Subgroups — C₂×#14530621 #17187552

Order 4352.

662 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763226	C₂	1	2	yes	maximal, complemented
2	#14530621	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11762630	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11762593	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×#63214	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₇×#443	C₂	1	1	yes	maximal, characteristic
7	#11763226	—	2	4	no	—
8	#11763226	C₂²	1	8	yes	complemented
9	C₂×#10587483	—	2	2	no	—
10	C₂×#10587483	C₂²	1	2	yes	—
11	C₂²×#63214	—	2	2	no	—
12	C₂²×#63214	C₂²	1	1	yes	characteristic
13	C₂²×#63214	C₂²	1	1	yes	characteristic
14	#11762630	C₂²	1	4	yes	complemented
15	#11762593	C₂²	1	8	yes	complemented
16	C₄×#63214	C₂²	1	4	yes	complemented
17	C₂×C₁₇×#135	C₄	1	1	yes	characteristic
18	C₂×C₁₇×#131	C₂²	1	2	yes	—
19	C₁₇×#443	C₄	1	4	yes	—
20	C₁₇×#443	C₂²	1	4	yes	—
21	C₂×C₁₇×#108	C₄	1	1	yes	characteristic
22	C₂×C₁₇×#104	C₄	1	2	yes	—
23	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
24	C₂×#41×C₁₇	C₂×C₄	1	2	yes	—
25	C₁₇×#135	C₈	1	4	yes	—
26	C₁₇×#135	C₂×C₄	1	4	yes	—
27	C₂×Q₁₆×C₁₇	C₈	1	1	yes	characteristic
28	C₂×SD₁₆×C₁₇	C₈	1	2	yes	—
29	C₂×D₈×C₁₇	C₈	1	1	yes	characteristic
30	C₁₇×#131	—	2	2	no	—
31	C₁₇×#131	—	2	2	no	—
32	C₁₇×#131	C₂×C₄	1	4	yes	—
33	C₁₇×#131	C₂³	1	4	yes	—
34	C₂×M₁₆×C₁₇	—	2	2	no	—
35	C₂×M₁₆×C₁₇	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₁₇	—	2	2	no	—
37	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
40	C₁₇×#108	C₂×C₄	1	4	yes	—
41	C₁₇×#104	C₂×C₄	1	8	yes	—
42	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₁₇	C₂³	1	2	yes	—
44	#10587483	—	2	8	no	—
45	#10587483	—	2	2	no	—
46	#10587483	C₂³	1	4	yes	—
47	C₂×#63214	—	2	8	no	—
48	C₂×#63214	—	2	2	no	—
49	C₂×#63214	—	2	1	no	—
50	C₂×#63214	—	2	1	no	—
51	C₂×#63214	D₄	1	1	yes	characteristic
52	C₂×#63214	D₄	1	1	yes	characteristic
53	C₂×#63214	D₄	1	2	yes	—
54	C₂×#63214	C₂³	1	2	yes	—
55	C₂×#63214	C₂³	1	2	yes	—
56	#41×C₁₇	—	2	2	no	—
57	#41×C₁₇	—	2	2	no	—
58	#41×C₁₇	C₂×C₈	1	4	yes	—
59	#41×C₁₇	C₂²×C₄	1	4	yes	—
60	C₂×Q₈×C₁₇	C₂×C₈	1	2	yes	—
61	C₂×D₄×C₁₇	—	2	2	no	—
62	C₂×D₄×C₁₇	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₁₇	—	2	1	no	—
64	C₂²×C₄×C₁₇	—	2	2	no	—
65	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
66	Q₁₆×C₁₇	C₂×C₈	1	4	yes	—
67	SD₁₆×C₁₇	C₂×C₈	1	8	yes	—
68	D₈×C₁₇	C₂×C₈	1	4	yes	—
69	M₁₆×C₁₇	—	2	2	no	—
70	M₁₆×C₁₇	—	2	4	no	—
71	M₁₆×C₁₇	—	2	4	no	—
72	M₁₆×C₁₇	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₁₇	—	2	2	no	—
74	C₂×C₈×C₁₇	—	2	4	no	—
75	C₂×C₈×C₁₇	—	2	4	no	—
76	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
77	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
78	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
79	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
84	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
85	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
86	C₄²×C₁₇	—	2	1	no	—
87	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
88	#63214	—	2	4	no	—
89	#63214	—	2	2	no	—
90	#63214	C₂×D₄	1	2	yes	—
91	#63214	C₂×D₄	1	2	yes	—
92	C₂×#2170	—	17	1	no	maximal, Sylow
93	C₂³×C₁₇	—	2	2	no	—
94	C₂³×C₁₇	#131	1	1	yes	characteristic
95	Q₈×C₁₇	—	2	2	no	—
96	Q₈×C₁₇	C₂²×C₈	1	4	yes	—
97	D₄×C₁₇	—	2	4	no	—
98	D₄×C₁₇	—	2	4	no	—
99	D₄×C₁₇	—	2	2	no	—
100	D₄×C₁₇	C₂²×C₈	1	4	yes	—
101	C₂×C₄×C₁₇	—	2	4	no	—
102	C₂×C₄×C₁₇	—	2	2	no	—
103	C₂×C₄×C₁₇	—	2	2	no	—
104	C₂×C₄×C₁₇	—	2	4	no	—
105	C₂×C₄×C₁₇	—	2	4	no	—
106	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
111	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
112	C₈×C₁₇	—	2	4	no	—
113	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
114	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
115	C₈×C₁₇	C₂²×C₈	1	2	yes	—
116	C₈×C₁₇	C₂²×C₈	1	2	yes	—
117	C₈×C₁₇	C₂²×D₄	1	1	yes	characteristic
118	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
119	C₂×#504	—	17	2	no	—
120	C₂×#443	D₁₇	1	1	yes	characteristic
121	#2170	—	17	8	no	—
122	C₂×#364	—	17	1	no	—
123	C₂×#350	—	17	2	no	—
124	C₂×C₄×C₁₆	—	17	1	no	—
125	C₂²×C₁₇	—	2	2	no	—
126	C₂²×C₁₇	—	2	4	no	—
127	C₂²×C₁₇	—	4	4	no	—
128	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
129	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
130	C₂²×C₁₇	C₂×#131	1	1	yes	characteristic
131	C₄×C₁₇	—	2	4	no	—
132	C₄×C₁₇	—	2	4	no	—
133	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
134	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
135	C₄×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
136	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
137	C₂×#135	Dic₁₇	1	1	yes	characteristic
138	C₂×#131	C₂×D₁₇	1	2	yes	—
139	#504	—	17	8	no	—
140	#504	—	34	4	no	—
141	C₂×M₃₂	—	17	2	no	—
142	C₂×M₃₂	—	34	2	no	—
143	C₂²×C₁₆	—	17	1	no	—
144	C₂²×C₁₆	—	17	1	no	—
145	C₂²×C₁₆	—	34	2	no	—
146	#443	Dic₁₇	1	4	yes	—
147	#443	C₂×D₁₇	1	4	yes	—
148	C₂×#108	Dic₁₇	1	1	yes	characteristic
149	C₂×#104	Dic₁₇	1	2	yes	—
150	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
151	#364	—	17	4	no	—
152	#350	—	17	8	no	—
153	C₄×C₁₆	—	17	4	no	—
154	C₂×C₁₇	—	2	1	no	—
155	C₂×C₁₇	—	2	1	no	—
156	C₂×C₁₇	—	4	4	no	—
157	C₂×C₁₇	#2170	1	2	yes	complemented
158	C₂×C₁₇	C₂×C₈×D₄	1	1	yes	characteristic
159	C₂×#41	C₂×Dic₁₇	1	2	yes	—
160	#135	#3622	1	4	yes	—
161	#135	C₂×Dic₁₇	1	4	yes	—
162	C₂×Q₁₆	#3622	1	1	yes	characteristic
163	C₂×SD₁₆	#3622	1	2	yes	—
164	C₂×D₈	#3622	1	1	yes	characteristic
165	#131	—	2	2	no	—
166	#131	—	2	2	no	—
167	#131	C₂×Dic₁₇	1	4	yes	—
168	#131	C₂²×D₁₇	1	4	yes	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	C₂×Dic₁₇	1	2	yes	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
173	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
174	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
175	M₃₂	—	17	4	no	—
176	M₃₂	—	34	8	no	—
177	M₃₂	—	34	2	no	—
178	C₂×C₁₆	—	17	2	no	—
179	C₂×C₁₆	—	17	1	no	—
180	C₂×C₁₆	—	17	1	no	—
181	C₂×C₁₆	—	17	2	no	—
182	C₂×C₁₆	—	17	2	no	—
183	C₂×C₁₆	—	34	8	no	—
184	C₂×C₁₆	—	34	2	no	—
185	C₂×C₁₆	—	34	1	no	—
186	C₂×C₁₆	—	34	1	no	—
187	#108	C₂×Dic₁₇	1	4	yes	—
188	#104	C₂×Dic₁₇	1	8	yes	—
189	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
190	C₄×C₈	C₂²×D₁₇	1	2	yes	—
191	C₁₇	C₂×#2170	1	1	yes	characteristic, complemented, Sylow
192	#41	—	2	2	no	—
193	#41	—	2	2	no	—
194	#41	C₂×#3622	1	4	yes	—
195	#41	C₂²×Dic₁₇	1	4	yes	—
196	C₂×Q₈	C₂×#3622	1	2	yes	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	C₂×#3622	1	2	yes	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
202	Q₁₆	C₂×#3622	1	4	yes	—
203	SD₁₆	C₂×#3622	1	8	yes	—
204	D₈	C₂×#3622	1	4	yes	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	2	4	no	—
207	M₁₆	—	2	4	no	—
208	M₁₆	C₂²×Dic₁₇	1	4	yes	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	C₂×#3622	1	1	yes	characteristic
213	C₂×C₈	C₂×#3622	1	1	yes	characteristic
214	C₂×C₈	C₂×#3622	1	2	yes	—
215	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
216	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
217	C₂×C₈	#63254	1	1	yes	characteristic
218	C₂×C₈	#63254	1	1	yes	characteristic
219	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
220	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
221	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
222	C₄²	—	2	1	no	—
223	C₄²	C₂²×Dic₁₇	1	2	yes	—
224	C₁₆	—	17	2	no	—
225	C₁₆	—	17	2	no	—
226	C₁₆	—	34	4	no	—
227	C₁₆	—	34	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	#10587634	1	1	yes	characteristic
230	Q₈	—	2	2	no	—
231	Q₈	C₂²×#3622	1	4	yes	—
232	D₄	—	2	4	no	—
233	D₄	—	2	4	no	—
234	D₄	—	2	2	no	—
235	D₄	C₂²×#3622	1	4	yes	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
242	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
243	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
244	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
245	C₂×C₄	#10587634	1	1	yes	characteristic
246	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
247	C₈	—	2	4	no	—
248	C₈	C₂²×#3622	1	2	yes	—
249	C₈	C₂²×#3622	1	2	yes	—
250	C₈	D₄×Dic₁₇	1	1	yes	characteristic
251	C₈	D₄×Dic₁₇	1	1	yes	characteristic
252	C₈	C₂×D₄×D₁₇	1	1	yes	characteristic
253	C₈	C₂×#63254	1	1	yes	characteristic
254	C₂²	—	2	2	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	4	4	no	—
257	C₂²	D₄×#3622	1	1	yes	characteristic
258	C₂²	D₄×#3622	1	1	yes	characteristic
259	C₂²	C₂×#10587634	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	2	4	no	—
262	C₄	D₄×#3622	1	1	yes	characteristic
263	C₄	D₄×#3622	1	1	yes	characteristic
264	C₄	C₂×D₄×Dic₁₇	1	1	yes	characteristic
265	C₄	C₂³×#3622	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	4	4	no	—
269	C₂	#14530621	1	2	yes	direct product factor
270	C₂	C₂×D₄×#3622	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₇
commutator	C₄×C₁₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₁₇×#443	C₂
socle	C₂²×C₁₇	C₈×D₄
prime radical	C₂×C₁₇×#135	C₄
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂×#443	D₁₇
hypocenter	C₁₇	C₂×#2170
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. C₂×#11763226	C₂	semidirect
2. #14530621	C₂	direct (N × H)
3. C₂×#11762630	C₂	semidirect
4. C₂×#11762593	C₂	semidirect
5. C₂×C₄×#63214	C₂	semidirect
8. #11763226	C₂²	semidirect
14. #11762630	C₂²	semidirect
15. #11762593	C₂²	semidirect
16. C₄×#63214	C₂²	semidirect
157. C₂×C₁₇	#2170	semidirect
191. C₁₇	C₂×#2170	semidirect
269. C₂	#14530621	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×#11763226	2
#14530621	2
C₂×#11762630	2
C₂×#11762593	2
C₂×C₄×#63214	2
C₂×C₁₇×#443	2
C₂×#2170	17

Subgroups — C2×#14530621 #17187552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14530621 #17187552