Subgroups — #16950672

Order 4352.

322 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762627	C₂	1	2	yes	maximal
2	#14528470	C₂	1	1	yes	maximal, characteristic
3	#14526862	C₂	1	1	yes	maximal, characteristic
4	#14526861	C₂	1	2	yes	maximal
5	C₁₇×#1374	C₂	1	1	yes	maximal, characteristic
6	C₂²×#63214	C₂²	1	2	yes	—
7	C₂×#10587474	C₄	1	1	yes	characteristic
8	C₂×#10587486	C₄	1	1	yes	characteristic
9	C₂×#10587486	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
12	#11762655	C₄	1	1	yes	characteristic
13	#11762627	—	2	4	no	—
14	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
22	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₁₇	—	2	2	no	—
24	C₂×C₁₆×C₁₇	—	2	2	no	—
25	C₂×C₁₆×C₁₇	—	2	2	no	—
26	C₄×C₈×C₁₇	—	2	2	no	—
27	#10587502	C₈	1	2	yes	—
28	#10587502	C₈	1	2	yes	—
29	#10587486	—	2	2	no	—
30	#10587486	—	2	2	no	—
31	C₈×Dic₁₇	—	2	2	no	—
32	C₈×Dic₁₇	D₄	1	2	yes	—
33	C₈×Dic₁₇	Q₈	1	2	yes	—
34	C₂×#63214	—	2	2	no	—
35	C₂×#63214	—	2	2	no	—
36	C₂×#63214	—	2	2	no	—
37	#10587474	—	2	2	no	—
38	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
39	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
42	C₂²×Dic₁₇	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₁₇	—	2	1	no	—
44	C₂×C₈×C₁₇	—	2	1	no	—
45	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
46	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₁₇	#41	1	2	yes	—
49	C₂×C₈×C₁₇	#41	1	2	yes	—
50	C₄²×C₁₇	—	2	1	no	—
51	C₄²×C₁₇	—	2	1	no	—
52	C₄×Dic₁₇	—	2	1	no	—
53	C₄×Dic₁₇	—	2	1	no	—
54	C₄×Dic₁₇	C₄⋊C₄	1	2	yes	—
55	C₄×Dic₁₇	M₁₆	1	2	yes	—
56	C₂×#3622	—	2	1	no	—
57	C₂×#3622	—	2	1	no	—
58	C₂×#3622	—	2	1	no	—
59	C₂×#3622	—	2	1	no	—
60	C₂×#3622	C₄⋊C₄	1	2	yes	—
61	C₂×#3622	M₁₆	1	2	yes	—
62	C₁₆×C₁₇	—	4	2	no	—
63	#63214	—	4	2	no	—
64	#12880	—	17	1	no	maximal, Sylow
65	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₇	—	2	2	no	—
67	C₂×C₄×C₁₇	—	2	2	no	—
68	C₂×C₄×C₁₇	—	2	2	no	—
69	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
71	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
75	C₂×Dic₁₇	—	2	2	no	—
76	C₂×Dic₁₇	—	2	2	no	—
77	C₂×Dic₁₇	—	2	1	no	—
78	C₂×Dic₁₇	—	2	1	no	—
79	C₂×Dic₁₇	—	2	2	no	—
80	C₂×Dic₁₇	—	2	2	no	—
81	C₂×Dic₁₇	C₄⋊C₈	1	4	yes	—
82	C₈×C₁₇	—	2	2	no	—
83	C₈×C₁₇	#124	1	2	yes	—
84	C₈×C₁₇	#125	1	2	yes	—
85	#3622	—	2	2	no	—
86	#3622	—	2	2	no	—
87	C₂×C₄⋊C₁₆	—	17	2	no	—
88	#1726	—	17	1	no	—
89	#1374	—	17	2	no	—
90	#1374	—	17	1	no	—
91	#1374	D₁₇	1	1	yes	characteristic
92	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
93	C₂²×C₁₇	#431	1	1	yes	characteristic
94	C₂²×C₁₇	#431	1	1	yes	characteristic
95	C₂²×C₁₇	#504	1	2	yes	—
96	C₂²×C₁₇	#504	1	2	yes	—
97	C₄×C₁₇	—	4	2	no	—
98	C₄×C₁₇	#382	1	1	yes	characteristic
99	C₄×C₁₇	#423	1	1	yes	characteristic
100	C₄×C₁₇	#432	1	1	yes	characteristic
101	C₄×C₁₇	#432	1	1	yes	characteristic
102	Dic₁₇	—	2	4	no	—
103	Dic₁₇	—	4	1	no	—
104	Dic₁₇	—	4	1	no	—
105	C₂²×C₁₆	—	17	2	no	—
106	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
107	C₂×#97	—	17	1	no	—
108	C₂×#97	—	17	1	no	—
109	C₂×#97	—	17	1	no	—
110	C₂×C₄×C₈	—	17	1	no	—
111	C₂×C₄×C₈	—	17	1	no	—
112	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
113	#376	—	17	1	no	—
114	C₄⋊C₁₆	—	34	4	no	—
115	C₂×C₁₇	#1831	1	1	yes	characteristic
116	C₂×C₁₇	#2151	1	2	yes	—
117	C₂×C₁₇	#2164	1	2	yes	—
118	C₂×C₁₇	#2164	1	1	yes	characteristic
119	C₂×C₁₇	#2164	1	1	yes	characteristic
120	C₂²×C₈	—	17	1	no	—
121	C₂²×C₈	—	17	1	no	—
122	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
124	C₂×C₄²	—	17	1	no	—
125	C₂×C₄²	—	17	1	no	—
126	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	—	2	2	no	—
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	34	2	no	—
131	C₂×C₁₆	—	34	2	no	—
132	C₂×C₁₆	—	34	2	no	—
133	#97	—	34	2	no	—
134	#97	—	34	2	no	—
135	#97	—	34	2	no	—
136	C₄×C₈	—	2	2	no	—
137	C₄×C₈	—	17	2	no	—
138	C₄×C₈	—	17	2	no	—
139	C₄×C₈	—	34	2	no	—
140	#95	—	17	2	no	—
141	#95	—	17	2	no	—
142	C₁₇	#12880	1	1	yes	characteristic, complemented, Sylow
143	C₂²×C₄	—	17	2	no	—
144	C₂²×C₄	—	17	1	no	—
145	C₂²×C₄	—	17	1	no	—
146	C₂²×C₄	C₈×D₁₇	1	2	yes	—
147	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	17	2	no	—
151	C₂×C₈	—	17	2	no	—
152	C₂×C₈	—	34	1	no	—
153	C₂×C₈	—	34	1	no	—
154	C₂×C₈	—	34	1	no	—
155	C₂×C₈	—	34	1	no	—
156	C₂×C₈	#63218	1	2	yes	—
157	C₂×C₈	#63252	1	2	yes	—
158	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
159	C₂×C₈	#63254	1	2	yes	—
160	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
161	C₄²	—	2	1	no	—
162	C₄²	—	2	1	no	—
163	C₄²	—	17	2	no	—
164	C₄²	—	17	2	no	—
165	C₄²	—	34	1	no	—
166	C₄²	—	34	1	no	—
167	C₁₆	—	4	2	no	—
168	C₁₆	—	68	2	no	—
169	C₂³	C₂×C₈×D₁₇	1	1	yes	characteristic
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	17	4	no	—
174	C₂×C₄	—	34	2	no	—
175	C₂×C₄	—	34	2	no	—
176	C₂×C₄	—	34	2	no	—
177	C₂×C₄	—	34	2	no	—
178	C₂×C₄	—	34	1	no	—
179	C₂×C₄	—	34	1	no	—
180	C₂×C₄	#10587558	1	1	yes	characteristic
181	C₂×C₄	#10587563	1	1	yes	characteristic
182	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
183	C₂×C₄	C₂×#63218	1	1	yes	characteristic
184	C₂×C₄	M₁₆×D₁₇	1	1	yes	characteristic
185	C₂×C₄	M₁₆×D₁₇	1	1	yes	characteristic
186	C₈	—	2	2	no	—
187	C₈	—	34	2	no	—
188	C₈	—	34	2	no	—
189	C₈	#10587571	1	2	yes	—
190	C₈	#10587575	1	2	yes	—
191	C₂²	C₄⋊C₈×D₁₇	1	1	yes	characteristic
192	C₂²	#11762859	1	1	yes	characteristic
193	C₂²	#11762860	1	1	yes	characteristic
194	C₂²	#11763225	1	2	yes	—
195	C₂²	#11763227	1	2	yes	—
196	C₄	—	4	2	no	—
197	C₄	—	34	4	no	—
198	C₄	—	68	1	no	—
199	C₄	—	68	1	no	—
200	C₄	#11762673	1	1	yes	characteristic
201	C₄	#11762831	1	1	yes	characteristic
202	C₄	#11762863	1	1	yes	characteristic
203	C₄	#11762864	1	1	yes	characteristic
204	C₂	#14528899	1	1	yes	characteristic
205	C₂	#14530541	1	2	yes	—
206	C₂	#14530588	1	2	yes	—
207	C₂	#14530589	1	1	yes	characteristic
208	C₂	#14530592	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₇
commutator, socle, prime radical	C₂³×C₁₇	C₂²×C₈
Fitting	C₁₇×#1374	C₂
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1374	D₁₇
hypocenter	C₁₇	#12880
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
142. C₁₇	#12880	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762627	2
#14528470	2
#14526862	2
#14526861	2
C₁₇×#1374	2
#12880	17