Subgroups — #103×Dic₁₇ #14528606

Order 2176.

332 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×Q₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#10587506	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×Dic₁₇	C₂	1	1	yes	maximal, characteristic
5	#11762572	C₂	1	1	yes	maximal, characteristic
6	#11762571	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₇×#103	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₁₇	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
11	Q₈×Dic₁₇	—	2	2	no	—
12	Q₈×Dic₁₇	C₄	1	4	yes	complemented
13	C₂×C₄⋊Dic₁₇	—	2	1	no	—
14	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₁₇	—	2	1	no	—
17	C₂×C₄×Dic₁₇	—	2	1	no	—
18	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
19	C₂²×#3622	C₂²	1	1	yes	characteristic
20	C₁₇×#103	C₄	1	4	yes	complemented
21	#10587506	C₄	1	4	yes	complemented
22	#10587502	—	2	1	no	—
23	C₈×Dic₁₇	—	2	2	no	—
24	C₂×Q₈×C₁₇	—	2	1	no	—
25	C₂×Q₈×C₁₇	—	2	1	no	—
26	C₂×Q₈×C₁₇	—	2	1	no	—
27	C₂×Q₈×C₁₇	C₂×C₄	1	2	yes	—
28	C₂×Q₈×C₁₇	C₂×C₄	1	2	yes	—
29	C₂×Q₈×C₁₇	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₇	—	2	1	no	—
31	C₂²×C₄×C₁₇	—	2	1	no	—
32	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
33	C₂²×Dic₁₇	—	2	1	no	—
34	C₂²×Dic₁₇	—	2	1	no	—
35	C₂²×Dic₁₇	D₄	1	1	yes	characteristic
36	C₂²×Dic₁₇	D₄	1	1	yes	characteristic
37	C₂×C₈×C₁₇	—	2	1	no	—
38	C₂×C₈×C₁₇	—	2	1	no	—
39	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
40	C₄⋊C₄×C₁₇	—	2	1	no	—
41	C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
42	C₄⋊Dic₁₇	—	2	1	no	—
43	C₄⋊Dic₁₇	—	2	1	no	—
44	C₄⋊Dic₁₇	—	2	4	no	—
45	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
46	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
47	C₄×Dic₁₇	—	2	4	no	—
48	C₄×Dic₁₇	—	4	2	no	—
49	C₄×Dic₁₇	C₂×C₄	1	2	yes	—
50	C₄×Dic₁₇	D₄	1	2	yes	—
51	C₂×#3622	—	2	1	no	—
52	C₂×#3622	—	2	1	no	—
53	C₂×#3622	C₂×C₄	1	2	yes	—
54	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
55	Q₈×C₁₇	—	2	2	no	—
56	Q₈×C₁₇	—	2	2	no	—
57	Q₈×C₁₇	—	2	2	no	—
58	Q₈×C₁₇	C₄²	1	4	yes	complemented
59	C₂×C₄×C₁₇	—	2	2	no	—
60	C₂×C₄×C₁₇	—	2	2	no	—
61	C₂×C₄×C₁₇	—	2	2	no	—
62	C₂×C₄×C₁₇	—	2	2	no	—
63	C₂×C₄×C₁₇	—	4	1	no	—
64	C₂×C₄×C₁₇	—	4	1	no	—
65	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
70	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
71	C₂×Dic₁₇	—	2	1	no	—
72	C₂×Dic₁₇	—	2	1	no	—
73	C₂×Dic₁₇	—	2	2	no	—
74	C₂×Dic₁₇	—	2	2	no	—
75	C₂×Dic₁₇	—	4	1	no	—
76	C₂×Dic₁₇	—	4	1	no	—
77	C₂×Dic₁₇	—	4	1	no	—
78	C₂×Dic₁₇	—	4	1	no	—
79	C₂×Dic₁₇	C₂²⋊C₄	1	2	yes	—
80	C₂×Dic₁₇	C₂²⋊C₄	1	2	yes	—
81	C₂×Dic₁₇	SD₁₆	1	2	yes	—
82	C₂×Dic₁₇	Q₁₆	1	2	yes	—
83	C₈×C₁₇	—	2	2	no	—
84	#3622	—	2	2	no	—
85	C₄×#103	—	17	1	no	maximal, Sylow
86	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
88	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
89	C₂²×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
90	C₂²×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
91	C₂²×C₁₇	#135	1	1	yes	characteristic
92	C₂²×C₁₇	#135	1	1	yes	characteristic
93	C₄×C₁₇	—	2	4	no	—
94	C₄×C₁₇	—	4	2	no	—
95	C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
96	C₄×C₁₇	#117	1	1	yes	characteristic
97	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
98	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
99	Dic₁₇	—	2	2	no	—
100	Dic₁₇	—	4	2	no	—
101	Dic₁₇	—	4	2	no	—
102	Dic₁₇	#103	1	4	yes	direct product factor
103	C₂×C₄×Q₈	—	17	1	no	—
104	C₂×#103	—	17	1	no	—
105	C₂×#103	D₁₇	1	1	yes	characteristic
106	C₂×C₄×C₈	—	17	1	no	—
107	C₄×C₄⋊C₄	—	17	1	no	—
108	#340	—	17	1	no	—
109	#340	—	17	1	no	—
110	C₂×C₁₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₁₇	C₂×#103	1	1	yes	characteristic
112	C₂×C₁₇	#416	1	1	yes	characteristic
113	C₂×C₁₇	C₄×SD₁₆	1	1	yes	characteristic
114	C₂×C₁₇	C₄×SD₁₆	1	1	yes	characteristic
115	C₂×C₁₇	C₄×Q₁₆	1	1	yes	characteristic
116	C₂×C₁₇	C₄×Q₁₆	1	1	yes	characteristic
117	C₂²×Q₈	C₂×D₁₇	1	1	yes	characteristic
118	C₂²×C₈	—	17	1	no	—
119	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
120	C₄×Q₈	—	17	4	no	—
121	C₄×Q₈	—	34	2	no	—
122	C₂×C₄⋊C₄	—	17	1	no	—
123	C₂×C₄⋊C₄	—	17	1	no	—
124	C₂×C₄⋊C₄	—	34	1	no	—
125	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
126	C₂×C₄²	—	17	1	no	—
127	C₂×C₄²	—	34	1	no	—
128	C₂×C₄²	—	34	1	no	—
129	#103	—	17	4	no	—
130	#103	Dic₁₇	1	4	yes	direct product factor
131	C₄×C₈	—	34	2	no	—
132	#95	—	34	1	no	—
133	C₁₇	C₄×#103	1	1	yes	characteristic, complemented, Sylow
134	C₂×Q₈	—	2	1	no	—
135	C₂×Q₈	—	2	1	no	—
136	C₂×Q₈	—	2	1	no	—
137	C₂×Q₈	C₄×D₁₇	1	2	yes	—
138	C₂×Q₈	C₄×D₁₇	1	2	yes	—
139	C₂×Q₈	C₂×Dic₁₇	1	2	yes	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	2	1	no	—
142	C₂²×C₄	—	17	1	no	—
143	C₂²×C₄	—	17	1	no	—
144	C₂²×C₄	—	34	1	no	—
145	C₂²×C₄	—	34	1	no	—
146	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	17	2	no	—
150	C₂×C₈	—	34	1	no	—
151	C₂×C₈	—	34	1	no	—
152	C₂×C₈	C₂×Dic₁₇	1	2	yes	—
153	C₄⋊C₄	—	2	1	no	—
154	C₄⋊C₄	—	17	2	no	—
155	C₄⋊C₄	—	17	2	no	—
156	C₄⋊C₄	—	34	4	no	—
157	C₄⋊C₄	—	34	1	no	—
158	C₄⋊C₄	—	34	1	no	—
159	C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
160	C₄²	—	17	2	no	—
161	C₄²	—	17	2	no	—
162	C₄²	—	34	4	no	—
163	C₄²	—	68	2	no	—
164	C₂³	D₄×D₁₇	1	1	yes	characteristic
165	Q₈	—	2	2	no	—
166	Q₈	—	2	2	no	—
167	Q₈	—	2	2	no	—
168	Q₈	C₄×Dic₁₇	1	4	yes	complemented
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	17	2	no	—
176	C₂×C₄	—	17	2	no	—
177	C₂×C₄	—	17	2	no	—
178	C₂×C₄	—	17	2	no	—
179	C₂×C₄	—	34	1	no	—
180	C₂×C₄	—	34	1	no	—
181	C₂×C₄	—	34	2	no	—
182	C₂×C₄	—	34	2	no	—
183	C₂×C₄	—	68	1	no	—
184	C₂×C₄	—	68	1	no	—
185	C₂×C₄	—	68	1	no	—
186	C₂×C₄	—	68	1	no	—
187	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
189	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
190	C₂×C₄	#63254	1	1	yes	characteristic
191	C₂×C₄	#63254	1	1	yes	characteristic
192	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
193	C₈	—	2	2	no	—
194	C₈	—	34	2	no	—
195	C₂²	C₂²⋊C₄×D₁₇	1	1	yes	characteristic
196	C₂²	#10587567	1	1	yes	characteristic
197	C₂²	SD₁₆×D₁₇	1	1	yes	characteristic
198	C₂²	#10587602	1	1	yes	characteristic
199	C₂²	Q₁₆×D₁₇	1	1	yes	characteristic
200	C₂²	#10587605	1	1	yes	characteristic
201	C₂²	D₄×Dic₁₇	1	1	yes	characteristic
202	C₄	—	2	4	no	—
203	C₄	—	4	2	no	—
204	C₄	—	17	4	no	—
205	C₄	—	34	2	no	—
206	C₄	—	68	2	no	—
207	C₄	—	68	2	no	—
208	C₄	#10587563	1	1	yes	characteristic
209	C₄	#10587567	1	1	yes	characteristic
210	C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
211	C₄	D₄×Dic₁₇	1	1	yes	characteristic
212	C₂	C₂²⋊C₄×Dic₁₇	1	1	yes	characteristic
213	C₂	#103×D₁₇	1	1	yes	characteristic
214	C₂	#11762804	1	1	yes	characteristic
215	C₂	#11762894	1	1	yes	characteristic
216	C₂	SD₁₆×Dic₁₇	1	1	yes	characteristic
217	C₂	#11762901	1	1	yes	characteristic
218	C₂	Q₁₆×Dic₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₇
commutator	C₄×C₁₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂×C₁₇×#103	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₄
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	C₂×#103	D₁₇
hypocenter	C₁₇	C₄×#103
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

Complements

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

N	H ≅ G/N	type
12. Q₈×Dic₁₇	C₄	semidirect
20. C₁₇×#103	C₄	semidirect
21. #10587506	C₄	semidirect
58. Q₈×C₁₇	C₄²	semidirect
102. Dic₁₇	#103	direct (N × H)
130. #103	Dic₁₇	direct (N × H)
133. C₁₇	C₄×#103	semidirect
168. Q₈	C₄×Dic₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×Dic₁₇	2
C₂×#10587506	2
C₂×C₈×Dic₁₇	2
C₄⋊C₄×Dic₁₇	2
#11762572	2
#11762571	2
C₂×C₁₇×#103	2
C₄×#103	17

Subgroups — #103×Dic17 #14528606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #103×Dic₁₇ #14528606