Subgroups — #17005185

Order 4352.

544 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14537879	C₂	1	2	yes	maximal
2	#14527656	C₂	1	4	yes	maximal, complemented
3	#14527635	C₂	1	4	yes	maximal, complemented
4	#14527547	C₂	1	1	yes	maximal, characteristic
5	#14527257	C₂	1	2	yes	maximal
6	#14527076	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#3170	C₂	1	1	yes	maximal, characteristic
8	#11762940	—	2	4	no	—
9	#11762940	C₂²	1	4	yes	—
10	Q₈×#3622	C₂²	1	4	yes	—
11	#11762870	—	2	4	no	—
12	#11762866	C₂²	1	2	yes	—
13	#11762862	C₂²	1	1	yes	characteristic
14	#11762862	C₂²	1	1	yes	characteristic
15	#11762562	C₂²	1	2	yes	—
16	#11762560	C₂²	1	2	yes	—
17	Q₈⋊#3622	C₂²	1	8	yes	—
18	#11762531	C₂²	1	2	yes	—
19	#11762531	C₂²	1	2	yes	—
20	C₁₇×#533	C₂²	1	2	yes	—
21	C₁₇×#500	C₄	1	1	yes	characteristic
22	C₁₇×#494	C₄	1	1	yes	characteristic
23	C₁₇×#487	C₄	1	2	yes	—
24	C₁₇×#467	C₄	1	2	yes	—
25	C₁₇×#467	C₄	1	2	yes	—
26	C₄×Q₁₆×C₁₇	C₂²	1	2	yes	—
27	C₄×Q₁₆×C₁₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
29	C₂×Q₁₆×C₁₇	C₂×C₄	1	2	yes	—
30	C₂×Q₁₆×C₁₇	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₁₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
34	C₁₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
35	C₁₇×#125	—	2	2	no	—
36	C₁₇×C₂²⋊Q₈	—	2	2	no	—
37	C₁₇×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
38	C₄×Q₈×C₁₇	—	2	2	no	—
39	C₄×Q₈×C₁₇	C₂³	1	4	yes	—
40	C₁₇×#117	—	2	2	no	—
41	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
42	C₁₇×#107	C₂×C₄	1	2	yes	—
43	C₁₇×#107	C₂×C₄	1	2	yes	—
44	C₁₇×#103	C₂×C₄	1	4	yes	—
45	C₁₇×#103	C₂×C₄	1	4	yes	—
46	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
49	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
50	C₂²⋊#3622	—	2	2	no	—
51	C₂²⋊#3622	—	2	1	no	—
52	C₂²⋊#3622	D₄	1	2	yes	—
53	C₄⋊#3622	—	2	4	no	—
54	C₄⋊#3622	—	2	4	no	—
55	C₄⋊#3622	—	2	1	no	—
56	C₄⋊#3622	D₄	1	2	yes	—
57	C₄⋊#3622	C₂³	1	4	yes	—
58	#10587474	—	2	4	no	—
59	#10587474	—	2	2	no	—
60	C₄×#3622	—	2	4	no	—
61	C₄×#3622	C₂³	1	2	yes	—
62	C₄×#3622	C₂³	1	2	yes	—
63	C₂×Q₈×C₁₇	—	2	2	no	—
64	C₂×Q₈×C₁₇	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
68	Q₁₆×C₁₇	—	4	2	no	—
69	Q₁₆×C₁₇	—	4	2	no	—
70	C₂×C₈×C₁₇	—	4	1	no	—
71	C₂×C₈×C₁₇	—	4	1	no	—
72	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
76	C₄⋊C₄×C₁₇	—	2	2	no	—
77	C₄⋊C₄×C₁₇	—	2	4	no	—
78	C₄⋊C₄×C₁₇	—	2	2	no	—
79	C₄⋊C₄×C₁₇	—	2	2	no	—
80	C₄⋊C₄×C₁₇	—	2	4	no	—
81	C₄⋊C₄×C₁₇	C₂²×C₄	1	4	yes	—
82	C₂²⋊C₄×C₁₇	—	2	4	no	—
83	C₄²×C₁₇	—	2	4	no	—
84	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₁₇	#41	1	1	yes	characteristic
87	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
88	C₂×#3622	—	2	4	no	—
89	C₂×#3622	—	2	2	no	—
90	C₂×#3622	C₂×D₄	1	4	yes	—
91	#17991	—	17	1	no	maximal, Sylow
92	C₂³×C₁₇	C₄×D₄	1	1	yes	characteristic
93	Q₈×C₁₇	—	2	4	no	—
94	Q₈×C₁₇	—	2	4	no	—
95	Q₈×C₁₇	—	8	2	no	—
96	C₂×C₄×C₁₇	—	2	1	no	—
97	C₂×C₄×C₁₇	—	2	1	no	—
98	C₂×C₄×C₁₇	—	2	2	no	—
99	C₂×C₄×C₁₇	—	2	4	no	—
100	C₂×C₄×C₁₇	—	2	4	no	—
101	C₂×C₄×C₁₇	—	4	1	no	—
102	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
108	C₈×C₁₇	—	2	1	no	—
109	C₈×C₁₇	—	2	1	no	—
110	C₈×C₁₇	—	2	1	no	—
111	C₈×C₁₇	—	2	1	no	—
112	#3622	—	2	4	no	—
113	#3622	—	4	2	no	—
114	#3622	—	4	4	no	—
115	#3170	D₁₇	1	1	yes	characteristic
116	#2985	—	17	2	no	—
117	#1587	—	17	4	no	—
118	#1584	—	17	4	no	—
119	#1568	—	17	1	no	—
120	#1497	—	17	2	no	—
121	#1452	—	17	1	no	—
122	C₂²×C₁₇	—	2	1	no	—
123	C₂²×C₁₇	—	2	1	no	—
124	C₂²×C₁₇	—	2	1	no	—
125	C₂²×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
126	C₄×C₁₇	—	2	1	no	—
127	C₄×C₁₇	—	2	1	no	—
128	C₄×C₁₇	—	2	1	no	—
129	C₄×C₁₇	—	2	1	no	—
130	C₄×C₁₇	—	4	4	no	—
131	C₄×C₁₇	—	4	4	no	—
132	C₄×C₁₇	#568	1	1	yes	characteristic
133	C₄×C₁₇	#568	1	1	yes	characteristic
134	C₄×C₁₇	#578	1	2	yes	—
135	#533	C₂×D₁₇	1	2	yes	—
136	#500	Dic₁₇	1	1	yes	characteristic
137	#494	Dic₁₇	1	1	yes	characteristic
138	#487	Dic₁₇	1	2	yes	—
139	#467	Dic₁₇	1	2	yes	—
140	#467	Dic₁₇	1	2	yes	—
141	#446	—	17	4	no	—
142	#446	—	34	4	no	—
143	C₈×Q₈	—	17	4	no	—
144	C₄×Q₁₆	C₂×D₁₇	1	2	yes	—
145	C₄×Q₁₆	C₂×D₁₇	1	2	yes	—
146	#433	—	34	4	no	—
147	#432	—	17	2	no	—
148	#431	—	17	1	no	—
149	#431	—	17	1	no	—
150	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
151	#335	—	17	2	no	—
152	#334	—	17	2	no	—
153	Q₈⋊C₈	—	17	8	no	—
154	#322	—	17	2	no	—
155	#322	—	17	2	no	—
156	C₂×C₁₇	—	4	1	no	—
157	C₂×C₁₇	#2932	1	1	yes	characteristic
158	C₂×C₁₇	#2950	1	1	yes	characteristic
159	C₂×C₁₇	#2957	1	1	yes	characteristic
160	C₂×Q₁₆	C₂×Dic₁₇	1	2	yes	—
161	C₂×Q₁₆	C₂×Dic₁₇	1	2	yes	—
162	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
163	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
164	C₄⋊Q₈	C₂×Dic₁₇	1	2	yes	—
165	C₄⋊Q₈	C₂×Dic₁₇	1	2	yes	—
166	#125	—	2	2	no	—
167	C₂²⋊Q₈	—	2	2	no	—
168	C₂²⋊Q₈	C₂×Dic₁₇	1	4	yes	—
169	C₄×Q₈	—	2	2	no	—
170	C₄×Q₈	C₂²×D₁₇	1	4	yes	—
171	#117	—	2	2	no	—
172	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
173	#107	C₂×Dic₁₇	1	2	yes	—
174	#107	C₂×Dic₁₇	1	2	yes	—
175	C₄⋊C₈	—	17	2	no	—
176	C₄⋊C₈	—	17	4	no	—
177	C₄⋊C₈	—	34	4	no	—
178	C₄⋊C₈	—	34	4	no	—
179	C₄⋊C₈	—	34	1	no	—
180	#103	C₂×Dic₁₇	1	4	yes	—
181	#103	C₂×Dic₁₇	1	4	yes	—
182	C₂²⋊C₈	—	17	2	no	—
183	C₂²⋊C₈	—	34	2	no	—
184	C₂²⋊C₈	—	34	1	no	—
185	#97	—	34	4	no	—
186	#97	—	34	2	no	—
187	C₄×C₈	—	17	2	no	—
188	C₄×C₈	—	17	2	no	—
189	C₄×C₈	—	34	4	no	—
190	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
191	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
192	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
194	C₁₇	#17991	1	1	yes	characteristic, complemented, Sylow
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	C₂²×Dic₁₇	1	4	yes	—
197	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
198	C₂²×C₄	#63254	1	1	yes	characteristic
199	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
200	Q₁₆	—	4	2	no	—
201	Q₁₆	—	4	2	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	17	4	no	—
205	C₂×C₈	—	34	4	no	—
206	C₂×C₈	—	34	2	no	—
207	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
208	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
209	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
210	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	C₂²×Dic₁₇	1	4	yes	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₄²	—	2	4	no	—
219	C₄²	D₄×D₁₇	1	1	yes	characteristic
220	C₄²	#63254	1	1	yes	characteristic
221	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
222	C₄²	C₂³×D₁₇	1	1	yes	characteristic
223	C₂³	D₄×Dic₁₇	1	1	yes	characteristic
224	Q₈	—	2	4	no	—
225	Q₈	—	2	4	no	—
226	Q₈	—	8	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
234	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
235	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
236	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
237	C₂×C₄	C₂×#63254	1	1	yes	characteristic
238	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	34	4	no	—
244	C₈	—	68	2	no	—
245	C₈	—	68	4	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	C₂×D₄×Dic₁₇	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	4	no	—
255	C₄	—	4	4	no	—
256	C₄	#11763576	1	1	yes	characteristic
257	C₄	#11763576	1	1	yes	characteristic
258	C₄	#11763622	1	2	yes	—
259	C₂	—	4	1	no	—
260	C₂	#14537502	1	1	yes	characteristic
261	C₂	#14537652	1	1	yes	characteristic
262	C₂	#14537701	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₄²	C₂³×D₁₇
Fitting	C₁₇×#3170	C₂
socle	C₂²×C₁₇	C₂×C₄×D₄
prime radical	C₂³×C₁₇	C₄×D₄
cosocle	C₄²×C₁₇	C₂⁴
hypercenter	#3170	D₁₇
hypocenter	C₁₇	#17991
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #14527656	C₂	semidirect
3. #14527635	C₂	semidirect
194. C₁₇	#17991	semidirect

Maximal subgroups

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

subgroup	index
#14537879	2
#14527656	2
#14527635	2
#14527547	2
#14527257	2
#14527076	2
C₁₇×#3170	2
#17991	17