Subgroups — #16950742

Order 4352.

342 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×#11762630	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762629	C₂	1	2	yes	maximal
3	#14530539	C₂	1	2	yes	maximal
4	#14528470	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×#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	#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	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
25	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
26	C₁₇×M₃₂	—	2	2	no	—
27	C₁₇×M₃₂	—	2	2	no	—
28	C₂×C₁₆×C₁₇	—	2	2	no	—
29	C₄×C₈×C₁₇	D₄	1	2	yes	—
30	C₄×C₈×C₁₇	Q₈	1	2	yes	—
31	#10587502	—	2	2	no	—
32	#10587486	—	2	2	no	—
33	#10587486	C₈	1	4	yes	—
34	C₈×Dic₁₇	D₄	1	2	yes	—
35	C₈×Dic₁₇	D₄	1	2	yes	—
36	C₈×Dic₁₇	Q₈	1	2	yes	—
37	C₈×Dic₁₇	Q₈	1	2	yes	—
38	#10587483	—	2	2	no	—
39	#10587483	—	2	2	no	—
40	C₂×#63214	—	2	2	no	—
41	#10587474	—	2	2	no	—
42	C₂²×C₄×C₁₇	—	2	1	no	—
43	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₁₇	—	2	1	no	—
45	C₂²×Dic₁₇	M₁₆	1	2	yes	—
46	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
49	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
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×Q₈	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	C₄²×C₁₇	—	2	1	no	—
56	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
57	C₄×Dic₁₇	—	2	1	no	—
58	C₄×Dic₁₇	C₄⋊C₄	1	2	yes	—
59	C₄×Dic₁₇	C₄⋊C₄	1	2	yes	—
60	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
61	C₂×#3622	—	2	1	no	—
62	C₂×#3622	—	2	1	no	—
63	C₂×#3622	C₄⋊C₄	1	2	yes	—
64	C₂×#3622	C₄⋊C₄	1	2	yes	—
65	C₂×#3622	C₂×C₈	1	2	yes	—
66	C₂×#3622	M₁₆	1	2	yes	—
67	C₁₆×C₁₇	—	4	2	no	—
68	#63214	—	4	2	no	—
69	#12901	—	17	1	no	maximal, Sylow
70	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₁₇	—	2	2	no	—
72	C₂×C₄×C₁₇	—	4	2	no	—
73	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
79	C₂×Dic₁₇	—	2	2	no	—
80	C₂×Dic₁₇	—	4	2	no	—
81	C₂×Dic₁₇	—	4	2	no	—
82	C₂×Dic₁₇	C₄⋊C₈	1	4	yes	—
83	C₈×C₁₇	C₄⋊C₈	1	4	yes	—
84	C₈×C₁₇	#127	1	1	yes	characteristic
85	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
86	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
87	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
88	#3622	—	2	2	no	—
89	#3622	C₄⋊C₈	1	4	yes	—
90	C₂×#364	—	17	2	no	—
91	C₂×#364	—	17	1	no	—
92	C₂×#364	D₁₇	1	1	yes	characteristic
93	#2150	—	17	2	no	—
94	#1726	—	17	1	no	—
95	C₂²×C₁₇	—	2	2	no	—
96	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
97	C₂²×C₁₇	#423	1	1	yes	characteristic
98	C₂²×C₁₇	#423	1	1	yes	characteristic
99	C₄×C₁₇	—	4	2	no	—
100	C₄×C₁₇	#384	1	1	yes	characteristic
101	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
103	C₄×C₁₇	#423	1	1	yes	characteristic
104	Dic₁₇	—	2	4	no	—
105	Dic₁₇	—	4	2	no	—
106	C₂×M₃₂	—	17	2	no	—
107	C₂×M₃₂	C₂×D₁₇	1	2	yes	—
108	C₂×#97	—	17	1	no	—
109	C₂×#97	—	17	1	no	—
110	C₂×#97	—	17	1	no	—
111	C₂×C₄×C₈	—	17	1	no	—
112	C₂×C₄×C₈	—	17	1	no	—
113	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
114	#376	—	17	1	no	—
115	#364	—	2	2	no	—
116	#364	—	34	4	no	—
117	#364	—	34	2	no	—
118	C₄⋊C₁₆	—	34	4	no	—
119	C₂×C₁₇	—	2	1	no	—
120	C₂×C₁₇	—	2	1	no	—
121	C₂×C₁₇	#1842	1	1	yes	characteristic
122	C₂×C₁₇	#2153	1	2	yes	—
123	C₂²×C₈	—	17	1	no	—
124	C₂²×C₈	—	17	1	no	—
125	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
127	C₂×C₄²	—	17	1	no	—
128	C₂×C₄²	—	17	1	no	—
129	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
130	M₃₂	—	2	2	no	—
131	M₃₂	—	2	2	no	—
132	M₃₂	—	34	2	no	—
133	M₃₂	—	34	2	no	—
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	34	2	no	—
136	#97	—	17	4	no	—
137	#97	—	34	2	no	—
138	#97	—	34	2	no	—
139	C₄×C₈	—	17	2	no	—
140	C₄×C₈	—	17	2	no	—
141	C₄×C₈	—	17	2	no	—
142	C₄×C₈	—	17	2	no	—
143	C₄×C₈	Dic₃₄	1	2	yes	—
144	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
145	#95	—	34	2	no	—
146	C₁₇	#12901	1	1	yes	characteristic, complemented, Sylow
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	17	2	no	—
149	C₂²×C₄	—	34	1	no	—
150	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
151	C₂×C₈	—	17	2	no	—
152	C₂×C₈	—	17	2	no	—
153	C₂×C₈	—	17	2	no	—
154	C₂×C₈	—	17	2	no	—
155	C₂×C₈	—	34	1	no	—
156	C₂×C₈	—	34	1	no	—
157	C₂×C₈	C₈×D₁₇	1	2	yes	—
158	C₂×C₈	#63218	1	2	yes	—
159	C₂×C₈	#63225	1	2	yes	—
160	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
161	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
162	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
163	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
164	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
165	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
166	C₄²	—	2	1	no	—
167	C₄²	—	17	2	no	—
168	C₄²	—	17	2	no	—
169	C₄²	—	17	2	no	—
170	C₄²	—	34	1	no	—
171	C₄²	#63225	1	2	yes	—
172	C₁₆	—	4	2	no	—
173	C₁₆	—	68	2	no	—
174	C₂³	M₁₆×D₁₇	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	17	4	no	—
178	C₂×C₄	—	34	2	no	—
179	C₂×C₄	—	68	2	no	—
180	C₂×C₄	—	68	2	no	—
181	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
182	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
183	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
184	C₂×C₄	C₂×#63218	1	1	yes	characteristic
185	C₂×C₄	M₁₆×D₁₇	1	1	yes	characteristic
186	C₂×C₄	C₂×#63225	1	1	yes	characteristic
187	C₈	—	17	4	no	—
188	C₈	—	34	2	no	—
189	C₈	#10587485	1	4	yes	—
190	C₈	#10587574	1	1	yes	characteristic
191	C₈	#10587574	1	1	yes	characteristic
192	C₈	#10587626	1	1	yes	characteristic
193	C₈	#10587630	1	1	yes	characteristic
194	C₂²	—	2	2	no	—
195	C₂²	C₄⋊C₈×D₁₇	1	1	yes	characteristic
196	C₂²	#11762831	1	1	yes	characteristic
197	C₂²	#11762832	1	1	yes	characteristic
198	C₄	—	4	2	no	—
199	C₄	—	34	4	no	—
200	C₄	—	68	2	no	—
201	C₄	#11762678	1	1	yes	characteristic
202	C₄	C₄⋊C₈×D₁₇	1	1	yes	characteristic
203	C₄	C₂×#10587485	1	1	yes	characteristic
204	C₄	#11762831	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	#14528935	1	1	yes	characteristic
208	C₂	#14530548	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	#12901
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
146. C₁₇	#12901	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
#14528470	2
C₂×C₁₇×#364	2
#12901	17