Subgroups — #16952926

Order 4352.

414 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#63214	C₂	1	1	yes	maximal, characteristic
2	#14528944	C₂	1	1	yes	maximal, characteristic
3	#14526875	C₂	1	4	yes	maximal
4	C₂×C₁₇×#500	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63214	C₂²	1	2	yes	—
6	C₂×#10587489	—	2	1	no	—
7	C₂×#10587489	C₂²	1	2	yes	—
8	#11762679	—	2	1	no	—
9	C₄×#63214	—	2	2	no	—
10	C₂×C₁₇×C₄⋊Q₈	—	2	1	no	—
11	C₁₇×#500	—	2	2	no	—
12	C₁₇×#500	C₄	1	4	yes	complemented
13	C₂×C₁₇×#107	—	2	1	no	—
14	C₂×C₁₇×#107	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₂²×Q₈×C₁₇	—	4	1	no	—
17	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
19	C₁₇×C₄⋊Q₈	—	2	4	no	—
20	C₁₇×C₄⋊Q₈	—	4	2	no	—
21	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
22	C₂×C₄⋊C₄×C₁₇	—	4	1	no	—
23	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₇	—	2	2	no	—
25	C₂×C₄⋊Dic₁₇	—	4	1	no	—
26	C₂×C₄⋊Dic₁₇	—	4	1	no	—
27	C₁₇×#107	—	2	4	no	—
28	C₁₇×#107	—	2	2	no	—
29	C₁₇×#107	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₁₇	D₄	1	2	yes	—
32	#10587489	—	2	4	no	—
33	#10587489	—	4	2	no	—
34	C₂×#63214	—	2	2	no	—
35	C₂×#63214	—	2	2	no	—
36	C₂×#63214	Q₈	1	4	yes	complemented
37	C₂×Q₈×C₁₇	—	4	4	no	—
38	C₂×Q₈×C₁₇	—	8	2	no	—
39	C₂×Q₈×C₁₇	—	8	2	no	—
40	C₂²×C₄×C₁₇	—	4	2	no	—
41	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₁₇	—	4	2	no	—
45	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
47	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
48	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
51	C₂×C₈×C₁₇	#41	1	2	yes	—
52	C₄⋊C₄×C₁₇	—	2	4	no	—
53	C₄⋊C₄×C₁₇	—	4	4	no	—
54	C₄⋊C₄×C₁₇	—	4	2	no	—
55	C₄²×C₁₇	C₂²⋊C₄	1	2	yes	—
56	C₄²×C₁₇	SD₁₆	1	2	yes	—
57	C₄⋊Dic₁₇	—	4	2	no	—
58	C₄⋊Dic₁₇	—	4	2	no	—
59	C₄⋊Dic₁₇	—	8	2	no	—
60	C₄⋊Dic₁₇	—	8	2	no	—
61	#63214	—	2	4	no	—
62	#13282	—	17	1	no	maximal, Sylow
63	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
64	Q₈×C₁₇	—	8	8	no	—
65	C₂×C₄×C₁₇	—	4	4	no	—
66	C₂×C₄×C₁₇	—	8	2	no	—
67	C₂×C₄×C₁₇	—	8	2	no	—
68	C₂×C₄×C₁₇	#102	1	2	yes	—
69	C₂×C₄×C₁₇	#102	1	2	yes	—
70	C₂×C₄×C₁₇	D₁₆	1	2	yes	—
71	C₂×C₄×C₁₇	SD₃₂	1	2	yes	—
72	C₂×C₄×C₁₇	SD₃₂	1	2	yes	—
73	C₂×C₄×C₁₇	Q₃₂	1	2	yes	—
74	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×Dic₁₇	—	8	2	no	—
81	C₂×Dic₁₇	—	8	2	no	—
82	C₂×Dic₁₇	—	8	2	no	—
83	C₈×C₁₇	#103	1	4	yes	—
84	C₈×C₁₇	C₄×Q₈	1	2	yes	—
85	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
86	C₂×#500	D₁₇	1	1	yes	characteristic
87	C₂×C₄×C₁₆	—	17	1	no	—
88	#1845	—	17	1	no	—
89	#1380	—	17	4	no	—
90	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
91	C₂²×C₁₇	#486	1	1	yes	characteristic
92	C₂²×C₁₇	#500	1	1	yes	characteristic
93	C₂²×C₁₇	C₂×D₁₆	1	1	yes	characteristic
94	C₂²×C₁₇	C₂×SD₃₂	1	1	yes	characteristic
95	C₂²×C₁₇	C₂×SD₃₂	1	1	yes	characteristic
96	C₂²×C₁₇	C₂×Q₃₂	1	1	yes	characteristic
97	C₄×C₁₇	—	8	4	no	—
98	C₄×C₁₇	#357	1	4	yes	—
99	C₄×C₁₇	#358	1	4	yes	—
100	C₄×C₁₇	#391	1	1	yes	characteristic
101	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
102	C₄×C₁₇	#487	1	1	yes	characteristic
103	C₄×C₁₇	#498	1	1	yes	characteristic
104	Dic₁₇	—	16	2	no	—
105	C₂×C₄⋊Q₈	—	2	1	no	—
106	C₂²×C₁₆	—	17	2	no	—
107	#500	—	2	2	no	—
108	#500	Dic₁₇	1	4	yes	complemented
109	C₂×#107	—	2	1	no	—
110	C₂×#107	—	17	2	no	—
111	C₂×#107	—	34	1	no	—
112	C₂×#107	C₂×D₁₇	1	2	yes	—
113	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
114	#384	—	34	1	no	—
115	C₄×C₁₆	—	34	2	no	—
116	C₂×C₁₇	#1990	1	1	yes	characteristic
117	C₂×C₁₇	C₂×#357	1	1	yes	characteristic
118	C₂×C₁₇	C₂×#358	1	1	yes	characteristic
119	C₂×C₁₇	#2240	1	1	yes	characteristic
120	C₂×C₁₇	#2241	1	1	yes	characteristic
121	C₂×C₁₇	#2252	1	1	yes	characteristic
122	C₂×C₁₇	#2254	1	1	yes	characteristic
123	C₂²×Q₈	—	4	1	no	—
124	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
126	C₄⋊Q₈	—	2	4	no	—
127	C₄⋊Q₈	—	4	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	4	1	no	—
130	C₂×C₄⋊C₄	—	34	2	no	—
131	C₂×C₄⋊C₄	—	68	1	no	—
132	C₂×C₄⋊C₄	—	68	1	no	—
133	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
134	C₂×C₁₆	—	17	4	no	—
135	C₂×C₁₆	—	34	2	no	—
136	C₂×C₁₆	—	34	2	no	—
137	#107	—	2	4	no	—
138	#107	—	2	2	no	—
139	#107	—	34	4	no	—
140	#107	—	68	2	no	—
141	#107	C₂×Dic₁₇	1	4	yes	—
142	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
143	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
144	C₁₇	#13282	1	1	yes	characteristic, complemented, Sylow
145	C₂×Q₈	—	4	4	no	—
146	C₂×Q₈	—	8	2	no	—
147	C₂×Q₈	—	8	2	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	—	68	2	no	—
150	C₂²×C₄	#63228	1	1	yes	characteristic
151	C₂²×C₄	#63228	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
153	C₂×C₈	#63230	1	2	yes	—
154	C₂×C₈	#63231	1	2	yes	—
155	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
156	C₂×C₈	Q₈×D₁₇	1	2	yes	—
157	C₂×C₈	#63256	1	2	yes	—
158	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
159	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	4	4	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	68	2	no	—
164	C₄⋊C₄	—	68	2	no	—
165	C₄⋊C₄	—	136	2	no	—
166	C₄⋊C₄	—	136	2	no	—
167	C₄²	#63229	1	2	yes	—
168	C₄²	C₂²⋊Dic₁₇	1	2	yes	—
169	C₁₆	—	34	4	no	—
170	C₂³	C₂×#63228	1	1	yes	characteristic
171	Q₈	—	8	8	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	136	2	no	—
176	C₂×C₄	—	136	2	no	—
177	C₂×C₄	—	136	2	no	—
178	C₂×C₄	#10587497	1	2	yes	—
179	C₂×C₄	#10587498	1	2	yes	—
180	C₂×C₄	#10587499	1	2	yes	—
181	C₂×C₄	#10587500	1	2	yes	—
182	C₂×C₄	#10587503	1	2	yes	—
183	C₂×C₄	#10587503	1	2	yes	—
184	C₂×C₄	C₂×#63229	1	1	yes	characteristic
185	C₂×C₄	C₂×#63230	1	1	yes	characteristic
186	C₂×C₄	C₂×#63231	1	1	yes	characteristic
187	C₂×C₄	#10587630	1	1	yes	characteristic
188	C₂×C₄	#10587633	1	1	yes	characteristic
189	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
190	C₈	#10587506	1	4	yes	—
191	C₈	Q₈×Dic₁₇	1	2	yes	—
192	C₈	#10587632	1	2	yes	—
193	C₂²	C₂×#10587503	1	1	yes	characteristic
194	C₂²	#11763157	1	1	yes	characteristic
195	C₂²	#11763212	1	1	yes	characteristic
196	C₂²	C₂×#10587497	1	1	yes	characteristic
197	C₂²	C₂×#10587499	1	1	yes	characteristic
198	C₂²	C₂×#10587498	1	1	yes	characteristic
199	C₂²	C₂×#10587500	1	1	yes	characteristic
200	C₄	—	8	4	no	—
201	C₄	—	272	2	no	—
202	C₄	#11762610	1	4	yes	—
203	C₄	#11762613	1	4	yes	—
204	C₄	#11762716	1	1	yes	characteristic
205	C₄	C₂×#10587506	1	1	yes	characteristic
206	C₄	#11763161	1	1	yes	characteristic
207	C₄	#11763205	1	1	yes	characteristic
208	C₂	#14529746	1	1	yes	characteristic
209	C₂	C₂×#11762610	1	1	yes	characteristic
210	C₂	C₂×#11762613	1	1	yes	characteristic
211	C₂	#14531009	1	1	yes	characteristic
212	C₂	#14531013	1	1	yes	characteristic
213	C₂	#14531057	1	1	yes	characteristic
214	C₂	#14531064	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#63228
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#500	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×#500	D₁₇
hypocenter	C₁₇	#13282
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
12. C₁₇×#500	C₄	semidirect
36. C₂×#63214	Q₈	semidirect
108. #500	Dic₁₇	semidirect
144. C₁₇	#13282	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#63214	2
#14528944	2
#14526875	2
C₂×C₁₇×#500	2
#13282	17