Subgroups — C₂×#14530582 #17187388

Order 4352.

726 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₁₇	C₂	1	1	yes	maximal, characteristic
2	#14530582	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊#63214	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11762588	C₂	1	2	yes	maximal
5	C₂×C₁₆×Dic₁₇	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₁₇	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₄²×D₁₇	C₄	1	1	yes	characteristic
10	C₂²×#63214	C₂²	1	2	yes	—
11	C₂×C₄×#3622	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×D₁₇	C₄	1	4	yes	—
15	C₄×C₈×D₁₇	C₂²	1	4	yes	—
16	C₄⋊#63214	C₂²	1	4	yes	complemented
17	#11762588	C₂²	1	8	yes	—
18	C₁₆×Dic₁₇	C₂²	1	8	yes	complemented
19	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
21	C₁₇×C₄⋊C₁₆	C₂²	1	4	yes	complemented
22	C₂²×C₄×D₁₇	C₈	1	1	yes	characteristic
23	C₂²×C₄×D₁₇	C₈	1	1	yes	characteristic
24	C₂²×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₁₇	C₈	1	1	yes	characteristic
29	C₂×C₄×Dic₁₇	C₈	1	1	yes	characteristic
30	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
32	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₁₇	—	2	4	no	—
34	C₂×C₈×D₁₇	—	2	2	no	—
35	C₂×C₈×D₁₇	—	2	2	no	—
36	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₁₇	C₂×C₄	1	4	yes	—
38	C₂×C₈×D₁₇	C₂³	1	2	yes	—
39	C₄²×D₁₇	C₈	1	4	yes	—
40	C₄²×D₁₇	C₂×C₄	1	4	yes	—
41	C₂×C₁₆×C₁₇	—	2	4	no	—
42	C₂×C₁₆×C₁₇	C₂³	1	4	yes	—
43	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₁₇	C₂³	1	2	yes	—
45	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
46	C₈×Dic₁₇	C₂×C₄	1	4	yes	—
47	C₈×Dic₁₇	C₂³	1	2	yes	—
48	C₂×#63214	—	2	4	no	—
49	C₂×#63214	C₂³	1	4	yes	—
50	C₄×#3622	C₂×C₄	1	4	yes	—
51	C₂³×D₁₇	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
56	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
57	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
58	C₂×C₄×D₁₇	—	2	2	no	—
59	C₂×C₄×D₁₇	—	2	2	no	—
60	C₂×C₄×D₁₇	—	2	4	no	—
61	C₂×C₄×D₁₇	C₁₆	1	8	yes	complemented
62	C₂×C₄×D₁₇	C₂×C₈	1	4	yes	—
63	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
64	C₂×C₄×D₁₇	C₂×C₈	1	4	yes	—
65	C₂×C₄×D₁₇	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₁₇	—	2	2	no	—
67	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₇	#41	1	2	yes	—
70	C₂×C₈×C₁₇	#41	1	2	yes	—
71	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
72	C₄²×C₁₇	C₂×C₈	1	2	yes	—
73	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
74	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
75	C₄×Dic₁₇	C₂×C₈	1	4	yes	—
76	C₄×Dic₁₇	C₂×C₈	1	4	yes	—
77	C₄×Dic₁₇	C₂²×C₄	1	2	yes	—
78	C₂×#3622	—	2	2	no	—
79	C₂×#3622	M₁₆	1	4	yes	—
80	C₂×#3622	C₂²×C₄	1	2	yes	—
81	C₂×#3622	C₂²×C₄	1	2	yes	—
82	C₈×D₁₇	—	2	4	no	—
83	C₈×D₁₇	—	2	2	no	—
84	C₈×D₁₇	—	2	2	no	—
85	C₈×D₁₇	—	2	8	no	—
86	C₁₆×C₁₇	—	2	4	no	—
87	#63214	—	2	4	no	—
88	C₂×#2162	—	17	1	no	maximal, Sylow
89	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
90	C₂²×D₁₇	—	2	2	no	—
91	C₂²×D₁₇	—	2	2	no	—
92	C₂²×D₁₇	C₂×C₁₆	1	4	yes	—
93	C₂²×D₁₇	C₂²×C₈	1	2	yes	—
94	C₂×C₄×C₁₇	—	2	2	no	—
95	C₂×C₄×C₁₇	C₂×C₁₆	1	4	yes	—
96	C₂×C₄×C₁₇	#117	1	2	yes	—
97	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
98	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
100	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
101	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
102	C₂×Dic₁₇	—	2	2	no	—
103	C₂×Dic₁₇	C₂×C₁₆	1	4	yes	—
104	C₂×Dic₁₇	M₃₂	1	4	yes	—
105	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
106	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
107	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
108	C₄×D₁₇	—	2	2	no	—
109	C₄×D₁₇	—	2	2	no	—
110	C₄×D₁₇	—	2	4	no	—
111	C₄×D₁₇	—	2	8	no	—
112	C₄×D₁₇	C₂×C₁₆	1	16	yes	complemented
113	C₈×C₁₇	—	2	2	no	—
114	C₈×C₁₇	#117	1	2	yes	—
115	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
116	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
117	#3622	—	2	2	no	—
118	#3622	C₂×M₁₆	1	4	yes	—
119	C₂²×C₄×C₈	—	17	1	no	—
120	#2162	—	17	8	no	—
121	C₂×C₄⋊C₁₆	—	17	1	no	—
122	C₂×C₄⋊C₁₆	D₁₇	1	1	yes	characteristic, complemented
123	C₂×C₂²⋊C₁₆	—	17	2	no	—
124	C₂×C₄×C₁₆	—	17	2	no	—
125	C₂²×C₁₇	#431	1	2	yes	—
126	C₂²×C₁₇	C₂²×C₁₆	1	2	yes	—
127	C₂²×C₁₇	C₂×M₃₂	1	2	yes	—
128	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
129	C₂×D₁₇	—	2	2	no	—
130	C₂×D₁₇	—	2	2	no	—
131	C₂×D₁₇	—	2	8	no	—
132	C₂×D₁₇	C₂²×C₁₆	1	4	yes	—
133	C₄×C₁₇	—	2	2	no	—
134	C₄×C₁₇	#431	1	2	yes	—
135	C₄×C₁₇	C₂²×C₁₆	1	4	yes	—
136	C₄×C₁₇	C₂×#117	1	1	yes	characteristic
137	C₄×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
138	Dic₁₇	—	2	2	no	—
139	Dic₁₇	C₂²×C₁₆	1	4	yes	—
140	Dic₁₇	C₂×M₃₂	1	4	yes	—
141	C₂³×C₈	—	17	1	no	—
142	C₂³×C₈	—	17	1	no	—
143	C₂²×C₄²	—	17	1	no	—
144	C₂²×C₁₆	—	17	2	no	—
145	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
146	C₂×C₄×C₈	—	17	1	no	—
147	C₂×C₄×C₈	—	17	1	no	—
148	C₂×C₄×C₈	—	17	1	no	—
149	C₂×C₄×C₈	—	17	4	no	—
150	C₂×C₄×C₈	—	17	4	no	—
151	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
152	C₄⋊C₁₆	—	17	4	no	—
153	C₄⋊C₁₆	C₂×D₁₇	1	4	yes	complemented
154	C₂²⋊C₁₆	—	17	8	no	—
155	C₄×C₁₆	—	17	8	no	—
156	C₂×C₁₇	#2162	1	4	yes	complemented
157	C₂×C₁₇	C₂×#431	1	1	yes	characteristic
158	C₂×C₁₇	C₂³×C₁₆	1	1	yes	characteristic
159	C₂×C₁₇	C₂²×M₃₂	1	1	yes	characteristic
160	D₁₇	—	2	4	no	—
161	C₂³×C₄	—	17	1	no	—
162	C₂³×C₄	—	17	1	no	—
163	C₂³×C₄	—	17	1	no	—
164	C₂²×C₈	—	17	1	no	—
165	C₂²×C₈	—	17	1	no	—
166	C₂²×C₈	—	17	4	no	—
167	C₂²×C₈	—	17	2	no	—
168	C₂²×C₈	—	17	2	no	—
169	C₂²×C₈	—	34	4	no	—
170	C₂²×C₈	—	34	2	no	—
171	C₂²×C₈	—	34	2	no	—
172	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
173	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
174	C₂×C₄²	—	17	1	no	—
175	C₂×C₄²	—	17	1	no	—
176	C₂×C₄²	—	17	1	no	—
177	C₂×C₄²	—	17	4	no	—
178	C₂×C₄²	—	17	4	no	—
179	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
180	C₂×C₁₆	—	2	4	no	—
181	C₂×C₁₆	—	17	4	no	—
182	C₂×C₁₆	—	34	4	no	—
183	C₂×C₁₆	C₂²×D₁₇	1	4	yes	—
184	C₄×C₈	—	17	2	no	—
185	C₄×C₈	—	17	4	no	—
186	C₄×C₈	—	17	2	no	—
187	C₄×C₈	—	17	4	no	—
188	C₄×C₈	C₄×D₁₇	1	2	yes	—
189	C₄×C₈	C₂²×D₁₇	1	2	yes	—
190	C₁₇	C₂×#2162	1	1	yes	characteristic, complemented, Sylow
191	C₂⁴	—	17	1	no	—
192	C₂²×C₄	—	17	1	no	—
193	C₂²×C₄	—	17	1	no	—
194	C₂²×C₄	—	17	8	no	—
195	C₂²×C₄	—	17	4	no	—
196	C₂²×C₄	—	17	1	no	—
197	C₂²×C₄	—	17	2	no	—
198	C₂²×C₄	—	17	4	no	—
199	C₂²×C₄	—	17	2	no	—
200	C₂²×C₄	—	34	4	no	—
201	C₂²×C₄	—	34	2	no	—
202	C₂²×C₄	—	34	2	no	—
203	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
204	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	17	2	no	—
208	C₂×C₈	—	17	4	no	—
209	C₂×C₈	—	17	2	no	—
210	C₂×C₈	—	34	8	no	—
211	C₂×C₈	—	34	2	no	—
212	C₂×C₈	—	34	2	no	—
213	C₂×C₈	—	34	2	no	—
214	C₂×C₈	—	34	4	no	—
215	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
216	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
217	C₂×C₈	#63254	1	2	yes	—
218	C₂×C₈	#63256	1	2	yes	—
219	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
220	C₄²	—	17	2	no	—
221	C₄²	—	17	2	no	—
222	C₄²	—	17	4	no	—
223	C₄²	—	17	4	no	—
224	C₄²	C₈×D₁₇	1	2	yes	—
225	C₄²	C₂×C₄×D₁₇	1	2	yes	—
226	C₁₆	—	2	4	no	—
227	C₁₆	—	34	4	no	—
228	C₂³	—	17	4	no	—
229	C₂³	—	17	2	no	—
230	C₂³	—	34	2	no	—
231	C₂³	—	34	2	no	—
232	C₂³	C₂×C₈×D₁₇	1	1	yes	characteristic
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	17	4	no	—
235	C₂×C₄	—	17	4	no	—
236	C₂×C₄	—	17	2	no	—
237	C₂×C₄	—	17	2	no	—
238	C₂×C₄	—	17	16	no	—
239	C₂×C₄	—	17	2	no	—
240	C₂×C₄	—	34	8	no	—
241	C₂×C₄	—	34	2	no	—
242	C₂×C₄	—	34	2	no	—
243	C₂×C₄	—	34	2	no	—
244	C₂×C₄	—	34	4	no	—
245	C₂×C₄	C₁₆×D₁₇	1	4	yes	—
246	C₂×C₄	#10587578	1	2	yes	—
247	C₂×C₄	C₂×C₈×D₁₇	1	2	yes	—
248	C₂×C₄	C₂×C₈×D₁₇	1	2	yes	—
249	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
250	C₂×C₄	M₁₆×D₁₇	1	2	yes	—
251	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	—	17	4	no	—
254	C₈	—	34	2	no	—
255	C₈	#10587578	1	2	yes	—
256	C₈	C₂×#63254	1	1	yes	characteristic
257	C₈	C₂×#63256	1	1	yes	characteristic
258	C₂²	—	17	4	no	—
259	C₂²	—	34	8	no	—
260	C₂²	—	34	2	no	—
261	C₂²	—	34	2	no	—
262	C₂²	#11762861	1	2	yes	—
263	C₂²	C₂×C₁₆×D₁₇	1	2	yes	—
264	C₂²	M₃₂×D₁₇	1	2	yes	—
265	C₂²	C₂²×C₈×D₁₇	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	17	4	no	—
268	C₄	—	17	4	no	—
269	C₄	—	34	2	no	—
270	C₄	#11762861	1	2	yes	—
271	C₄	C₂×C₁₆×D₁₇	1	4	yes	—
272	C₄	C₂×#10587578	1	1	yes	characteristic
273	C₄	C₂×M₁₆×D₁₇	1	1	yes	characteristic
274	C₂	—	34	4	no	—
275	C₂	#14530582	1	4	yes	direct product factor
276	C₂	C₂×#11762861	1	1	yes	characteristic
277	C₂	C₂²×C₁₆×D₁₇	1	1	yes	characteristic
278	C₂	C₂×M₃₂×D₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₁₇
commutator	C₂×C₁₇	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₁₇×C₄⋊C₁₆	C₂
socle	C₂³×C₁₇	C₂²×C₈
prime radical	C₂³×D₁₇	C₂×C₈
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₁₇
hypocenter	C₁₇	C₂×#2162
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #14530582	C₂	direct (N × H)
3. C₂×C₄⋊#63214	C₂	semidirect
5. C₂×C₁₆×Dic₁₇	C₂	semidirect
6. C₂×C₁₇×C₄⋊C₁₆	C₂	semidirect
16. C₄⋊#63214	C₂²	semidirect
18. C₁₆×Dic₁₇	C₂²	semidirect
21. C₁₇×C₄⋊C₁₆	C₂²	semidirect
61. C₂×C₄×D₁₇	C₁₆	semidirect
112. C₄×D₁₇	C₂×C₁₆	semidirect
122. C₂×C₄⋊C₁₆	D₁₇	semidirect
153. C₄⋊C₁₆	C₂×D₁₇	semidirect
156. C₂×C₁₇	#2162	semidirect
190. C₁₇	C₂×#2162	semidirect
275. C₂	#14530582	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×C₄×C₈×D₁₇	2
#14530582	2
C₂×C₄⋊#63214	2
C₂×#11762588	2
C₂×C₁₆×Dic₁₇	2
C₂×C₁₇×C₄⋊C₁₆	2
C₂×#2162	17

Subgroups — C2×#14530582 #17187388

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14530582 #17187388