Subgroups — C₂×#14528401 #17022018

Order 4352.

878 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763214	C₂	1	1	yes	maximal, characteristic
2	C₂×#11763212	C₂	1	2	yes	maximal
3	C₂×#11763188	C₂	1	2	yes	maximal
4	#14528401	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₈×#3622	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₇×#500	C₂	1	1	yes	maximal, characteristic
7	C₂²×#63231	—	2	4	no	—
8	C₂×#10587555	C₂²	1	2	yes	—
9	#11763214	—	2	2	no	—
10	#11763214	C₂²	1	4	yes	—
11	#11763212	—	2	4	no	—
12	#11763212	C₂²	1	8	yes	—
13	#11763188	—	2	4	no	—
14	#11763188	C₂²	1	8	yes	—
15	C₂×#10587489	—	2	2	no	—
16	C₂×#10587478	—	2	4	no	—
17	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
18	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
19	C₈×#3622	C₂²	1	4	yes	—
20	C₂×C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
21	C₁₇×#500	—	2	2	no	—
22	C₁₇×#500	C₂²	1	4	yes	—
23	C₂×C₁₇×#107	—	2	2	no	—
24	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
25	C₂²×Q₈×C₁₇	—	2	2	no	—
26	C₂²×Dic₃₄	—	2	2	no	—
27	C₂²×C₈×C₁₇	D₄	1	2	yes	—
28	C₁₇×C₄⋊Q₈	—	2	2	no	—
29	C₁₇×C₄⋊Q₈	—	2	2	no	—
30	C₁₇×C₄⋊Q₈	—	2	2	no	—
31	C₁₇×C₄⋊Q₈	C₂³	1	4	yes	—
32	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
34	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
35	C₂×#63231	—	2	16	no	—
36	C₂×#63231	—	4	16	no	—
37	C₂×C₄⋊Dic₁₇	—	2	2	no	—
38	C₂×C₄⋊Dic₁₇	—	2	2	no	—
39	C₂²×#3622	D₄	1	2	yes	—
40	C₂²×#3622	D₄	1	2	yes	—
41	#10587555	—	2	2	no	—
42	#10587555	—	2	2	no	—
43	#10587555	—	2	2	no	—
44	#10587555	C₂³	1	4	yes	—
45	C₁₇×#107	—	2	8	no	—
46	C₄×C₈×C₁₇	D₄	1	2	yes	—
47	C₄×C₈×C₁₇	C₂³	1	2	yes	—
48	#10587489	—	2	8	no	—
49	#10587478	—	2	16	no	—
50	C₄×#3622	D₄	1	2	yes	—
51	C₄×#3622	D₄	1	2	yes	—
52	C₄×#3622	C₂³	1	2	yes	—
53	C₄×#3622	C₂³	1	2	yes	—
54	C₂×Q₈×C₁₇	—	2	4	no	—
55	C₂×Q₈×C₁₇	—	4	2	no	—
56	C₂×Q₈×C₁₇	—	4	8	no	—
57	C₂²×C₄×C₁₇	—	4	2	no	—
58	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₁₇	—	4	2	no	—
62	C₂×Dic₃₄	—	2	4	no	—
63	C₂×Dic₃₄	—	4	8	no	—
64	C₂×Dic₃₄	—	4	2	no	—
65	C₂×C₈×C₁₇	Q₁₆	1	8	yes	complemented
66	C₂×C₈×C₁₇	C₂×D₄	1	4	yes	—
67	C₄⋊C₄×C₁₇	—	2	4	no	—
68	C₄⋊C₄×C₁₇	—	2	4	no	—
69	C₄⋊C₄×C₁₇	—	4	2	no	—
70	C₄⋊C₄×C₁₇	—	4	2	no	—
71	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
75	#63231	—	4	32	no	—
76	C₄⋊Dic₁₇	—	2	4	no	—
77	C₄⋊Dic₁₇	—	2	4	no	—
78	C₄⋊Dic₁₇	—	4	2	no	—
79	C₄⋊Dic₁₇	—	4	2	no	—
80	C₂×#3622	D₈	1	8	yes	—
81	C₂×#3622	Q₁₆	1	8	yes	complemented
82	C₂×#3622	C₂×D₄	1	4	yes	—
83	C₂×#3622	C₂×D₄	1	4	yes	—
84	C₂×#1713	—	17	1	no	maximal, Sylow
85	C₂³×C₁₇	#127	1	1	yes	characteristic
86	Q₈×C₁₇	—	4	8	no	—
87	Q₈×C₁₇	—	8	4	no	—
88	C₂×C₄×C₁₇	—	4	4	no	—
89	C₂×C₄×C₁₇	—	8	4	no	—
90	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
91	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
92	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
93	C₂×C₄×C₁₇	C₂×D₈	1	4	yes	—
94	C₂×C₄×C₁₇	C₂×Q₁₆	1	4	yes	—
95	C₂×C₄×C₁₇	C₂×Q₁₆	1	4	yes	—
96	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
99	C₂×Dic₁₇	—	4	4	no	—
100	C₂×Dic₁₇	—	8	4	no	—
101	Dic₃₄	—	4	8	no	—
102	Dic₃₄	—	8	4	no	—
103	C₈×C₁₇	C₂×Q₁₆	1	8	yes	complemented
104	#3622	C₂×D₈	1	8	yes	—
105	#3622	C₂×Q₁₆	1	8	yes	complemented
106	C₂×#500	—	17	1	no	—
107	C₂×#500	—	17	2	no	—
108	C₂×#500	D₁₇	1	1	yes	characteristic
109	C₂×#494	—	17	2	no	—
110	#1713	—	17	8	no	—
111	C₂×C₈²	—	17	1	no	—
112	C₂²×C₁₇	#493	1	2	yes	—
113	C₂²×C₁₇	#494	1	2	yes	—
114	C₂²×C₁₇	#494	1	2	yes	—
115	C₂²×C₁₇	C₂×#127	1	1	yes	characteristic
116	C₄×C₁₇	—	8	4	no	—
117	C₄×C₁₇	#493	1	2	yes	—
118	C₄×C₁₇	#494	1	2	yes	—
119	C₄×C₁₇	#494	1	2	yes	—
120	C₄×C₁₇	C₂²×D₈	1	2	yes	—
121	C₄×C₁₇	C₂²×Q₁₆	1	2	yes	—
122	C₄×C₁₇	C₂²×Q₁₆	1	2	yes	—
123	Dic₁₇	—	8	4	no	—
124	C₂²×Q₁₆	—	34	4	no	—
125	C₂×C₄⋊Q₈	—	17	2	no	—
126	C₂×C₄⋊Q₈	C₂×D₁₇	1	2	yes	—
127	#500	—	2	2	no	—
128	#500	—	17	8	no	—
129	#500	—	17	4	no	—
130	#500	—	34	4	no	—
131	#500	—	34	2	no	—
132	#500	C₂×D₁₇	1	4	yes	—
133	#494	—	17	8	no	—
134	#494	—	34	4	no	—
135	C₂×#107	—	2	2	no	—
136	C₂×#107	—	34	4	no	—
137	C₂×#107	—	34	2	no	—
138	C₂×C₄×C₈	—	17	1	no	—
139	C₂×C₄×C₈	—	17	1	no	—
140	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
141	C₈²	—	17	4	no	—
142	C₂×C₁₇	#1713	1	4	yes	complemented
143	C₂×C₁₇	C₂×#493	1	1	yes	characteristic
144	C₂×C₁₇	C₂×#494	1	1	yes	characteristic
145	C₂×C₁₇	C₂×#494	1	1	yes	characteristic
146	C₂²×Q₈	—	2	2	no	—
147	C₂²×Q₈	—	34	2	no	—
148	C₂×Q₁₆	—	34	16	no	—
149	C₂×Q₁₆	—	68	16	no	—
150	C₂²×C₈	—	17	2	no	—
151	C₂²×C₈	—	17	2	no	—
152	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
153	C₄⋊Q₈	—	2	2	no	—
154	C₄⋊Q₈	—	2	2	no	—
155	C₄⋊Q₈	—	2	2	no	—
156	C₄⋊Q₈	—	17	4	no	—
157	C₄⋊Q₈	—	34	2	no	—
158	C₄⋊Q₈	—	34	2	no	—
159	C₄⋊Q₈	—	34	2	no	—
160	C₄⋊Q₈	C₂²×D₁₇	1	4	yes	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	34	2	no	—
164	C₂×C₄⋊C₄	—	34	2	no	—
165	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
166	#107	—	2	8	no	—
167	#107	—	34	16	no	—
168	#107	—	34	8	no	—
169	C₄×C₈	—	17	2	no	—
170	C₄×C₈	—	17	2	no	—
171	C₄×C₈	—	17	2	no	—
172	C₄×C₈	—	17	2	no	—
173	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
174	C₄×C₈	C₂²×D₁₇	1	2	yes	—
175	C₁₇	C₂×#1713	1	1	yes	characteristic, complemented, Sylow
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	—	4	8	no	—
179	C₂×Q₈	—	34	4	no	—
180	C₂×Q₈	—	68	2	no	—
181	C₂×Q₈	—	68	8	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	68	2	no	—
184	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
185	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
187	Q₁₆	—	68	32	no	—
188	C₂×C₈	—	17	4	no	—
189	C₂×C₈	—	17	4	no	—
190	C₂×C₈	—	17	8	no	—
191	C₂×C₈	—	17	8	no	—
192	C₂×C₈	#63231	1	8	yes	complemented
193	C₂×C₈	C₂×C₂²⋊D₁₇	1	4	yes	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	34	4	no	—
199	C₄⋊C₄	—	34	4	no	—
200	C₄⋊C₄	—	68	2	no	—
201	C₄⋊C₄	—	68	2	no	—
202	C₄²	D₄×D₁₇	1	1	yes	characteristic
203	C₄²	D₄×D₁₇	1	1	yes	characteristic
204	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
205	C₄²	C₂³×D₁₇	1	1	yes	characteristic
206	C₂³	#10587626	1	1	yes	characteristic
207	Q₈	—	4	8	no	—
208	Q₈	—	8	4	no	—
209	Q₈	—	68	8	no	—
210	Q₈	—	136	4	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	8	4	no	—
213	C₂×C₄	—	68	4	no	—
214	C₂×C₄	—	136	4	no	—
215	C₂×C₄	D₈×D₁₇	1	4	yes	—
216	C₂×C₄	Q₁₆×D₁₇	1	4	yes	—
217	C₂×C₄	#10587626	1	1	yes	characteristic
218	C₂×C₄	#10587626	1	1	yes	characteristic
219	C₂×C₄	#10587626	1	1	yes	characteristic
220	C₂×C₄	C₂×#63231	1	4	yes	—
221	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
222	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
223	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
224	C₈	—	17	8	no	—
225	C₈	—	17	8	no	—
226	C₈	C₂×#63231	1	8	yes	complemented
227	C₂²	#11763184	1	2	yes	—
228	C₂²	#11763187	1	2	yes	—
229	C₂²	#11763188	1	2	yes	—
230	C₂²	C₂×#10587626	1	1	yes	characteristic
231	C₄	—	8	4	no	—
232	C₄	—	136	4	no	—
233	C₄	#11763184	1	2	yes	—
234	C₄	#11763187	1	2	yes	—
235	C₄	#11763188	1	2	yes	—
236	C₄	C₂×D₈×D₁₇	1	2	yes	—
237	C₄	C₂×Q₁₆×D₁₇	1	2	yes	—
238	C₄	C₂²×#63231	1	2	yes	—
239	C₂	#14528401	1	4	yes	direct product factor
240	C₂	C₂×#11763184	1	1	yes	characteristic
241	C₂	C₂×#11763187	1	1	yes	characteristic
242	C₂	C₂×#11763188	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10587626
commutator, cosocle	C₄²×C₁₇	C₂⁴
Frattini	C₄²	C₂³×D₁₇
Fitting	C₂×C₁₇×#500	C₂
socle, prime radical	C₂³×C₁₇	#127
hypercenter	C₂×#500	D₁₇
hypocenter	C₁₇	C₂×#1713
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
4. #14528401	C₂	direct (N × H)
65. C₂×C₈×C₁₇	Q₁₆	semidirect
81. C₂×#3622	Q₁₆	semidirect
103. C₈×C₁₇	C₂×Q₁₆	semidirect
105. #3622	C₂×Q₁₆	semidirect
142. C₂×C₁₇	#1713	semidirect
175. C₁₇	C₂×#1713	semidirect
192. C₂×C₈	#63231	semidirect
226. C₈	C₂×#63231	semidirect
239. C₂	#14528401	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×#11763214	2
C₂×#11763212	2
C₂×#11763188	2
#14528401	2
C₂×C₈×#3622	2
C₂×C₁₇×#500	2
C₂×#1713	17

Subgroups — C2×#14528401 #17022018

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14528401 #17022018