Subgroups — #16939035

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	#14529720	C₂	1	1	yes	maximal, characteristic
2	#14528786	C₂	1	2	yes	maximal
3	#14526667	C₂	1	2	yes	maximal
4	#14526619	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₃₄	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊#3622	C₂²	1	2	yes	—
8	C₂×C₄⋊#3622	C₂²	1	1	yes	characteristic
9	C₂×#10587481	—	2	2	no	—
10	#11762571	—	2	2	no	—
11	#11762563	—	2	1	no	—
12	#11762563	—	2	1	no	—
13	C₂×C₁₇×#107	—	2	1	no	—
14	C₂×C₁₇×#107	—	2	1	no	—
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₁₇×#384	C₂²	1	2	yes	—
17	C₂²×Dic₃₄	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
21	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
22	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
23	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
25	C₂×#63225	—	2	1	no	—
26	C₂×C₄×Dic₁₇	—	2	1	no	—
27	C₂²×#3622	—	2	2	no	—
28	C₂²×#3622	—	2	1	no	—
29	C₄×Dic₃₄	—	2	2	no	—
30	C₄×Dic₃₄	—	2	2	no	—
31	C₁₇×#107	—	4	2	no	—
32	C₁₇×#107	—	4	2	no	—
33	C₄×C₈×C₁₇	—	2	2	no	—
34	#10587481	—	4	4	no	—
35	C₄⋊#3622	—	2	2	no	—
36	C₄⋊#3622	—	2	2	no	—
37	C₄⋊#3622	—	2	2	no	—
38	C₂²×C₄×C₁₇	—	4	2	no	—
39	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
42	C₂²×Dic₁₇	—	2	1	no	—
43	C₂²×Dic₁₇	—	2	1	no	—
44	C₂×Dic₃₄	—	2	2	no	—
45	C₂×Dic₃₄	—	2	2	no	—
46	C₂×Dic₃₄	—	2	1	no	—
47	C₂×Dic₃₄	D₈	1	2	yes	—
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	SD₁₆	1	2	yes	—
53	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
54	C₄⋊C₄×C₁₇	—	4	2	no	—
55	C₄⋊C₄×C₁₇	—	4	2	no	—
56	C₄⋊C₄×C₁₇	—	4	2	no	—
57	C₄⋊C₄×C₁₇	—	4	2	no	—
58	C₄⋊C₄×C₁₇	—	8	2	no	—
59	C₄²×C₁₇	#41	1	2	yes	—
60	C₄²×C₁₇	#41	1	2	yes	—
61	C₄⋊Dic₁₇	—	2	2	no	—
62	#63225	—	4	2	no	—
63	C₄×Dic₁₇	—	4	1	no	—
64	C₄×Dic₁₇	—	4	1	no	—
65	C₂×#3622	—	4	2	no	—
66	C₂×#3622	—	4	1	no	—
67	C₂×#3622	—	4	2	no	—
68	C₂×#3622	—	4	1	no	—
69	C₂×#3622	—	4	2	no	—
70	C₂×#3622	—	4	1	no	—
71	#10617	—	17	1	no	maximal, Sylow
72	C₂³×C₁₇	#121	1	1	yes	characteristic
73	C₂×C₄×C₁₇	—	2	1	no	—
74	C₂×C₄×C₁₇	—	2	1	no	—
75	C₂×C₄×C₁₇	—	8	2	no	—
76	C₂×C₄×C₁₇	—	8	2	no	—
77	C₂×C₄×C₁₇	—	8	2	no	—
78	C₂×C₄×C₁₇	#123	1	2	yes	—
79	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
80	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
81	C₂×C₄×C₁₇	#126	1	2	yes	—
82	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
85	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
86	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
87	C₂×C₄×C₁₇	#137	1	2	yes	—
88	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
89	C₂×Dic₁₇	—	2	2	no	—
90	C₂×Dic₁₇	—	4	1	no	—
91	C₂×Dic₁₇	—	4	1	no	—
92	C₂×Dic₁₇	—	4	1	no	—
93	C₂×Dic₁₇	—	4	1	no	—
94	C₂×Dic₁₇	—	4	1	no	—
95	Dic₃₄	—	2	4	no	—
96	Dic₃₄	—	4	2	no	—
97	C₈×C₁₇	—	2	2	no	—
98	C₈×C₁₇	—	2	2	no	—
99	#3622	—	8	2	no	—
100	#3622	—	8	1	no	—
101	#1984	—	17	1	no	—
102	#1845	D₁₇	1	1	yes	characteristic
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	#466	1	1	yes	characteristic
110	C₂²×C₁₇	#470	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	—	16	2	no	—
115	C₄×C₁₇	#480	1	2	yes	—
116	C₄×C₁₇	#483	1	1	yes	characteristic
117	C₄×C₁₇	#484	1	1	yes	characteristic
118	C₄×C₁₇	#485	1	2	yes	—
119	C₄×C₁₇	#487	1	1	yes	characteristic
120	C₄×C₁₇	#489	1	1	yes	characteristic
121	Dic₁₇	—	4	2	no	—
122	Dic₁₇	—	8	1	no	—
123	C₂×C₄×Q₈	—	17	1	no	—
124	C₂×#107	—	2	1	no	—
125	C₂×#107	—	2	1	no	—
126	C₂×C₄⋊C₈	—	17	2	no	—
127	C₂×C₄⋊C₈	—	17	1	no	—
128	C₂×#103	—	34	2	no	—
129	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
130	#384	C₂×D₁₇	1	2	yes	—
131	#340	—	34	2	no	—
132	#336	—	34	1	no	—
133	#336	—	34	1	no	—
134	C₂×C₁₇	#1668	1	1	yes	characteristic
135	C₂×C₁₇	#1674	1	1	yes	characteristic
136	C₂×C₁₇	#1690	1	1	yes	characteristic
137	C₂×C₁₇	#1696	1	1	yes	characteristic
138	C₂×C₁₇	#2072	1	1	yes	characteristic
139	C₂×C₁₇	#2075	1	1	yes	characteristic
140	C₂×C₁₇	#2078	1	1	yes	characteristic
141	C₂²×Q₈	—	17	1	no	—
142	C₂²×C₈	—	34	2	no	—
143	C₂²×C₈	—	34	1	no	—
144	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
145	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
146	C₄×Q₈	—	34	2	no	—
147	C₄×Q₈	—	34	2	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄⋊C₄	—	17	1	no	—
152	C₂×C₄⋊C₄	—	34	1	no	—
153	C₂×C₄²	—	34	1	no	—
154	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
155	#107	—	4	2	no	—
156	#107	—	4	2	no	—
157	C₄⋊C₈	—	34	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	#10617	1	1	yes	characteristic, complemented, Sylow
163	C₂×Q₈	—	17	2	no	—
164	C₂×Q₈	—	34	1	no	—
165	C₂×Q₈	—	34	2	no	—
166	C₂×Q₈	—	34	2	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	—	34	1	no	—
169	C₂²×C₄	—	34	1	no	—
170	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
171	C₂²×C₄	#63254	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	—	68	2	no	—
178	C₂×C₈	—	68	2	no	—
179	C₂×C₈	—	68	2	no	—
180	C₂×C₈	—	68	1	no	—
181	C₂×C₈	—	68	1	no	—
182	C₂×C₈	—	68	1	no	—
183	C₂×C₈	#63229	1	2	yes	—
184	C₂×C₈	#63231	1	2	yes	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	8	2	no	—
190	C₄⋊C₄	—	34	2	no	—
191	C₄⋊C₄	—	68	2	no	—
192	C₄²	—	68	1	no	—
193	C₄²	—	68	1	no	—
194	C₄²	#63252	1	2	yes	—
195	C₄²	#63256	1	2	yes	—
196	C₂³	#10587624	1	1	yes	characteristic
197	Q₈	—	34	4	no	—
198	Q₈	—	68	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	8	2	no	—
202	C₂×C₄	—	8	2	no	—
203	C₂×C₄	—	8	2	no	—
204	C₂×C₄	—	34	2	no	—
205	C₂×C₄	—	68	1	no	—
206	C₂×C₄	—	68	1	no	—
207	C₂×C₄	—	68	1	no	—
208	C₂×C₄	—	68	1	no	—
209	C₂×C₄	—	68	1	no	—
210	C₂×C₄	#10587580	1	2	yes	—
211	C₂×C₄	#10587584	1	2	yes	—
212	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
213	C₂×C₄	#10587598	1	1	yes	characteristic
214	C₂×C₄	#10587604	1	2	yes	—
215	C₂×C₄	#10587615	1	1	yes	characteristic
216	C₂×C₄	#10587618	1	1	yes	characteristic
217	C₂×C₄	C₂×#63229	1	1	yes	characteristic
218	C₂×C₄	#10587628	1	1	yes	characteristic
219	C₂×C₄	C₂×#63231	1	1	yes	characteristic
220	C₂×C₄	#10587633	1	1	yes	characteristic
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	—	136	2	no	—
224	C₈	—	136	1	no	—
225	C₂²	#11762735	1	1	yes	characteristic
226	C₂²	#11763006	1	1	yes	characteristic
227	C₂²	#11763016	1	1	yes	characteristic
228	C₂²	#11763042	1	1	yes	characteristic
229	C₂²	#11763064	1	1	yes	characteristic
230	C₂²	#11763076	1	1	yes	characteristic
231	C₂²	#11763081	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	16	2	no	—
234	C₄	—	68	2	no	—
235	C₄	—	136	1	no	—
236	C₄	#11763124	1	2	yes	—
237	C₄	#11763139	1	1	yes	characteristic
238	C₄	#11763144	1	1	yes	characteristic
239	C₄	#11763152	1	2	yes	—
240	C₄	#11763161	1	1	yes	characteristic
241	C₄	#11763171	1	1	yes	characteristic
242	C₂	#14528141	1	1	yes	characteristic
243	C₂	#14528175	1	1	yes	characteristic
244	C₂	#14528275	1	1	yes	characteristic
245	C₂	#14528309	1	1	yes	characteristic
246	C₂	#14530193	1	1	yes	characteristic
247	C₂	#14530208	1	1	yes	characteristic
248	C₂	#14530225	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10587624
commutator, cosocle	C₂×C₄²×C₁₇	C₂³
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1845	C₂
socle, prime radical	C₂³×C₁₇	#121
hypercenter	#1845	D₁₇
hypocenter	C₁₇	#10617
LCS	G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
162. C₁₇	#10617	semidirect

Maximal subgroups

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

subgroup	index
#14529720	2
#14528786	2
#14526667	2
#14526619	2
C₁₇×#1845	2
#10617	17