Subgroups — #16950739

Order 4352.

382 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762630	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762629	C₂	1	2	yes	maximal
3	#14530539	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₇×#364	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587483	C₂²	1	2	yes	—
7	C₂×C₄×#3622	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
12	C₄²×Dic₁₇	C₄	1	1	yes	characteristic
13	#11762630	—	2	2	no	—
14	#11762629	—	2	4	no	—
15	#11762627	—	2	4	no	—
16	C₂×C₁₇×M₃₂	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₁₇×#364	—	2	2	no	—
19	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₇	—	2	2	no	—
23	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
26	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
27	C₁₇×M₃₂	—	2	2	no	—
28	C₁₇×M₃₂	—	2	2	no	—
29	C₂×C₁₆×C₁₇	—	2	2	no	—
30	C₄×C₈×C₁₇	D₄	1	2	yes	—
31	C₄×C₈×C₁₇	Q₈	1	2	yes	—
32	C₈×Dic₁₇	—	2	2	no	—
33	C₈×Dic₁₇	C₈	1	2	yes	—
34	C₈×Dic₁₇	C₈	1	2	yes	—
35	C₈×Dic₁₇	D₄	1	2	yes	—
36	C₈×Dic₁₇	D₄	1	2	yes	—
37	C₈×Dic₁₇	Q₈	1	2	yes	—
38	C₈×Dic₁₇	Q₈	1	2	yes	—
39	#10587483	—	2	2	no	—
40	#10587483	—	2	2	no	—
41	C₂×#63214	—	2	2	no	—
42	C₄×#3622	—	2	2	no	—
43	C₂²×C₄×C₁₇	—	2	1	no	—
44	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
45	C₂²×Dic₁₇	—	2	1	no	—
46	C₂²×Dic₁₇	M₁₆	1	2	yes	—
47	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
50	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₁₇	—	2	1	no	—
57	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
58	C₄×Dic₁₇	—	2	8	no	—
59	C₄×Dic₁₇	—	2	1	no	—
60	C₄×Dic₁₇	C₄⋊C₄	1	2	yes	—
61	C₄×Dic₁₇	C₄⋊C₄	1	2	yes	—
62	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
63	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
64	C₂×#3622	—	2	1	no	—
65	C₂×#3622	—	2	1	no	—
66	C₂×#3622	C₄⋊C₄	1	2	yes	—
67	C₂×#3622	C₄⋊C₄	1	2	yes	—
68	C₂×#3622	C₂×C₈	1	2	yes	—
69	C₂×#3622	M₁₆	1	2	yes	—
70	C₁₆×C₁₇	—	4	2	no	—
71	#63214	—	4	2	no	—
72	#12900	—	17	1	no	maximal, Sylow
73	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₁₇	—	2	4	no	—
75	C₂×C₄×C₁₇	—	2	2	no	—
76	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
82	C₂×Dic₁₇	—	2	4	no	—
83	C₂×Dic₁₇	—	2	2	no	—
84	C₂×Dic₁₇	—	2	4	no	—
85	C₂×Dic₁₇	C₄⋊C₈	1	2	yes	—
86	C₂×Dic₁₇	C₄⋊C₈	1	2	yes	—
87	C₈×C₁₇	C₄⋊C₈	1	4	yes	—
88	C₈×C₁₇	#127	1	1	yes	characteristic
89	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
90	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
91	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
92	#3622	—	2	2	no	—
93	#3622	C₄⋊C₈	1	4	yes	—
94	C₂×#364	—	17	2	no	—
95	C₂×#364	—	17	1	no	—
96	C₂×#364	D₁₇	1	1	yes	characteristic
97	#2150	—	17	2	no	—
98	C₄²×C₈	—	17	1	no	—
99	C₂²×C₁₇	—	2	2	no	—
100	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
101	C₂²×C₁₇	#423	1	1	yes	characteristic
102	C₂²×C₁₇	#423	1	1	yes	characteristic
103	C₄×C₁₇	—	2	4	no	—
104	C₄×C₁₇	#384	1	1	yes	characteristic
105	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
106	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
107	C₄×C₁₇	#423	1	1	yes	characteristic
108	Dic₁₇	—	2	4	no	—
109	Dic₁₇	—	2	2	no	—
110	Dic₁₇	#364	1	4	yes	complemented
111	C₂×M₃₂	—	17	2	no	—
112	C₂×M₃₂	C₂×D₁₇	1	2	yes	—
113	C₂×C₄×C₈	—	17	1	no	—
114	C₂×C₄×C₈	—	17	1	no	—
115	C₂×C₄×C₈	—	17	1	no	—
116	C₂×C₄×C₈	—	17	1	no	—
117	C₂×C₄×C₈	—	17	1	no	—
118	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
119	C₄³	—	17	1	no	—
120	#364	—	2	2	no	—
121	#364	—	34	4	no	—
122	#364	—	34	2	no	—
123	C₄⋊C₁₆	—	34	4	no	—
124	C₂×C₁₇	—	2	1	no	—
125	C₂×C₁₇	—	2	1	no	—
126	C₂×C₁₇	#1842	1	1	yes	characteristic
127	C₂×C₁₇	C₂×#364	1	2	yes	—
128	C₂²×C₈	—	17	1	no	—
129	C₂²×C₈	—	17	1	no	—
130	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
131	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
132	C₂×C₄²	—	17	1	no	—
133	C₂×C₄²	—	17	1	no	—
134	C₂×C₄²	—	34	2	no	—
135	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
136	M₃₂	—	2	2	no	—
137	M₃₂	—	2	2	no	—
138	M₃₂	—	34	2	no	—
139	M₃₂	—	34	2	no	—
140	C₂×C₁₆	—	2	2	no	—
141	C₂×C₁₆	—	34	2	no	—
142	C₄×C₈	—	17	2	no	—
143	C₄×C₈	—	17	2	no	—
144	C₄×C₈	—	17	2	no	—
145	C₄×C₈	—	17	2	no	—
146	C₄×C₈	—	17	2	no	—
147	C₄×C₈	—	17	2	no	—
148	C₄×C₈	—	34	2	no	—
149	C₄×C₈	—	34	2	no	—
150	C₄×C₈	Dic₃₄	1	2	yes	—
151	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
152	C₁₇	#12900	1	1	yes	characteristic, complemented, Sylow
153	C₂²×C₄	—	2	1	no	—
154	C₂²×C₄	—	17	2	no	—
155	C₂²×C₄	—	34	1	no	—
156	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
157	C₂×C₈	—	17	2	no	—
158	C₂×C₈	—	17	2	no	—
159	C₂×C₈	—	17	2	no	—
160	C₂×C₈	—	17	2	no	—
161	C₂×C₈	—	34	1	no	—
162	C₂×C₈	—	34	1	no	—
163	C₂×C₈	C₈×D₁₇	1	2	yes	—
164	C₂×C₈	#63218	1	2	yes	—
165	C₂×C₈	#63225	1	2	yes	—
166	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
167	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
168	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
169	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
170	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
171	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
172	C₄²	—	2	1	no	—
173	C₄²	—	17	2	no	—
174	C₄²	—	17	1	no	—
175	C₄²	—	17	1	no	—
176	C₄²	—	17	2	no	—
177	C₄²	—	34	8	no	—
178	C₄²	—	34	1	no	—
179	C₄²	#63225	1	2	yes	—
180	C₁₆	—	4	2	no	—
181	C₁₆	—	68	2	no	—
182	C₂³	M₁₆×D₁₇	1	1	yes	characteristic
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	17	2	no	—
186	C₂×C₄	—	17	2	no	—
187	C₂×C₄	—	34	4	no	—
188	C₂×C₄	—	34	4	no	—
189	C₂×C₄	—	34	2	no	—
190	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
191	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
192	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
193	C₂×C₄	C₂×#63218	1	1	yes	characteristic
194	C₂×C₄	M₁₆×D₁₇	1	1	yes	characteristic
195	C₂×C₄	C₂×#63225	1	1	yes	characteristic
196	C₈	—	17	4	no	—
197	C₈	—	34	2	no	—
198	C₈	#10587485	1	4	yes	—
199	C₈	#10587574	1	1	yes	characteristic
200	C₈	#10587574	1	1	yes	characteristic
201	C₈	#10587626	1	1	yes	characteristic
202	C₈	#10587630	1	1	yes	characteristic
203	C₂²	—	2	2	no	—
204	C₂²	C₄⋊C₈×D₁₇	1	1	yes	characteristic
205	C₂²	#11762831	1	1	yes	characteristic
206	C₂²	#11762832	1	1	yes	characteristic
207	C₄	—	2	4	no	—
208	C₄	—	17	4	no	—
209	C₄	—	34	4	no	—
210	C₄	—	34	2	no	—
211	C₄	#11762678	1	1	yes	characteristic
212	C₄	C₄⋊C₈×D₁₇	1	1	yes	characteristic
213	C₄	C₂×#10587485	1	1	yes	characteristic
214	C₄	#11762831	1	1	yes	characteristic
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	#14528935	1	1	yes	characteristic
218	C₂	D₁₇×#364	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#364	C₂
socle	C₂²×C₁₇	C₂×C₄⋊C₈
prime radical	C₂³×C₁₇	C₂×M₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×#364	D₁₇
hypocenter	C₁₇	#12900
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
110. Dic₁₇	#364	semidirect
152. C₁₇	#12900	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762630	2
C₂×#11762629	2
#14530539	2
C₄×C₈×Dic₁₇	2
C₂×C₁₇×#364	2
#12900	17