Subgroups — #16939230

Order 4352.

350 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14529542	C₂	1	2	yes	maximal
2	#14528944	C₂	1	1	yes	maximal, characteristic
3	#14526668	C₂	1	2	yes	maximal
4	#14526619	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1939	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587489	—	2	1	no	—
7	C₂×#10587489	—	2	1	no	—
8	C₂×#10587478	—	2	2	no	—
9	C₂×C₄⋊#3622	C₂²	1	2	yes	—
10	#11762679	C₂²	1	2	yes	—
11	#11762571	—	2	2	no	—
12	C₂×C₁₇×C₄⋊C₈	—	2	1	no	—
13	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
16	C₁₇×#336	—	2	1	no	—
17	C₂²×C₈×C₁₇	—	2	1	no	—
18	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₇	—	2	1	no	—
23	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₇	—	2	2	no	—
25	C₂×C₄⋊Dic₁₇	—	2	2	no	—
26	C₂×C₄⋊Dic₁₇	—	2	2	no	—
27	C₂²×#3622	—	2	2	no	—
28	C₁₇×C₄⋊C₈	—	2	2	no	—
29	C₁₇×C₄⋊C₈	—	4	2	no	—
30	C₄×C₈×C₁₇	—	2	2	no	—
31	C₁₇×#95	—	2	1	no	—
32	#10587489	—	4	2	no	—
33	#10587489	—	4	2	no	—
34	#10587478	—	4	4	no	—
35	C₄⋊#3622	—	2	2	no	—
36	C₄⋊#3622	Q₈	1	4	yes	—
37	C₂²×C₄×C₁₇	—	2	1	no	—
38	C₂²×C₄×C₁₇	—	2	1	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	—	4	2	no	—
43	C₂×C₈×C₁₇	—	2	1	no	—
44	C₂×C₈×C₁₇	—	2	1	no	—
45	C₂×C₈×C₁₇	—	2	1	no	—
46	C₂×C₈×C₁₇	—	2	1	no	—
47	C₂×C₈×C₁₇	—	4	1	no	—
48	C₂×C₈×C₁₇	—	4	1	no	—
49	C₂×C₈×C₁₇	—	4	1	no	—
50	C₂×C₈×C₁₇	D₈	1	2	yes	—
51	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
52	C₄⋊C₄×C₁₇	—	2	1	no	—
53	C₄⋊C₄×C₁₇	—	2	2	no	—
54	C₄⋊C₄×C₁₇	D₈	1	2	yes	—
55	C₄²×C₁₇	—	4	1	no	—
56	C₄²×C₁₇	—	4	1	no	—
57	C₄²×C₁₇	C₂×Q₈	1	2	yes	—
58	C₄²×C₁₇	#41	1	2	yes	—
59	C₄⋊Dic₁₇	—	4	2	no	—
60	C₄⋊Dic₁₇	—	4	2	no	—
61	C₄⋊Dic₁₇	—	4	2	no	—
62	C₄⋊Dic₁₇	—	4	2	no	—
63	C₄⋊Dic₁₇	—	8	2	no	—
64	C₂×#3622	—	2	4	no	—
65	C₂×#3622	—	4	2	no	—
66	C₂×#3622	—	4	2	no	—
67	#10649	—	17	1	no	maximal, Sylow
68	C₂³×C₁₇	#121	1	1	yes	characteristic
69	C₂×C₄×C₁₇	—	2	1	no	—
70	C₂×C₄×C₁₇	—	2	1	no	—
71	C₂×C₄×C₁₇	—	2	2	no	—
72	C₂×C₄×C₁₇	—	4	1	no	—
73	C₂×C₄×C₁₇	—	4	1	no	—
74	C₂×C₄×C₁₇	—	4	1	no	—
75	C₂×C₄×C₁₇	—	4	1	no	—
76	C₂×C₄×C₁₇	—	4	1	no	—
77	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
78	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
79	C₂×C₄×C₁₇	#125	1	2	yes	—
80	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
84	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
85	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
86	C₂×C₄×C₁₇	#137	1	2	yes	—
87	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
88	C₂×Dic₁₇	—	8	2	no	—
89	C₂×Dic₁₇	—	8	2	no	—
90	C₂×Dic₁₇	—	8	2	no	—
91	C₈×C₁₇	—	2	2	no	—
92	C₈×C₁₇	—	2	2	no	—
93	C₈×C₁₇	—	8	1	no	—
94	#3622	—	4	4	no	—
95	#1947	—	17	2	no	—
96	#1939	D₁₇	1	1	yes	characteristic
97	#1845	—	17	1	no	—
98	#1296	—	17	2	no	—
99	#1277	—	17	1	no	—
100	C₂²×C₁₇	#398	1	1	yes	characteristic
101	C₂²×C₁₇	#459	1	1	yes	characteristic
102	C₂²×C₁₇	#462	1	1	yes	characteristic
103	C₂²×C₁₇	#466	1	1	yes	characteristic
104	C₂²×C₁₇	#470	1	1	yes	characteristic
105	C₂²×C₁₇	#472	1	1	yes	characteristic
106	C₂²×C₁₇	#473	1	1	yes	characteristic
107	C₄×C₁₇	—	2	2	no	—
108	C₄×C₁₇	—	4	2	no	—
109	C₄×C₁₇	—	8	1	no	—
110	C₄×C₁₇	#474	1	2	yes	—
111	C₄×C₁₇	#479	1	2	yes	—
112	C₄×C₁₇	#480	1	1	yes	characteristic
113	C₄×C₁₇	#484	1	1	yes	characteristic
114	C₄×C₁₇	#500	1	1	yes	characteristic
115	C₄×C₁₇	#501	1	1	yes	characteristic
116	Dic₁₇	—	16	2	no	—
117	C₂×#107	—	34	1	no	—
118	C₂×#107	—	34	1	no	—
119	C₂×#107	—	34	2	no	—
120	C₂×C₄⋊C₈	—	2	1	no	—
121	C₂×C₄⋊C₈	—	17	2	no	—
122	C₂×C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
123	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
124	#384	—	17	2	no	—
125	C₄×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
126	#340	—	34	2	no	—
127	#336	—	2	1	no	—
128	C₂×C₁₇	#1667	1	1	yes	characteristic
129	C₂×C₁₇	#1674	1	1	yes	characteristic
130	C₂×C₁₇	#1689	1	1	yes	characteristic
131	C₂×C₁₇	#1696	1	1	yes	characteristic
132	C₂×C₁₇	#2083	1	1	yes	characteristic
133	C₂×C₁₇	#2086	1	1	yes	characteristic
134	C₂×C₁₇	#2089	1	1	yes	characteristic
135	C₂²×C₈	—	2	1	no	—
136	C₂²×C₈	—	34	2	no	—
137	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
138	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
139	C₂×C₄⋊C₄	—	34	2	no	—
140	C₂×C₄⋊C₄	—	34	2	no	—
141	C₂×C₄⋊C₄	—	34	2	no	—
142	C₂×C₄⋊C₄	D₆₈	1	1	yes	characteristic
143	C₂×C₄⋊C₄	D₆₈	1	1	yes	characteristic
144	C₂×C₄²	—	2	1	no	—
145	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
146	#107	—	68	2	no	—
147	#107	—	68	2	no	—
148	#107	—	68	4	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	4	2	no	—
151	C₄⋊C₈	—	17	4	no	—
152	C₄⋊C₈	—	34	2	no	—
153	C₄×C₈	—	2	2	no	—
154	#95	—	2	1	no	—
155	C₁₇	#10649	1	1	yes	characteristic, complemented, Sylow
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	68	2	no	—
159	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
160	C₂²×C₄	#63252	1	1	yes	characteristic
161	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	34	4	no	—
170	C₂×C₈	—	68	2	no	—
171	C₂×C₈	—	68	2	no	—
172	C₂×C₈	#63228	1	2	yes	—
173	C₂×C₈	#63231	1	2	yes	—
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	68	2	no	—
177	C₄⋊C₄	—	68	2	no	—
178	C₄⋊C₄	—	68	2	no	—
179	C₄⋊C₄	—	68	2	no	—
180	C₄⋊C₄	—	136	2	no	—
181	C₄⋊C₄	D₁₃₆	1	2	yes	—
182	C₄²	—	4	1	no	—
183	C₄²	—	4	1	no	—
184	C₄²	#63254	1	2	yes	—
185	C₄²	Q₈×D₁₇	1	2	yes	—
186	C₂³	#10587616	1	1	yes	characteristic
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	136	2	no	—
196	C₂×C₄	—	136	2	no	—
197	C₂×C₄	—	136	2	no	—
198	C₂×C₄	#10587575	1	2	yes	—
199	C₂×C₄	#10587583	1	2	yes	—
200	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
201	C₂×C₄	#10587590	1	1	yes	characteristic
202	C₂×C₄	#10587595	1	2	yes	—
203	C₂×C₄	C₂×#63228	1	1	yes	characteristic
204	C₂×C₄	#10587621	1	1	yes	characteristic
205	C₂×C₄	C₂×#63231	1	1	yes	characteristic
206	C₂×C₄	#10587630	1	1	yes	characteristic
207	C₂×C₄	#10587635	1	1	yes	characteristic
208	C₂×C₄	#10587637	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₈	—	8	1	no	—
212	C₈	—	68	4	no	—
213	C₂²	#11762739	1	1	yes	characteristic
214	C₂²	#11763004	1	1	yes	characteristic
215	C₂²	#11763019	1	1	yes	characteristic
216	C₂²	#11763045	1	1	yes	characteristic
217	C₂²	#11763067	1	1	yes	characteristic
218	C₂²	#11763079	1	1	yes	characteristic
219	C₂²	#11763084	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	4	2	no	—
222	C₄	—	8	1	no	—
223	C₄	—	272	2	no	—
224	C₄	#11763090	1	2	yes	—
225	C₄	#11763117	1	2	yes	—
226	C₄	#11763120	1	1	yes	characteristic
227	C₄	#11763142	1	1	yes	characteristic
228	C₄	#11763212	1	1	yes	characteristic
229	C₄	#11763219	1	1	yes	characteristic
230	C₂	#14528139	1	1	yes	characteristic
231	C₂	#14528178	1	1	yes	characteristic
232	C₂	#14528273	1	1	yes	characteristic
233	C₂	#14528312	1	1	yes	characteristic
234	C₂	#14530255	1	1	yes	characteristic
235	C₂	#14530272	1	1	yes	characteristic
236	C₂	#14530289	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10587616
commutator, cosocle	C₂×C₄²×C₁₇	C₂³
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1939	C₂
socle, prime radical	C₂³×C₁₇	#121
hypercenter	#1939	D₁₇
hypocenter	C₁₇	#10649
LCS	G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1939

Complements

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

N	H ≅ G/N	type
155. C₁₇	#10649	semidirect

Maximal subgroups

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

subgroup	index
#14529542	2
#14528944	2
#14526668	2
#14526619	2
C₁₇×#1939	2
#10649	17