Subgroups — #16927583

Order 4352.

362 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528956	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762562	C₂	1	1	yes	maximal, characteristic
3	#14527474	C₂	1	2	yes	maximal
4	#14526658	C₂	1	2	yes	maximal
5	C₁₇×#2960	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587490	—	2	1	no	—
7	C₂×#10587490	—	2	1	no	—
8	#11762833	C₂²	1	2	yes	—
9	C₂×C₄⋊#3622	C₂²	1	2	yes	—
10	#11762562	—	2	1	no	—
11	#11762562	—	2	1	no	—
12	#11762558	—	2	2	no	—
13	#11762555	—	2	2	no	—
14	C₁₇×#533	C₂²	1	1	yes	characteristic
15	C₁₇×#446	—	2	1	no	—
16	C₁₇×#446	C₄	1	2	yes	—
17	C₈×Q₈×C₁₇	—	2	1	no	—
18	C₈×Q₈×C₁₇	C₄	1	2	yes	—
19	C₁₇×#424	—	2	1	no	—
20	C₁₇×#423	C₂²	1	1	yes	characteristic
21	C₁₇×#405	—	2	1	no	—
22	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
23	C₂×M₁₆×C₁₇	—	2	1	no	—
24	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
26	C₁₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
27	C₁₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
28	C₁₇×#125	—	2	1	no	—
29	C₁₇×C₂²⋊Q₈	—	2	1	no	—
30	C₁₇×C₂²⋊Q₈	Q₈	1	2	yes	—
31	C₄×Q₈×C₁₇	—	2	1	no	—
32	C₄×Q₈×C₁₇	C₂×C₄	1	2	yes	—
33	C₁₇×#117	—	2	1	no	—
34	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
35	C₂×#63223	—	2	2	no	—
36	C₂²×#3622	—	2	2	no	—
37	C₁₇×C₄⋊C₈	—	2	2	no	—
38	C₁₇×C₄⋊C₈	—	2	2	no	—
39	C₁₇×C₄⋊C₈	—	2	1	no	—
40	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
41	C₁₇×#97	—	2	2	no	—
42	C₄×C₈×C₁₇	—	2	2	no	—
43	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
47	#10587490	—	4	2	no	—
48	#10587490	—	4	2	no	—
49	C₄⋊#3622	—	2	2	no	—
50	C₄⋊#3622	—	2	2	no	—
51	C₄⋊#3622	—	2	2	no	—
52	C₄⋊#3622	—	2	2	no	—
53	C₂×Q₈×C₁₇	—	2	1	no	—
54	C₂×Q₈×C₁₇	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
58	M₁₆×C₁₇	—	4	1	no	—
59	M₁₆×C₁₇	—	4	1	no	—
60	C₂×C₈×C₁₇	—	2	1	no	—
61	C₂×C₈×C₁₇	—	2	1	no	—
62	C₂×C₈×C₁₇	—	2	2	no	—
63	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
68	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
69	C₄⋊C₄×C₁₇	—	2	2	no	—
70	C₄⋊C₄×C₁₇	—	2	1	no	—
71	C₄⋊C₄×C₁₇	—	2	1	no	—
72	C₄⋊C₄×C₁₇	—	2	1	no	—
73	C₄⋊C₄×C₁₇	—	2	2	no	—
74	C₄⋊C₄×C₁₇	C₄⋊C₄	1	2	yes	—
75	C₂²⋊C₄×C₁₇	—	2	2	no	—
76	C₄²×C₁₇	—	2	2	no	—
77	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₇	#41	1	1	yes	characteristic
81	#63223	—	4	2	no	—
82	#63223	—	4	2	no	—
83	C₂×#3622	—	4	2	no	—
84	C₂×#3622	—	4	2	no	—
85	C₂×#3622	—	8	2	no	—
86	#8685	—	17	1	no	maximal, Sylow
87	C₂³×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
88	Q₈×C₁₇	—	4	2	no	—
89	Q₈×C₁₇	—	4	1	no	—
90	Q₈×C₁₇	—	4	1	no	—
91	C₂×C₄×C₁₇	—	2	1	no	—
92	C₂×C₄×C₁₇	—	2	1	no	—
93	C₂×C₄×C₁₇	—	2	2	no	—
94	C₂×C₄×C₁₇	—	2	2	no	—
95	C₂×C₄×C₁₇	—	4	1	no	—
96	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
99	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
100	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
101	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
104	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
106	C₈×C₁₇	—	2	1	no	—
107	C₈×C₁₇	—	2	1	no	—
108	C₈×C₁₇	—	4	2	no	—
109	C₈×C₁₇	#103	1	2	yes	—
110	C₈×C₁₇	#103	1	2	yes	—
111	#3622	—	8	2	no	—
112	#3622	—	8	2	no	—
113	#2960	D₁₇	1	1	yes	characteristic
114	#1848	—	17	1	no	—
115	C₂×#335	—	17	1	no	—
116	#1550	—	17	2	no	—
117	#1292	—	17	2	no	—
118	C₂²×C₁₇	—	2	1	no	—
119	C₂²×C₁₇	—	2	1	no	—
120	C₂²×C₁₇	#380	1	1	yes	characteristic
121	C₂²×C₁₇	#475	1	1	yes	characteristic
122	C₂²×C₁₇	#477	1	1	yes	characteristic
123	C₄×C₁₇	—	2	1	no	—
124	C₄×C₁₇	—	2	1	no	—
125	C₄×C₁₇	—	2	1	no	—
126	C₄×C₁₇	—	4	2	no	—
127	C₄×C₁₇	—	4	2	no	—
128	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
129	C₄×C₁₇	#417	1	1	yes	characteristic
130	C₄×C₁₇	#460	1	1	yes	characteristic
131	C₄×C₁₇	#462	1	1	yes	characteristic
132	C₄×C₁₇	#472	1	1	yes	characteristic
133	C₄×C₁₇	#473	1	1	yes	characteristic
134	#533	C₂×D₁₇	1	1	yes	characteristic
135	#446	—	2	1	no	—
136	#446	Dic₁₇	1	2	yes	—
137	C₈×Q₈	—	2	1	no	—
138	C₈×Q₈	Dic₁₇	1	2	yes	—
139	C₂×#108	—	34	1	no	—
140	C₂×#108	—	34	1	no	—
141	#424	—	2	1	no	—
142	#423	—	17	2	no	—
143	#423	C₂×D₁₇	1	1	yes	characteristic
144	C₂×C₄⋊C₈	—	17	2	no	—
145	#405	—	2	1	no	—
146	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
147	#335	—	34	1	no	—
148	#335	—	34	1	no	—
149	#333	—	34	2	no	—
150	#332	—	34	2	no	—
151	C₂×C₁₇	—	2	1	no	—
152	C₂×C₁₇	—	2	1	no	—
153	C₂×C₁₇	#1803	1	1	yes	characteristic
154	C₂×C₁₇	#1813	1	1	yes	characteristic
155	C₂×C₁₇	#1816	1	1	yes	characteristic
156	C₂×M₁₆	—	2	1	no	—
157	C₂×M₁₆	—	34	2	no	—
158	C₂²×C₈	—	34	2	no	—
159	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
160	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
161	C₄⋊Q₈	D₆₈	1	1	yes	characteristic
162	C₄⋊Q₈	D₆₈	1	1	yes	characteristic
163	#125	—	2	1	no	—
164	C₂²⋊Q₈	—	2	1	no	—
165	C₂²⋊Q₈	Dic₃₄	1	2	yes	—
166	C₄×Q₈	—	2	1	no	—
167	C₄×Q₈	C₂×Dic₁₇	1	2	yes	—
168	#117	—	2	1	no	—
169	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
170	#108	—	68	2	no	—
171	#108	—	68	2	no	—
172	C₄⋊C₈	—	2	2	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	2	1	no	—
175	C₄⋊C₈	—	34	2	no	—
176	C₄⋊C₈	—	34	2	no	—
177	C₄⋊C₈	—	34	2	no	—
178	C₄⋊C₈	—	34	2	no	—
179	C₄⋊C₈	C₂×Dic₁₇	1	2	yes	—
180	#97	—	2	2	no	—
181	C₄×C₈	—	2	2	no	—
182	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
183	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
184	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
185	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
186	C₁₇	#8685	1	1	yes	characteristic, complemented, Sylow
187	C₂×Q₈	—	2	1	no	—
188	C₂×Q₈	C₄⋊Dic₁₇	1	2	yes	—
189	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
190	C₂²×C₄	#63252	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
192	M₁₆	—	4	1	no	—
193	M₁₆	—	4	1	no	—
194	M₁₆	—	68	2	no	—
195	M₁₆	—	68	2	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	68	2	no	—
200	C₂×C₈	—	68	2	no	—
201	C₂×C₈	—	136	2	no	—
202	C₂×C₈	#63230	1	2	yes	—
203	C₂×C₈	#63231	1	2	yes	—
204	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
205	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
207	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	C₄⋊Dic₁₇	1	2	yes	—
214	C₂²⋊C₄	—	2	2	no	—
215	C₄²	—	2	2	no	—
216	C₄²	C₂×D₆₈	1	1	yes	characteristic
217	C₄²	#63252	1	1	yes	characteristic
218	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
219	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
220	C₂³	#10587610	1	1	yes	characteristic
221	Q₈	—	4	2	no	—
222	Q₈	—	4	1	no	—
223	Q₈	—	4	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	#10587610	1	1	yes	characteristic
230	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
231	C₂×C₄	#10587612	1	1	yes	characteristic
232	C₂×C₄	#10587616	1	1	yes	characteristic
233	C₂×C₄	#10587616	1	1	yes	characteristic
234	C₂×C₄	C₂×#63230	1	1	yes	characteristic
235	C₂×C₄	C₂×#63231	1	1	yes	characteristic
236	C₂×C₄	#10587635	1	1	yes	characteristic
237	C₂×C₄	#10587637	1	1	yes	characteristic
238	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	4	2	no	—
242	C₈	—	136	2	no	—
243	C₈	—	136	2	no	—
244	C₈	#10587506	1	2	yes	—
245	C₈	#10587506	1	2	yes	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	#11762669	1	1	yes	characteristic
249	C₂²	#11763094	1	1	yes	characteristic
250	C₂²	#11763104	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	2	no	—
255	C₄	—	4	2	no	—
256	C₄	C₂×#10587506	1	1	yes	characteristic
257	C₄	#11762810	1	1	yes	characteristic
258	C₄	#11763009	1	1	yes	characteristic
259	C₄	#11763019	1	1	yes	characteristic
260	C₄	#11763079	1	1	yes	characteristic
261	C₄	#11763084	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	#14528789	1	1	yes	characteristic
265	C₂	#14528834	1	1	yes	characteristic
266	C₂	#14528848	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11762669
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#2960	C₂
socle	C₂²×C₁₇	#380
prime radical	C₂³×C₁₇	C₂²⋊Q₈
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#2960	D₁₇
hypocenter	C₁₇	#8685
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
186. C₁₇	#8685	semidirect

Maximal subgroups

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

subgroup	index
#14528956	2
C₂×#11762562	2
#14527474	2
#14526658	2
C₁₇×#2960	2
#8685	17