Subgroups — #16952957

Order 4352.

374 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763214	C₂	1	1	yes	maximal, characteristic
2	#14528956	C₂	1	1	yes	maximal, characteristic
3	#14526893	C₂	1	4	yes	maximal
4	C₄×C₁₇×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587555	—	2	1	no	—
6	#11763214	—	2	1	no	—
7	#11763214	—	2	1	no	—
8	#11763214	C₄	1	2	yes	—
9	#11763214	C₄	1	2	yes	—
10	C₂×#10587490	—	2	1	no	—
11	C₂×#10587490	C₂²	1	2	yes	—
12	C₂×#10587489	—	2	1	no	—
13	C₂×#10587489	C₂²	1	2	yes	—
14	#11762833	—	2	1	no	—
15	C₂×C₁₇×M₃₂	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×#346	—	2	1	no	—
18	C₄×C₁₆×C₁₇	—	2	1	no	—
19	C₂²×Dic₃₄	—	4	1	no	—
20	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₇	—	2	2	no	—
24	C₂×C₄⋊Dic₁₇	—	4	1	no	—
25	C₂×#63223	—	2	2	no	—
26	#10587555	—	2	2	no	—
27	#10587555	—	2	2	no	—
28	#10587555	—	4	1	no	—
29	#10587555	—	4	1	no	—
30	C₁₇×M₃₂	—	2	2	no	—
31	C₁₇×M₃₂	Q₈	1	4	yes	complemented
32	C₂×C₁₆×C₁₇	—	2	2	no	—
33	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
36	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
37	#10587490	—	2	4	no	—
38	#10587490	—	4	2	no	—
39	#10587489	—	2	2	no	—
40	#10587489	—	2	2	no	—
41	#10587489	—	2	2	no	—
42	#10587489	C₂×C₄	1	4	yes	—
43	C₄⋊#3622	—	4	1	no	—
44	C₄⋊#3622	—	4	1	no	—
45	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₁₇	—	4	2	no	—
49	C₂×Dic₃₄	—	4	2	no	—
50	C₂×Dic₃₄	—	4	2	no	—
51	C₂×Dic₃₄	—	8	4	no	—
52	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
55	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
56	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₁₇	#41	1	2	yes	—
60	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₁₇	D₈	1	1	yes	characteristic
63	C₄²×C₁₇	D₈	1	1	yes	characteristic
64	C₄⋊Dic₁₇	—	2	4	no	—
65	C₄⋊Dic₁₇	—	4	2	no	—
66	C₄⋊Dic₁₇	—	4	2	no	—
67	C₄⋊Dic₁₇	—	4	2	no	—
68	#63223	—	4	2	no	—
69	#63223	—	4	2	no	—
70	C₂×#3622	—	4	2	no	—
71	C₁₆×C₁₇	—	2	4	no	—
72	#13290	—	17	1	no	maximal, Sylow
73	C₂³×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₁₇	—	2	2	no	—
75	C₂×C₄×C₁₇	—	2	2	no	—
76	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
77	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
78	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
79	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
82	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
85	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
86	C₂×Dic₁₇	—	4	4	no	—
87	C₂×Dic₁₇	—	8	4	no	—
88	Dic₃₄	—	8	4	no	—
89	Dic₃₄	—	8	4	no	—
90	C₈×C₁₇	#103	1	2	yes	—
91	C₈×C₁₇	#103	1	2	yes	—
92	C₈×C₁₇	C₄×Q₈	1	2	yes	—
93	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
94	#3622	—	8	2	no	—
95	C₂×#500	—	17	1	no	—
96	C₄×M₃₂	D₁₇	1	1	yes	characteristic
97	#1848	—	17	1	no	—
98	#1387	—	17	4	no	—
99	C₂²×C₁₇	—	2	1	no	—
100	C₂²×C₁₇	—	2	1	no	—
101	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
102	C₂²×C₁₇	#487	1	1	yes	characteristic
103	C₂²×C₁₇	#498	1	1	yes	characteristic
104	C₄×C₁₇	—	2	1	no	—
105	C₄×C₁₇	—	2	1	no	—
106	C₄×C₁₇	#361	1	2	yes	—
107	C₄×C₁₇	#362	1	2	yes	—
108	C₄×C₁₇	#391	1	1	yes	characteristic
109	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
110	C₄×C₁₇	#486	1	1	yes	characteristic
111	C₄×C₁₇	#500	1	1	yes	characteristic
112	Dic₁₇	—	8	4	no	—
113	C₂×C₄⋊Q₈	—	34	1	no	—
114	C₂×M₃₂	C₂×D₁₇	1	2	yes	—
115	#500	—	17	2	no	—
116	#500	—	17	2	no	—
117	#500	—	34	1	no	—
118	#500	—	34	1	no	—
119	C₂×#108	—	17	2	no	—
120	C₂×#108	—	34	1	no	—
121	C₂×#107	—	17	2	no	—
122	C₂×#107	—	34	1	no	—
123	#423	—	34	1	no	—
124	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
125	#346	—	2	1	no	—
126	C₄×C₁₆	—	2	1	no	—
127	C₂×C₁₇	—	2	1	no	—
128	C₂×C₁₇	—	2	1	no	—
129	C₂×C₁₇	#1990	1	1	yes	characteristic
130	C₂×C₁₇	C₂×#361	1	1	yes	characteristic
131	C₂×C₁₇	C₂×#362	1	1	yes	characteristic
132	C₂²×Q₈	—	68	1	no	—
133	C₂×M₁₆	—	34	2	no	—
134	C₂²×C₈	D₆₈	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
136	C₄⋊Q₈	—	34	2	no	—
137	C₄⋊Q₈	—	34	2	no	—
138	C₄⋊Q₈	—	68	1	no	—
139	C₄⋊Q₈	—	68	1	no	—
140	C₂×C₄⋊C₄	—	34	2	no	—
141	C₂×C₄⋊C₄	—	68	1	no	—
142	C₂×C₄²	D₆₈	1	1	yes	characteristic
143	M₃₂	—	2	2	no	—
144	M₃₂	Dic₃₄	1	4	yes	complemented
145	C₂×C₁₆	—	2	2	no	—
146	#108	—	34	4	no	—
147	#108	—	68	2	no	—
148	#107	—	17	4	no	—
149	#107	—	34	2	no	—
150	#107	—	34	2	no	—
151	#107	—	34	2	no	—
152	C₄⋊C₈	—	68	1	no	—
153	C₄⋊C₈	—	68	1	no	—
154	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
155	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
156	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
158	C₁₇	#13290	1	1	yes	characteristic, complemented, Sylow
159	C₂×Q₈	—	68	2	no	—
160	C₂×Q₈	—	68	2	no	—
161	C₂×Q₈	—	136	4	no	—
162	C₂²×C₄	—	68	2	no	—
163	C₂²×C₄	#63219	1	1	yes	characteristic
164	C₂²×C₄	#63219	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
166	M₁₆	—	68	2	no	—
167	M₁₆	—	68	2	no	—
168	C₂×C₈	—	68	2	no	—
169	C₂×C₈	#63219	1	2	yes	—
170	C₂×C₈	Dic₆₈	1	2	yes	—
171	C₂×C₈	#63227	1	1	yes	characteristic
172	C₂×C₈	#63227	1	1	yes	characteristic
173	C₂×C₈	C₂×Dic₃₄	1	2	yes	—
174	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
175	C₂×C₈	#63252	1	2	yes	—
176	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
177	C₄⋊C₄	—	34	4	no	—
178	C₄⋊C₄	—	68	2	no	—
179	C₄⋊C₄	—	68	2	no	—
180	C₄⋊C₄	—	68	2	no	—
181	C₄²	D₁₃₆	1	1	yes	characteristic
182	C₄²	D₁₃₆	1	1	yes	characteristic
183	C₄²	#63227	1	1	yes	characteristic
184	C₄²	#63227	1	1	yes	characteristic
185	C₁₆	—	2	4	no	—
186	C₂³	C₂×#63219	1	1	yes	characteristic
187	Q₈	—	136	4	no	—
188	Q₈	—	136	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	68	4	no	—
192	C₂×C₄	—	136	4	no	—
193	C₂×C₄	#10587492	1	1	yes	characteristic
194	C₂×C₄	#10587492	1	1	yes	characteristic
195	C₂×C₄	#10587492	1	1	yes	characteristic
196	C₂×C₄	#10587492	1	1	yes	characteristic
197	C₂×C₄	#10587555	1	1	yes	characteristic
198	C₂×C₄	#10587561	1	1	yes	characteristic
199	C₂×C₄	C₂×#63219	1	1	yes	characteristic
200	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
201	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
202	C₂×C₄	C₂×#63227	1	1	yes	characteristic
203	C₈	—	136	2	no	—
204	C₈	#10587487	1	2	yes	—
205	C₈	#10587487	1	2	yes	—
206	C₈	C₄×Dic₃₄	1	2	yes	—
207	C₈	#10587610	1	2	yes	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	C₂×#10587492	1	1	yes	characteristic
211	C₂²	#11763160	1	1	yes	characteristic
212	C₂²	#11763207	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	136	4	no	—
216	C₄	#11762623	1	2	yes	—
217	C₄	#11762626	1	2	yes	—
218	C₄	#11762715	1	1	yes	characteristic
219	C₄	C₂×#10587487	1	1	yes	characteristic
220	C₄	#11763156	1	1	yes	characteristic
221	C₄	#11763214	1	1	yes	characteristic
222	C₂	—	2	1	no	—
223	C₂	—	2	1	no	—
224	C₂	#14529745	1	1	yes	characteristic
225	C₂	C₂×#11762623	1	1	yes	characteristic
226	C₂	C₂×#11762626	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#10587492
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₄×C₁₇×M₃₂	C₂
socle	C₂²×C₁₇	C₂×#102
prime radical	C₂³×C₁₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₄×M₃₂	D₁₇
hypocenter	C₁₇	#13290
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
31. C₁₇×M₃₂	Q₈	semidirect
144. M₃₂	Dic₃₄	semidirect
158. C₁₇	#13290	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763214	2
#14528956	2
#14526893	2
C₄×C₁₇×M₃₂	2
#13290	17