Subgroups — #17005167

Order 4352.

624 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14537860	C₂	1	2	yes	maximal, complemented
2	#14527649	C₂	1	4	yes	maximal, complemented
3	#14527614	C₂	1	4	yes	maximal, complemented
4	#14527547	C₂	1	1	yes	maximal, characteristic, complemented
5	#14527253	C₂	1	2	yes	maximal, complemented
6	#14527076	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#3169	C₂	1	1	yes	maximal, characteristic
8	#11762887	C₂²	1	4	yes	—
9	#11762881	—	2	4	no	—
10	D₄×#3622	C₂²	1	4	yes	—
11	#11762866	C₂²	1	2	yes	—
12	#11762862	C₂²	1	1	yes	characteristic
13	#11762862	C₂²	1	1	yes	characteristic
14	#11762776	—	2	4	no	—
15	#11762562	C₂²	1	2	yes	—
16	#11762560	C₂²	1	2	yes	—
17	D₄⋊#3622	C₂²	1	8	yes	—
18	#11762531	C₂²	1	2	yes	—
19	#11762531	C₂²	1	2	yes	—
20	C₁₇×#532	C₂²	1	2	yes	—
21	C₁₇×#500	C₄	1	1	yes	characteristic
22	C₁₇×#493	C₄	1	1	yes	characteristic
23	C₁₇×#486	C₄	1	2	yes	—
24	C₁₇×#466	C₄	1	2	yes	—
25	C₁₇×#466	C₄	1	2	yes	—
26	C₄×D₈×C₁₇	C₂²	1	2	yes	—
27	C₄×D₈×C₁₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
29	C₂×#41×C₁₇	—	2	2	no	—
30	C₂×D₈×C₁₇	C₂×C₄	1	2	yes	—
31	C₂×D₈×C₁₇	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
34	C₁₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
35	C₁₇×#127	C₂×C₄	1	2	yes	—
36	C₁₇×#124	—	2	2	no	—
37	C₁₇×#121	—	2	2	no	—
38	C₁₇×#121	C₂×C₄	1	4	yes	—
39	C₄×D₄×C₁₇	—	2	2	no	—
40	C₄×D₄×C₁₇	C₂³	1	4	yes	—
41	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
42	C₂×#63223	—	2	4	no	—
43	C₂²×#3622	—	2	4	no	—
44	C₁₇×#107	C₂×C₄	1	2	yes	—
45	C₁₇×#107	C₂×C₄	1	2	yes	—
46	C₁₇×#102	C₂×C₄	1	4	yes	—
47	C₁₇×#102	C₂×C₄	1	4	yes	—
48	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
52	C₂²⋊#3622	—	2	4	no	—
53	C₂²⋊#3622	—	2	4	no	—
54	C₂²⋊#3622	—	2	2	no	—
55	C₂²⋊#3622	—	2	1	no	—
56	C₂²⋊#3622	D₄	1	2	yes	—
57	C₄⋊#3622	—	2	1	no	—
58	C₄⋊#3622	D₄	1	2	yes	—
59	C₄⋊#3622	C₂³	1	4	yes	—
60	#10587474	—	2	2	no	—
61	C₄×#3622	C₂³	1	2	yes	—
62	C₄×#3622	C₂³	1	2	yes	—
63	#41×C₁₇	—	8	4	no	—
64	C₂×Q₈×C₁₇	—	2	2	no	—
65	C₂×D₄×C₁₇	—	2	4	no	—
66	C₂×D₄×C₁₇	—	2	2	no	—
67	C₂×D₄×C₁₇	—	2	2	no	—
68	C₂×D₄×C₁₇	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₁₇	—	2	4	no	—
70	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
73	D₈×C₁₇	—	4	2	no	—
74	D₈×C₁₇	—	4	2	no	—
75	C₂×C₈×C₁₇	—	4	1	no	—
76	C₂×C₈×C₁₇	—	4	1	no	—
77	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
81	C₄⋊C₄×C₁₇	—	2	2	no	—
82	C₄⋊C₄×C₁₇	C₂²×C₄	1	4	yes	—
83	C₂²⋊C₄×C₁₇	—	2	4	no	—
84	C₂²⋊C₄×C₁₇	—	2	4	no	—
85	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₇	#41	1	1	yes	characteristic
88	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
89	#63223	—	8	4	no	—
90	C₂×#3622	—	2	4	no	—
91	C₂×#3622	—	2	2	no	—
92	C₂×#3622	—	8	4	no	—
93	C₂×#3622	C₂×D₄	1	4	yes	—
94	#17989	—	17	1	no	maximal, Sylow
95	C₂³×C₁₇	—	2	4	no	—
96	C₂³×C₁₇	C₄×D₄	1	1	yes	characteristic
97	Q₈×C₁₇	—	8	2	no	—
98	D₄×C₁₇	—	2	4	no	—
99	D₄×C₁₇	—	2	4	no	—
100	D₄×C₁₇	—	8	2	no	—
101	D₄×C₁₇	—	8	2	no	—
102	D₄×C₁₇	—	8	4	no	—
103	C₂×C₄×C₁₇	—	2	1	no	—
104	C₂×C₄×C₁₇	—	2	1	no	—
105	C₂×C₄×C₁₇	—	2	2	no	—
106	C₂×C₄×C₁₇	—	2	4	no	—
107	C₂×C₄×C₁₇	—	4	1	no	—
108	C₂×C₄×C₁₇	—	4	4	no	—
109	C₂×C₄×C₁₇	—	4	4	no	—
110	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
116	C₈×C₁₇	—	2	1	no	—
117	C₈×C₁₇	—	2	1	no	—
118	C₈×C₁₇	—	2	1	no	—
119	C₈×C₁₇	—	2	1	no	—
120	#3622	—	2	4	no	—
121	#3622	—	4	2	no	—
122	#3622	—	4	4	no	—
123	#3169	D₁₇	1	1	yes	characteristic
124	#2981	—	17	2	no	—
125	#1586	—	17	4	no	—
126	#1581	—	17	4	no	—
127	#1568	—	17	1	no	—
128	#1496	—	17	2	no	—
129	#1452	—	17	1	no	—
130	C₂²×C₁₇	—	2	1	no	—
131	C₂²×C₁₇	—	2	1	no	—
132	C₂²×C₁₇	—	2	1	no	—
133	C₂²×C₁₇	—	4	4	no	—
134	C₂²×C₁₇	—	4	4	no	—
135	C₂²×C₁₇	—	4	4	no	—
136	C₂²×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
137	C₄×C₁₇	—	2	1	no	—
138	C₄×C₁₇	—	2	1	no	—
139	C₄×C₁₇	—	2	1	no	—
140	C₄×C₁₇	—	2	1	no	—
141	C₄×C₁₇	—	4	4	no	—
142	C₄×C₁₇	#568	1	1	yes	characteristic
143	C₄×C₁₇	#568	1	1	yes	characteristic
144	C₄×C₁₇	#576	1	2	yes	—
145	#532	C₂×D₁₇	1	2	yes	—
146	#500	Dic₁₇	1	1	yes	characteristic
147	#493	Dic₁₇	1	1	yes	characteristic
148	#486	Dic₁₇	1	2	yes	—
149	#466	Dic₁₇	1	2	yes	—
150	#466	Dic₁₇	1	2	yes	—
151	C₄×D₈	C₂×D₁₇	1	2	yes	—
152	C₄×D₈	C₂×D₁₇	1	2	yes	—
153	#436	—	17	4	no	—
154	#435	—	34	4	no	—
155	C₈×D₄	—	17	4	no	—
156	#432	—	17	2	no	—
157	#431	—	17	1	no	—
158	#431	—	17	1	no	—
159	#408	—	34	4	no	—
160	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
161	#335	—	17	2	no	—
162	#334	—	17	2	no	—
163	D₄⋊C₈	—	17	8	no	—
164	#322	—	17	2	no	—
165	#322	—	17	2	no	—
166	C₂×C₁₇	—	4	1	no	—
167	C₂×C₁₇	—	8	4	no	—
168	C₂×C₁₇	#2932	1	1	yes	characteristic
169	C₂×C₁₇	#2948	1	1	yes	characteristic
170	C₂×C₁₇	#2957	1	1	yes	characteristic
171	C₂×#41	—	2	2	no	—
172	C₂×D₈	C₂×Dic₁₇	1	2	yes	—
173	C₂×D₈	C₂×Dic₁₇	1	2	yes	—
174	C₂×M₁₆	—	34	4	no	—
175	C₂²×C₈	—	34	4	no	—
176	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
177	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
178	C₄⋊Q₈	C₂×Dic₁₇	1	2	yes	—
179	#127	C₂×Dic₁₇	1	2	yes	—
180	#124	—	2	2	no	—
181	#121	—	2	2	no	—
182	#121	C₂×Dic₁₇	1	4	yes	—
183	C₄×D₄	—	2	2	no	—
184	C₄×D₄	C₂²×D₁₇	1	4	yes	—
185	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
186	#107	C₂×Dic₁₇	1	2	yes	—
187	#107	C₂×Dic₁₇	1	2	yes	—
188	C₄⋊C₈	—	17	2	no	—
189	C₄⋊C₈	—	17	4	no	—
190	C₄⋊C₈	—	34	1	no	—
191	#102	C₂×Dic₁₇	1	4	yes	—
192	#102	C₂×Dic₁₇	1	4	yes	—
193	C₂²⋊C₈	—	17	2	no	—
194	C₂²⋊C₈	—	34	4	no	—
195	C₂²⋊C₈	—	34	4	no	—
196	C₂²⋊C₈	—	34	2	no	—
197	C₂²⋊C₈	—	34	1	no	—
198	#97	—	34	2	no	—
199	C₄×C₈	—	17	2	no	—
200	C₄×C₈	—	17	2	no	—
201	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
202	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
203	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
204	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
205	C₁₇	#17989	1	1	yes	characteristic, complemented, Sylow
206	#41	—	8	4	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	C₂²×Dic₁₇	1	4	yes	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
214	C₂²×C₄	#63254	1	1	yes	characteristic
215	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
216	D₈	—	4	2	no	—
217	D₈	—	4	2	no	—
218	M₁₆	—	136	4	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	17	4	no	—
222	C₂×C₈	—	34	4	no	—
223	C₂×C₈	—	34	2	no	—
224	C₂×C₈	—	136	4	no	—
225	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
226	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
227	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
228	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	C₂²×Dic₁₇	1	4	yes	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₄²	D₄×D₁₇	1	1	yes	characteristic
234	C₄²	#63254	1	1	yes	characteristic
235	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
236	C₄²	C₂³×D₁₇	1	1	yes	characteristic
237	C₂³	—	2	4	no	—
238	C₂³	D₄×Dic₁₇	1	1	yes	characteristic
239	Q₈	—	8	2	no	—
240	D₄	—	2	4	no	—
241	D₄	—	2	4	no	—
242	D₄	—	8	2	no	—
243	D₄	—	8	2	no	—
244	D₄	—	8	4	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
253	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
254	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
255	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
256	C₂×C₄	C₂×#63254	1	1	yes	characteristic
257	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	34	4	no	—
263	C₈	—	68	2	no	—
264	C₈	—	68	4	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	4	4	no	—
271	C₂²	C₂×D₄×Dic₁₇	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	4	no	—
277	C₄	#11763576	1	1	yes	characteristic
278	C₄	#11763576	1	1	yes	characteristic
279	C₄	#11763610	1	2	yes	—
280	C₂	—	4	1	no	—
281	C₂	—	8	4	no	—
282	C₂	#14537502	1	1	yes	characteristic
283	C₂	#14537636	1	1	yes	characteristic
284	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, prime radical	C₁₇×#3169	C₂
socle	C₂²×C₁₇	C₂×C₄×D₄
cosocle	C₄²×C₁₇	C₂⁴
hypercenter	#3169	D₁₇
hypocenter	C₁₇	#17989
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #14537860	C₂	semidirect
2. #14527649	C₂	semidirect
3. #14527614	C₂	semidirect
4. #14527547	C₂	semidirect
5. #14527253	C₂	semidirect
6. #14527076	C₂	semidirect
205. C₁₇	#17989	semidirect

Maximal subgroups

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

subgroup	index
#14537860	2
#14527649	2
#14527614	2
#14527547	2
#14527253	2
#14527076	2
C₁₇×#3169	2
#17989	17