Subgroups — #16927573

Order 4352.

362 conjugacy classes of subgroups.

The subgroup lattice has 262 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×#11762560	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	#11762560	—	2	1	no	—
11	#11762560	—	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	#8683	—	17	1	no	maximal, Sylow
87	C₂³×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
88	Q₈×C₁₇	—	4	2	no	—
89	Q₈×C₁₇	—	4	2	no	—
90	C₂×C₄×C₁₇	—	2	2	no	—
91	C₂×C₄×C₁₇	—	2	2	no	—
92	C₂×C₄×C₁₇	—	2	2	no	—
93	C₂×C₄×C₁₇	—	4	1	no	—
94	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
97	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
98	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
99	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
102	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
104	C₈×C₁₇	—	2	1	no	—
105	C₈×C₁₇	—	2	1	no	—
106	C₈×C₁₇	—	4	2	no	—
107	C₈×C₁₇	#103	1	2	yes	—
108	C₈×C₁₇	#103	1	2	yes	—
109	#3622	—	8	2	no	—
110	#3622	—	8	2	no	—
111	#2960	D₁₇	1	1	yes	characteristic
112	#1848	—	17	1	no	—
113	C₂×#334	—	17	1	no	—
114	#1550	—	17	2	no	—
115	#1292	—	17	2	no	—
116	C₂²×C₁₇	—	2	1	no	—
117	C₂²×C₁₇	—	2	1	no	—
118	C₂²×C₁₇	#380	1	1	yes	characteristic
119	C₂²×C₁₇	#475	1	1	yes	characteristic
120	C₂²×C₁₇	#477	1	1	yes	characteristic
121	C₄×C₁₇	—	2	1	no	—
122	C₄×C₁₇	—	2	1	no	—
123	C₄×C₁₇	—	2	1	no	—
124	C₄×C₁₇	—	4	2	no	—
125	C₄×C₁₇	—	4	2	no	—
126	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
127	C₄×C₁₇	#417	1	1	yes	characteristic
128	C₄×C₁₇	#460	1	1	yes	characteristic
129	C₄×C₁₇	#462	1	1	yes	characteristic
130	C₄×C₁₇	#471	1	2	yes	—
131	#533	C₂×D₁₇	1	1	yes	characteristic
132	#446	—	2	1	no	—
133	#446	Dic₁₇	1	2	yes	—
134	C₈×Q₈	—	2	1	no	—
135	C₈×Q₈	Dic₁₇	1	2	yes	—
136	C₂×#108	—	34	1	no	—
137	C₂×#108	—	34	1	no	—
138	#424	—	2	1	no	—
139	#423	—	17	2	no	—
140	#423	C₂×D₁₇	1	1	yes	characteristic
141	C₂×C₄⋊C₈	—	17	2	no	—
142	#405	—	2	1	no	—
143	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
144	#334	—	34	1	no	—
145	#334	—	34	1	no	—
146	#333	—	34	2	no	—
147	#332	—	34	2	no	—
148	C₂×C₁₇	—	2	1	no	—
149	C₂×C₁₇	—	2	1	no	—
150	C₂×C₁₇	#1803	1	1	yes	characteristic
151	C₂×C₁₇	#1813	1	1	yes	characteristic
152	C₂×C₁₇	#1815	1	1	yes	characteristic
153	C₂×M₁₆	—	2	1	no	—
154	C₂×M₁₆	—	34	2	no	—
155	C₂²×C₈	—	34	2	no	—
156	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
157	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
158	C₄⋊Q₈	D₆₈	1	1	yes	characteristic
159	C₄⋊Q₈	D₆₈	1	1	yes	characteristic
160	#125	—	2	1	no	—
161	C₂²⋊Q₈	—	2	1	no	—
162	C₂²⋊Q₈	Dic₃₄	1	2	yes	—
163	C₄×Q₈	—	2	1	no	—
164	C₄×Q₈	C₂×Dic₁₇	1	2	yes	—
165	#117	—	2	1	no	—
166	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
167	#108	—	68	2	no	—
168	#108	—	68	2	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	1	no	—
172	C₄⋊C₈	—	34	2	no	—
173	C₄⋊C₈	—	34	2	no	—
174	C₄⋊C₈	—	34	2	no	—
175	C₄⋊C₈	—	34	2	no	—
176	C₄⋊C₈	C₂×Dic₁₇	1	2	yes	—
177	#97	—	2	2	no	—
178	C₄×C₈	—	2	2	no	—
179	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
180	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
183	C₁₇	#8683	1	1	yes	characteristic, complemented, Sylow
184	C₂×Q₈	—	2	1	no	—
185	C₂×Q₈	C₄⋊Dic₁₇	1	2	yes	—
186	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
187	C₂²×C₄	#63252	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
189	M₁₆	—	4	1	no	—
190	M₁₆	—	4	1	no	—
191	M₁₆	—	68	2	no	—
192	M₁₆	—	68	2	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	68	2	no	—
197	C₂×C₈	—	68	2	no	—
198	C₂×C₈	—	136	2	no	—
199	C₂×C₈	#63230	1	2	yes	—
200	C₂×C₈	#63231	1	2	yes	—
201	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	C₄⋊Dic₁₇	1	2	yes	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	C₂×D₆₈	1	1	yes	characteristic
214	C₄²	#63252	1	1	yes	characteristic
215	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
216	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
217	C₂³	#10587610	1	1	yes	characteristic
218	Q₈	—	4	2	no	—
219	Q₈	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	#10587610	1	1	yes	characteristic
225	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
226	C₂×C₄	#10587612	1	1	yes	characteristic
227	C₂×C₄	#10587616	1	1	yes	characteristic
228	C₂×C₄	#10587616	1	1	yes	characteristic
229	C₂×C₄	C₂×#63230	1	1	yes	characteristic
230	C₂×C₄	C₂×#63231	1	1	yes	characteristic
231	C₂×C₄	#10587635	1	1	yes	characteristic
232	C₂×C₄	#10587637	1	1	yes	characteristic
233	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	4	2	no	—
237	C₈	—	136	2	no	—
238	C₈	—	136	2	no	—
239	C₈	#10587506	1	2	yes	—
240	C₈	#10587506	1	2	yes	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	#11762669	1	1	yes	characteristic
244	C₂²	#11763094	1	1	yes	characteristic
245	C₂²	#11763104	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	2	no	—
251	C₄	C₂×#10587506	1	1	yes	characteristic
252	C₄	#11762810	1	1	yes	characteristic
253	C₄	#11763009	1	1	yes	characteristic
254	C₄	#11763019	1	1	yes	characteristic
255	C₄	#11763072	1	2	yes	—
256	C₂	—	2	1	no	—
257	C₂	—	2	1	no	—
258	C₂	#14528789	1	1	yes	characteristic
259	C₂	#14528834	1	1	yes	characteristic
260	C₂	#14528843	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	#8683
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
183. C₁₇	#8683	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×#11762560	2
#14527474	2
#14526658	2
C₁₇×#2960	2
#8683	17