Subgroups — #16939039

Order 4352.

366 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528944	C₂	1	1	yes	maximal, characteristic
2	#14528789	C₂	1	2	yes	maximal
3	#14526668	C₂	1	2	yes	maximal
4	#14526619	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1984	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587489	—	2	1	no	—
7	C₂×#10587489	—	2	1	no	—
8	C₂×C₄⋊#3622	C₂²	1	2	yes	—
9	C₂×#10587506	—	2	2	no	—
10	#11762679	C₂²	1	2	yes	—
11	#11762571	—	2	2	no	—
12	C₂×C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₁₇×#336	—	2	1	no	—
16	C₁₇×#336	—	2	1	no	—
17	C₂²×Q₈×C₁₇	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₇	—	2	1	no	—
19	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
21	C₄×Q₈×C₁₇	—	2	2	no	—
22	C₄×Q₈×C₁₇	—	2	2	no	—
23	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
24	C₂×C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₇	—	2	1	no	—
26	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₇	—	2	2	no	—
28	C₂×C₄⋊Dic₁₇	—	2	2	no	—
29	C₂×C₄⋊Dic₁₇	—	2	2	no	—
30	C₂²×#3622	—	2	2	no	—
31	C₁₇×C₄⋊C₈	—	2	2	no	—
32	C₄×C₈×C₁₇	—	2	2	no	—
33	#10587506	—	4	4	no	—
34	#10587489	—	4	2	no	—
35	#10587489	—	4	2	no	—
36	C₄⋊#3622	—	2	2	no	—
37	C₄⋊#3622	—	2	2	no	—
38	C₂×Q₈×C₁₇	—	2	1	no	—
39	C₂×Q₈×C₁₇	—	2	2	no	—
40	C₂×Q₈×C₁₇	—	2	2	no	—
41	C₂×Q₈×C₁₇	Q₁₆	1	2	yes	—
42	C₂²×C₄×C₁₇	—	2	1	no	—
43	C₂²×C₄×C₁₇	—	2	1	no	—
44	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
47	C₂²×Dic₁₇	—	4	2	no	—
48	C₂×C₈×C₁₇	—	2	1	no	—
49	C₂×C₈×C₁₇	—	2	1	no	—
50	C₂×C₈×C₁₇	—	2	1	no	—
51	C₂×C₈×C₁₇	—	2	1	no	—
52	C₂×C₈×C₁₇	—	4	1	no	—
53	C₂×C₈×C₁₇	—	4	1	no	—
54	C₂×C₈×C₁₇	—	4	1	no	—
55	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
56	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
57	C₄⋊C₄×C₁₇	—	2	2	no	—
58	C₄⋊C₄×C₁₇	—	4	2	no	—
59	C₄²×C₁₇	—	4	1	no	—
60	C₄²×C₁₇	—	4	1	no	—
61	C₄²×C₁₇	#41	1	2	yes	—
62	C₄²×C₁₇	#41	1	2	yes	—
63	C₄⋊Dic₁₇	—	4	2	no	—
64	C₄⋊Dic₁₇	—	4	2	no	—
65	C₄⋊Dic₁₇	—	4	2	no	—
66	C₄⋊Dic₁₇	—	4	2	no	—
67	C₄⋊Dic₁₇	—	8	2	no	—
68	C₂×#3622	—	4	2	no	—
69	C₂×#3622	—	4	2	no	—
70	C₂×#3622	—	4	2	no	—
71	#10618	—	17	1	no	maximal, Sylow
72	C₂³×C₁₇	#121	1	1	yes	characteristic
73	Q₈×C₁₇	—	2	4	no	—
74	Q₈×C₁₇	—	4	2	no	—
75	C₂×C₄×C₁₇	—	2	1	no	—
76	C₂×C₄×C₁₇	—	2	1	no	—
77	C₂×C₄×C₁₇	—	2	2	no	—
78	C₂×C₄×C₁₇	—	4	1	no	—
79	C₂×C₄×C₁₇	—	4	1	no	—
80	C₂×C₄×C₁₇	—	4	1	no	—
81	C₂×C₄×C₁₇	—	4	1	no	—
82	C₂×C₄×C₁₇	—	4	1	no	—
83	C₂×C₄×C₁₇	#123	1	2	yes	—
84	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
85	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
86	C₂×C₄×C₁₇	#126	1	2	yes	—
87	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
90	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
91	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
92	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
93	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
94	C₂×Dic₁₇	—	8	2	no	—
95	C₂×Dic₁₇	—	8	2	no	—
96	C₂×Dic₁₇	—	8	2	no	—
97	C₈×C₁₇	—	2	2	no	—
98	C₈×C₁₇	—	2	2	no	—
99	C₈×C₁₇	—	8	1	no	—
100	#3622	—	8	2	no	—
101	#1984	D₁₇	1	1	yes	characteristic
102	#1845	—	17	1	no	—
103	#1803	—	17	2	no	—
104	#1296	—	17	2	no	—
105	#1277	—	17	1	no	—
106	C₂²×C₁₇	#397	1	1	yes	characteristic
107	C₂²×C₁₇	#460	1	1	yes	characteristic
108	C₂²×C₁₇	#462	1	1	yes	characteristic
109	C₂²×C₁₇	#467	1	1	yes	characteristic
110	C₂²×C₁₇	#469	1	1	yes	characteristic
111	C₂²×C₁₇	#472	1	1	yes	characteristic
112	C₂²×C₁₇	#473	1	1	yes	characteristic
113	C₄×C₁₇	—	2	2	no	—
114	C₄×C₁₇	—	4	2	no	—
115	C₄×C₁₇	—	8	1	no	—
116	C₄×C₁₇	#482	1	2	yes	—
117	C₄×C₁₇	#483	1	1	yes	characteristic
118	C₄×C₁₇	#484	1	1	yes	characteristic
119	C₄×C₁₇	#484	1	2	yes	—
120	C₄×C₁₇	#487	1	1	yes	characteristic
121	C₄×C₁₇	#489	1	1	yes	characteristic
122	Dic₁₇	—	16	2	no	—
123	C₂×C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
124	C₂×#107	—	34	1	no	—
125	C₂×#107	—	34	1	no	—
126	C₂×C₄⋊C₈	—	17	2	no	—
127	C₂×C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
128	C₂×#103	—	34	2	no	—
129	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
130	#384	—	17	2	no	—
131	#340	—	34	2	no	—
132	#336	—	2	1	no	—
133	#336	—	2	1	no	—
134	C₂×C₁₇	#1670	1	1	yes	characteristic
135	C₂×C₁₇	#1676	1	1	yes	characteristic
136	C₂×C₁₇	#1688	1	1	yes	characteristic
137	C₂×C₁₇	#1694	1	1	yes	characteristic
138	C₂×C₁₇	#2072	1	1	yes	characteristic
139	C₂×C₁₇	#2076	1	1	yes	characteristic
140	C₂×C₁₇	#2078	1	1	yes	characteristic
141	C₂²×Q₈	D₆₈	1	1	yes	characteristic
142	C₂²×C₈	—	2	1	no	—
143	C₂²×C₈	—	34	2	no	—
144	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
145	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
146	C₄×Q₈	—	2	2	no	—
147	C₄×Q₈	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	—	34	2	no	—
150	C₂×C₄⋊C₄	—	34	2	no	—
151	C₂×C₄⋊C₄	—	34	2	no	—
152	C₂×C₄⋊C₄	D₆₈	1	1	yes	characteristic
153	C₂×C₄²	—	2	1	no	—
154	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
155	#107	—	68	2	no	—
156	#107	—	68	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	34	2	no	—
159	C₄⋊C₈	—	34	2	no	—
160	#103	—	68	4	no	—
161	C₄×C₈	—	2	2	no	—
162	C₁₇	#10618	1	1	yes	characteristic, complemented, Sylow
163	C₂×Q₈	—	2	1	no	—
164	C₂×Q₈	—	2	2	no	—
165	C₂×Q₈	—	2	2	no	—
166	C₂×Q₈	Dic₆₈	1	2	yes	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	68	2	no	—
170	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
171	C₂²×C₄	#63252	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	4	1	no	—
180	C₂×C₈	—	68	2	no	—
181	C₂×C₈	—	68	2	no	—
182	C₂×C₈	—	68	2	no	—
183	C₂×C₈	#63230	1	2	yes	—
184	C₂×C₈	#63231	1	2	yes	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	68	2	no	—
188	C₄⋊C₄	—	68	2	no	—
189	C₄⋊C₄	—	68	2	no	—
190	C₄⋊C₄	—	68	2	no	—
191	C₄⋊C₄	—	136	2	no	—
192	C₄²	—	4	1	no	—
193	C₄²	—	4	1	no	—
194	C₄²	#63254	1	2	yes	—
195	C₄²	#63254	1	2	yes	—
196	C₂³	#10587616	1	1	yes	characteristic
197	Q₈	—	2	4	no	—
198	Q₈	—	4	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	136	2	no	—
208	C₂×C₄	—	136	2	no	—
209	C₂×C₄	—	136	2	no	—
210	C₂×C₄	#10587565	1	2	yes	—
211	C₂×C₄	#10587572	1	2	yes	—
212	C₂×C₄	#10587590	1	1	yes	characteristic
213	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
214	C₂×C₄	#10587594	1	2	yes	—
215	C₂×C₄	#10587621	1	1	yes	characteristic
216	C₂×C₄	#10587622	1	1	yes	characteristic
217	C₂×C₄	C₂×#63230	1	1	yes	characteristic
218	C₂×C₄	C₂×#63231	1	1	yes	characteristic
219	C₂×C₄	#10587635	1	1	yes	characteristic
220	C₂×C₄	#10587637	1	1	yes	characteristic
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	—	8	1	no	—
224	C₈	—	136	2	no	—
225	C₂²	#11762734	1	1	yes	characteristic
226	C₂²	#11763009	1	1	yes	characteristic
227	C₂²	#11763019	1	1	yes	characteristic
228	C₂²	#11763050	1	1	yes	characteristic
229	C₂²	#11763062	1	1	yes	characteristic
230	C₂²	#11763079	1	1	yes	characteristic
231	C₂²	#11763084	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	4	2	no	—
234	C₄	—	8	1	no	—
235	C₄	—	272	2	no	—
236	C₄	#11763134	1	2	yes	—
237	C₄	#11763137	1	1	yes	characteristic
238	C₄	#11763142	1	1	yes	characteristic
239	C₄	#11763145	1	2	yes	—
240	C₄	#11763159	1	1	yes	characteristic
241	C₄	#11763168	1	1	yes	characteristic
242	C₂	#14528154	1	1	yes	characteristic
243	C₂	#14528188	1	1	yes	characteristic
244	C₂	#14528268	1	1	yes	characteristic
245	C₂	#14528302	1	1	yes	characteristic
246	C₂	#14530192	1	1	yes	characteristic
247	C₂	#14530212	1	1	yes	characteristic
248	C₂	#14530224	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10587616
commutator, cosocle	C₂×C₄²×C₁₇	C₂³
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1984	C₂
socle, prime radical	C₂³×C₁₇	#121
hypercenter	#1984	D₁₇
hypocenter	C₁₇	#10618
LCS	G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1984

Complements

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

N	H ≅ G/N	type
162. C₁₇	#10618	semidirect

Maximal subgroups

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

subgroup	index
#14528944	2
#14528789	2
#14526668	2
#14526619	2
C₁₇×#1984	2
#10618	17