₂×Dic₁₇	C₂⁵	1	8	yes	—
105	C₂×Dic₁₇	C₂⁵	1	8	yes	—
106	C₄×D₁₇	—	2	128	no	—
107	Dic₃₄	—	2	128	no	—
108	C₂⁴×Q₈	—	17	1	no	—
109	C₂⁵×C₄	—	17	1	no	—
110	C₂²×C₂²⋊Q₈	—	17	56	no	—
111	C₂³×C₄⋊C₄	—	17	1	no	—
112	C₂³×C₄⋊C₄	—	17	1	no	—
113	C₂³×C₄⋊C₄	D₁₇	1	1	yes	characteristic, complemented
114	C₂³×C₂²⋊C₄	—	17	1	no	—
115	C₂³×C₂²⋊C₄	—	17	1	no	—
116	C₂²×C₁₇	C₂×C₂²⋊Q₈	1	112	yes	complemented
117	C₂²×C₁₇	C₂³×D₄	1	14	yes	—
118	C₂²×C₁₇	C₂³×Q₈	1	14	yes	—
119	C₂²×C₁₇	C₂²×#41	1	14	yes	—
120	C₂²×C₁₇	C₂⁶	1	1	yes	characteristic
121	C₂×D₁₇	—	2	8	no	—
122	C₂×D₁₇	—	2	8	no	—
123	C₂×D₁₇	—	2	224	no	—
124	C₂×D₁₇	C₂³×Q₈	1	16	yes	complemented
125	C₄×C₁₇	—	2	8	no	—
126	C₄×C₁₇	—	2	8	no	—
127	C₄×C₁₇	—	2	8	no	—
128	Dic₁₇	—	2	8	no	—
129	Dic₁₇	—	2	8	no	—
130	Dic₁₇	C₂³×D₄	1	16	yes	—
131	C₂⁶	—	17	1	no	—
132	C₂³×Q₈	—	17	28	no	—
133	C₂³×Q₈	—	34	16	no	—
134	C₂⁴×C₄	—	17	1	no	—
135	C₂⁴×C₄	—	17	1	no	—
136	C₂⁴×C₄	—	17	1	no	—
137	C₂⁴×C₄	—	17	28	no	—
138	C₂⁴×C₄	—	34	16	no	—
139	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
140	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
141	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
142	C₂×C₂²⋊Q₈	—	17	448	no	—
143	C₂²×C₄⋊C₄	—	17	28	no	—
144	C₂²×C₄⋊C₄	—	17	28	no	—
145	C₂²×C₄⋊C₄	C₂×D₁₇	1	28	yes	complemented
146	C₂²×C₂²⋊C₄	—	17	28	no	—
147	C₂²×C₂²⋊C₄	—	17	28	no	—
148	C₂×C₁₇	C₂²×C₂²⋊Q₈	1	28	yes	complemented
149	C₂×C₁₇	C₂⁴×D₄	1	1	yes	characteristic
150	C₂×C₁₇	C₂⁴×Q₈	1	1	yes	characteristic
151	C₂×C₁₇	C₂³×#41	1	1	yes	characteristic
152	D₁₇	—	2	16	no	—
153	C₂⁵	—	17	16	no	—
154	C₂⁵	—	17	14	no	—
155	C₂⁵	—	34	8	no	—
156	C₂⁵	—	34	8	no	—
157	C₂⁵	C₂²×D₁₇	1	1	yes	characteristic
158	C₂²×Q₈	—	17	112	no	—
159	C₂²×Q₈	—	34	224	no	—
160	C₂³×C₄	—	2	8	no	—
161	C₂³×C₄	—	2	8	no	—
162	C₂³×C₄	—	2	8	no	—
163	C₂³×C₄	—	17	16	no	—
164	C₂³×C₄	—	17	14	no	—
165	C₂³×C₄	—	17	14	no	—
166	C₂³×C₄	—	17	14	no	—
167	C₂³×C₄	—	17	112	no	—
168	C₂³×C₄	—	34	224	no	—
169	C₂³×C₄	—	34	8	no	—
170	C₂³×C₄	—	34	8	no	—
171	C₂³×C₄	C₂²×D₁₇	1	14	yes	—
172	C₂³×C₄	C₂²×D₁₇	1	14	yes	—
173	C₂³×C₄	C₂²×D₁₇	1	14	yes	—
174	C₂²⋊Q₈	—	17	512	no	—
175	C₂×C₄⋊C₄	—	17	112	no	—
176	C₂×C₄⋊C₄	—	17	112	no	—
177	C₂×C₄⋊C₄	C₂²×D₁₇	1	112	yes	complemented
178	C₂×C₂²⋊C₄	—	17	112	no	—
179	C₂×C₂²⋊C₄	—	17	112	no	—
180	C₁₇	C₂³×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
181	C₂⁴	—	17	112	no	—
182	C₂⁴	—	17	28	no	—
183	C₂⁴	—	34	56	no	—
184	C₂⁴	—	34	56	no	—
185	C₂⁴	—	34	128	no	—
186	C₂⁴	#63252	1	8	yes	—
187	C₂⁴	D₄×D₁₇	1	8	yes	—
188	C₂⁴	Q₈×D₁₇	1	8	yes	—
189	C₂⁴	C₂³×D₁₇	1	7	yes	—
190	C₂×Q₈	—	17	64	no	—
191	C₂×Q₈	—	34	448	no	—
192	C₂²×C₄	—	2	56	no	—
193	C₂²×C₄	—	2	56	no	—
194	C₂²×C₄	—	2	56	no	—
195	C₂²×C₄	—	17	112	no	—
196	C₂²×C₄	—	17	28	no	—
197	C₂²×C₄	—	17	28	no	—
198	C₂²×C₄	—	17	28	no	—
199	C₂²×C₄	—	17	64	no	—
200	C₂²×C₄	—	34	448	no	—
201	C₂²×C₄	—	34	56	no	—
202	C₂²×C₄	—	34	56	no	—
203	C₂²×C₄	C₂³×D₁₇	1	28	yes	—
204	C₂²×C₄	C₂³×D₁₇	1	28	yes	—
205	C₂²×C₄	C₂³×D₁₇	1	28	yes	—
206	C₄⋊C₄	—	17	64	no	—
207	C₄⋊C₄	—	17	64	no	—
208	C₄⋊C₄	C₂³×D₁₇	1	64	yes	complemented
209	C₂²⋊C₄	—	17	64	no	—
210	C₂²⋊C₄	—	17	64	no	—
211	C₂³	—	17	112	no	—
212	C₂³	—	17	8	no	—
213	C₂³	—	34	448	no	—
214	C₂³	—	34	56	no	—
215	C₂³	—	34	56	no	—
216	C₂³	#10587582	1	64	yes	direct product factor
217	C₂³	C₂×#63252	1	28	yes	—
218	C₂³	C₂×D₄×D₁₇	1	28	yes	—
219	C₂³	C₂×Q₈×D₁₇	1	28	yes	—
220	C₂³	C₂⁴×D₁₇	1	7	yes	—
221	Q₈	—	34	128	no	—
222	C₂×C₄	—	2	56	no	—
223	C₂×C₄	—	2	56	no	—
224	C₂×C₄	—	2	56	no	—
225	C₂×C₄	—	17	112	no	—
226	C₂×C₄	—	17	8	no	—
227	C₂×C₄	—	17	8	no	—
228	C₂×C₄	—	17	8	no	—
229	C₂×C₄	—	34	128	no	—
230	C₂×C₄	—	34	56	no	—
231	C₂×C₄	—	34	56	no	—
232	C₂×C₄	C₂⁴×D₁₇	1	8	yes	—
233	C₂×C₄	C₂⁴×D₁₇	1	8	yes	—
234	C₂×C₄	C₂⁴×D₁₇	1	8	yes	—
235	C₂²	—	17	16	no	—
236	C₂²	—	34	224	no	—
237	C₂²	—	34	8	no	—
238	C₂²	—	34	8	no	—
239	C₂²	C₂×#10587582	1	112	yes	direct product factor
240	C₂²	C₂²×D₄×D₁₇	1	14	yes	—
241	C₂²	C₂²×Q₈×D₁₇	1	14	yes	—
242	C₂²	C₂²×#63252	1	14	yes	—
243	C₂²	C₂⁵×D₁₇	1	1	yes	characteristic
244	C₄	—	2	8	no	—
245	C₄	—	2	8	no	—
246	C₄	—	2	8	no	—
247	C₄	—	17	16	no	—
248	C₄	—	34	8	no	—
249	C₄	—	34	8	no	—
250	C₂	—	34	16	no	—
251	C₂	C₂²×#10587582	1	28	yes	direct product factor
252	C₂	C₂³×D₄×D₁₇	1	1	yes	characteristic
253	C₂	C₂³×Q₈×D₁₇	1	1	yes	characteristic
254	C₂	C₂³×#63252	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂⁴×Dic₃₄	C₂	semidirect
3. C₂²×#10587582	C₂	direct (N × H)
4. C₂³×#63225	C₂	semidirect
5. C₂³×#63225	C₂	semidirect
8. C₂³×C₄⋊C₄×C₁₇	C₂	semidirect
11. C₂³×Dic₃₄	C₂²	semidirect
17. C₂×#10587582	C₂²	direct (N × H)
18. C₂²×#63225	C₂²	semidirect
19. C₂²×#63225	C₂²	semidirect
25. C₂²×C₄⋊C₄×C₁₇	C₂²	semidirect
29. C₂⁴×D₁₇	Q₈	semidirect
46. C₂²×Dic₃₄	C₂³	semidirect
47. C₂×C₄⋊C₄×C₁₇	C₂³	semidirect
50. C₂×#63225	C₂³	semidirect
51. C₂×#63225	C₂³	semidirect
52. #10587582	C₂³	direct (N × H)
60. C₂³×D₁₇	C₂×Q₈	semidirect
77. C₂×Dic₃₄	C₂⁴	semidirect
78. C₄⋊C₄×C₁₇	C₂⁴	semidirect
81. #63225	C₂⁴	semidirect
82. #63225	C₂⁴	semidirect
84. C₂³×C₁₇	C₂²⋊Q₈	semidirect
92. C₂²×D₁₇	C₂²×Q₈	semidirect
113. C₂³×C₄⋊C₄	D₁₇	semidirect
116. C₂²×C₁₇	C₂×C₂²⋊Q₈	semidirect
124. C₂×D₁₇	C₂³×Q₈	semidirect
145. C₂²×C₄⋊C₄	C₂×D₁₇	semidirect
148. C₂×C₁₇	C₂²×C₂²⋊Q₈	semidirect
177. C₂×C₄⋊C₄	C₂²×D₁₇	semidirect
180. C₁₇	C₂³×C₂²⋊Q₈	semidirect
208. C₄⋊C₄	C₂³×D₁₇	semidirect
216. C₂³	#10587582	direct (N × H)
239. C₂²	C₂×#10587582	direct (N × H)
251. C₂	C₂²×#10587582	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
C₂⁴×C₄×D₁₇	2
C₂²×#10587582	2
C₂³×#63225	2
C₂³×#63227	2
C₂³×C₄⋊C₄×C₁₇	2
C₂³×C₂²⋊Q₈	17

Subgroups — C23×#10587582 #17940966

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#10587582 #17940966