Subgroups — #17189812

Order 4352.

490 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539298	C₂	1	1	yes	maximal, characteristic
2	#14531064	C₂	1	2	yes	maximal, complemented
3	#14531060	C₂	1	4	yes	maximal, complemented
4	#14531057	C₂	1	2	yes	maximal, complemented
5	#14530566	C₂	1	4	yes	maximal
6	C₂×C₄×#63214	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#3160	C₂	1	1	yes	maximal, characteristic
8	#11763366	—	2	1	no	—
9	C₂²×#63214	C₂²	1	2	yes	—
10	#11763214	C₂²	1	2	yes	—
11	#11763211	—	2	1	no	—
12	#11763211	C₂²	1	2	yes	—
13	#11763207	—	2	1	no	—
14	#11762847	—	2	1	no	—
15	#11762847	C₂²	1	2	yes	—
16	#11762636	C₂²	1	8	yes	—
17	#11762634	C₂²	1	8	yes	—
18	C₄×#63214	C₂²	1	1	yes	characteristic
19	C₄×#63214	C₂²	1	2	yes	—
20	C₄×#63214	C₂²	1	1	yes	characteristic
21	C₁₇×#533	—	2	1	no	—
22	C₁₇×#500	C₂²	1	2	yes	—
23	C₁₇×#499	—	2	1	no	—
24	C₁₇×#499	C₂²	1	2	yes	—
25	C₁₇×#498	—	2	1	no	—
26	C₁₇×#427	—	2	1	no	—
27	C₁₇×#427	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
29	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
31	C₁₇×C₄⋊Q₈	—	2	2	no	—
32	C₁₇×#125	—	2	2	no	—
33	C₁₇×C₂²⋊Q₈	—	4	2	no	—
34	C₄×Q₈×C₁₇	—	4	2	no	—
35	C₁₇×#117	—	2	2	no	—
36	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
37	#10587612	—	2	2	no	—
38	#10587610	—	4	2	no	—
39	#10587556	—	2	2	no	—
40	#10587555	—	2	2	no	—
41	C₄×Dic₃₄	—	4	2	no	—
42	C₁₇×#107	—	2	2	no	—
43	C₁₇×#107	C₂³	1	4	yes	—
44	C₁₇×#106	—	2	2	no	—
45	C₁₇×#106	—	2	2	no	—
46	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
49	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
50	#10587489	—	2	2	no	—
51	#10587489	C₂³	1	4	yes	—
52	#10587488	—	2	2	no	—
53	#10587488	—	2	2	no	—
54	C₂×#63214	Q₈	1	8	yes	complemented
55	C₂×#63214	C₂³	1	2	yes	—
56	C₂×#63214	C₂³	1	2	yes	—
57	C₂×Q₈×C₁₇	—	4	2	no	—
58	C₂²×C₄×C₁₇	D₈	1	2	yes	—
59	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
60	C₂×Dic₃₄	—	4	2	no	—
61	C₂×C₈×C₁₇	Q₁₆	1	4	yes	—
62	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₁₇	C₂×Q₈	1	4	yes	—
66	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
67	C₄⋊C₄×C₁₇	—	2	4	no	—
68	C₄⋊C₄×C₁₇	—	4	4	no	—
69	C₄⋊C₄×C₁₇	—	4	2	no	—
70	C₂²⋊C₄×C₁₇	—	4	2	no	—
71	C₄²×C₁₇	—	4	2	no	—
72	C₄²×C₁₇	D₈	1	2	yes	—
73	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
75	C₂²⋊Dic₁₇	—	4	2	no	—
76	C₄⋊Dic₁₇	—	2	4	no	—
77	C₄⋊Dic₁₇	—	4	2	no	—
78	#63225	—	4	4	no	—
79	C₄×Dic₁₇	—	4	2	no	—
80	#63214	C₂×Q₈	1	4	yes	—
81	#63214	C₂×Q₈	1	4	yes	—
82	#33726	—	17	1	no	maximal, Sylow
83	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
84	Q₈×C₁₇	—	8	4	no	—
85	C₂×C₄×C₁₇	—	2	4	no	—
86	C₂×C₄×C₁₇	—	4	4	no	—
87	C₂×C₄×C₁₇	C₄⋊Q₈	1	2	yes	—
88	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
92	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
93	C₂×Dic₁₇	—	4	4	no	—
94	Dic₃₄	—	8	4	no	—
95	C₈×C₁₇	C₄⋊Q₈	1	2	yes	—
96	C₈×C₁₇	C₂×Q₁₆	1	2	yes	—
97	C₈×C₁₇	C₂×Q₁₆	1	2	yes	—
98	C₈×C₁₇	C₂²×Q₈	1	2	yes	—
99	#3160	—	17	1	no	—
100	#3160	D₁₇	1	1	yes	characteristic
101	#2254	—	17	2	no	—
102	#2253	—	17	4	no	—
103	#2252	—	17	2	no	—
104	#2158	—	17	4	no	—
105	C₂×C₄×C₁₆	—	17	1	no	—
106	C₂²×C₁₇	—	2	2	no	—
107	C₂²×C₁₇	#500	1	2	yes	—
108	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
109	C₄×C₁₇	—	2	2	no	—
110	C₄×C₁₇	—	8	4	no	—
111	C₄×C₁₇	#500	1	2	yes	—
112	C₄×C₁₇	#508	1	4	yes	—
113	C₄×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
114	C₄×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
115	Dic₁₇	—	8	4	no	—
116	#533	—	2	1	no	—
117	#533	—	34	1	no	—
118	C₂²×C₁₆	—	17	2	no	—
119	#500	—	17	2	no	—
120	#500	C₂×D₁₇	1	2	yes	—
121	#499	—	2	1	no	—
122	#499	—	17	2	no	—
123	#499	—	34	1	no	—
124	#499	C₂×D₁₇	1	2	yes	—
125	#498	—	2	1	no	—
126	#498	—	34	1	no	—
127	#427	—	2	1	no	—
128	#427	—	17	2	no	—
129	#427	—	34	1	no	—
130	#427	C₂×D₁₇	1	2	yes	—
131	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
132	#367	—	17	8	no	—
133	#366	—	17	8	no	—
134	C₄×C₁₆	—	17	1	no	—
135	C₄×C₁₆	—	17	2	no	—
136	C₄×C₁₆	—	17	1	no	—
137	C₂×C₁₇	—	2	2	no	—
138	C₂×C₁₇	C₂×#500	1	1	yes	characteristic
139	C₂×C₁₇	C₂×#508	1	2	yes	—
140	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
141	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
142	C₄⋊Q₈	—	2	2	no	—
143	C₄⋊Q₈	—	34	2	no	—
144	#125	—	2	2	no	—
145	#125	—	34	2	no	—
146	C₂²⋊Q₈	—	4	2	no	—
147	C₂²⋊Q₈	—	68	2	no	—
148	C₄×Q₈	—	4	2	no	—
149	C₄×Q₈	—	68	2	no	—
150	#117	—	2	2	no	—
151	#117	—	34	2	no	—
152	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
153	C₂×C₁₆	—	17	8	no	—
154	C₂×C₁₆	—	17	2	no	—
155	C₂×C₁₆	—	17	2	no	—
156	#107	—	2	2	no	—
157	#107	—	17	4	no	—
158	#107	—	34	2	no	—
159	#107	C₂²×D₁₇	1	4	yes	—
160	#106	—	2	2	no	—
161	#106	—	2	2	no	—
162	#106	—	34	2	no	—
163	#106	—	34	2	no	—
164	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
165	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
166	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
167	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
168	C₁₇	#33726	1	1	yes	characteristic, complemented, Sylow
169	C₂×Q₈	—	4	2	no	—
170	C₂×Q₈	—	68	2	no	—
171	C₂²×C₄	#63228	1	2	yes	—
172	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
173	C₂×C₈	#63231	1	4	yes	—
174	C₂×C₈	Q₈×D₁₇	1	4	yes	—
175	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
176	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
177	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
178	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	4	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	34	4	no	—
183	C₄⋊C₄	—	68	4	no	—
184	C₄⋊C₄	—	68	2	no	—
185	C₂²⋊C₄	—	4	2	no	—
186	C₂²⋊C₄	—	68	2	no	—
187	C₄²	—	4	2	no	—
188	C₄²	—	68	2	no	—
189	C₄²	#63228	1	2	yes	—
190	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
191	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
192	C₁₆	—	17	4	no	—
193	C₁₆	—	17	4	no	—
194	C₂³	C₂×#63228	1	1	yes	characteristic
195	Q₈	—	8	4	no	—
196	Q₈	—	136	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	68	4	no	—
200	C₂×C₄	C₂×#63228	1	2	yes	—
201	C₂×C₄	C₂×#63228	1	2	yes	—
202	C₂×C₄	C₂×#63228	1	1	yes	characteristic
203	C₂×C₄	C₂×#63231	1	2	yes	—
204	C₂×C₄	#10587630	1	2	yes	—
205	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
206	C₈	C₂×#63231	1	2	yes	—
207	C₈	C₂×#63231	1	2	yes	—
208	C₈	#10587630	1	2	yes	—
209	C₈	C₂×Q₈×D₁₇	1	2	yes	—
210	C₂²	—	2	2	no	—
211	C₂²	#11763212	1	2	yes	—
212	C₂²	C₂²×#63228	1	1	yes	characteristic
213	C₄	—	2	2	no	—
214	C₄	—	8	4	no	—
215	C₄	—	136	4	no	—
216	C₄	#11763212	1	2	yes	—
217	C₄	#11763239	1	4	yes	—
218	C₄	C₂×#10587630	1	1	yes	characteristic
219	C₄	C₂²×#63231	1	1	yes	characteristic
220	C₂	—	2	2	no	—
221	C₂	C₂×#11763212	1	1	yes	characteristic
222	C₂	C₂×#11763239	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#63228
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₁₇×#3160	C₂
socle	C₂²×C₁₇	C₂²×D₈
prime radical	C₂³×C₁₇	C₂×D₈
hypercenter	#3160	D₁₇
hypocenter	C₁₇	#33726
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #14531064	C₂	semidirect
3. #14531060	C₂	semidirect
4. #14531057	C₂	semidirect
54. C₂×#63214	Q₈	semidirect
168. C₁₇	#33726	semidirect

Maximal subgroups

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

subgroup	index
#14539298	2
#14531064	2
#14531060	2
#14531057	2
#14530566	2
C₂×C₄×#63214	2
C₁₇×#3160	2
#33726	17