Subgroups — #16952349

Order 4352.

284 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	#14530549	C₂	1	2	yes	maximal
2	C₂×#11762630	C₂	1	1	yes	maximal, characteristic
3	#14528663	C₂	1	1	yes	maximal, characteristic
4	#14526871	C₂	1	2	yes	maximal
5	C₄×C₁₇×M₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587483	C₂²	1	2	yes	—
7	C₂×#10587490	C₄	1	1	yes	characteristic
8	#11762847	C₄	1	1	yes	characteristic
9	#11762764	C₂²	1	2	yes	—
10	#11762630	—	2	1	no	—
11	#11762630	—	2	1	no	—
12	#11762629	—	2	4	no	—
13	#11762569	C₄	1	2	yes	—
14	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
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×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	#10587612	C₂×C₄	1	2	yes	—
23	C₂×#63223	C₂×C₄	1	2	yes	—
24	C₁₇×M₃₂	—	2	1	no	—
25	C₁₇×M₃₂	—	2	1	no	—
26	C₁₇×M₃₂	D₄	1	2	yes	—
27	C₁₇×M₃₂	Q₈	1	2	yes	—
28	C₂×C₁₆×C₁₇	—	2	1	no	—
29	C₂×C₁₆×C₁₇	—	2	1	no	—
30	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
32	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
34	#10587490	C₈	1	2	yes	—
35	#10587490	C₈	1	2	yes	—
36	#10587489	—	2	1	no	—
37	#10587488	—	2	1	no	—
38	#10587486	—	2	2	no	—
39	C₈×Dic₁₇	—	2	2	no	—
40	#10587483	—	2	2	no	—
41	#10587483	—	2	2	no	—
42	C₂×#63214	—	2	2	no	—
43	C₂²×C₄×C₁₇	—	2	1	no	—
44	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈×C₁₇	M₁₆	1	1	yes	characteristic
50	C₂×C₈×C₁₇	M₁₆	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	#41	1	1	yes	characteristic
56	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
57	C₄²×C₁₇	—	2	1	no	—
58	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
59	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
60	C₂²⋊Dic₁₇	—	2	2	no	—
61	C₄⋊Dic₁₇	—	2	2	no	—
62	C₄×Dic₁₇	—	2	2	no	—
63	#63223	—	2	2	no	—
64	#63223	C₂×C₈	1	4	yes	—
65	C₂×#3622	—	2	2	no	—
66	C₁₆×C₁₇	—	2	2	no	—
67	C₁₆×C₁₇	—	2	2	no	—
68	#63214	—	4	2	no	—
69	#13164	—	17	1	no	maximal, Sylow
70	C₂³×C₁₇	#131	1	1	yes	characteristic
71	C₂×C₄×C₁₇	—	2	1	no	—
72	C₂×C₄×C₁₇	—	2	1	no	—
73	C₂×C₄×C₁₇	—	4	2	no	—
74	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
80	C₂×Dic₁₇	—	4	2	no	—
81	C₈×C₁₇	C₄⋊C₈	1	1	yes	characteristic
82	C₈×C₁₇	C₄⋊C₈	1	1	yes	characteristic
83	C₈×C₁₇	C₄⋊C₈	1	1	yes	characteristic
84	C₈×C₁₇	C₄⋊C₈	1	1	yes	characteristic
85	C₈×C₁₇	#121	1	1	yes	characteristic
86	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
87	C₈×C₁₇	#125	1	1	yes	characteristic
88	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
89	#3622	—	2	4	no	—
90	#2153	—	17	2	no	—
91	C₂×#364	—	17	1	no	—
92	C₄×M₃₂	D₁₇	1	1	yes	characteristic
93	#1772	—	17	1	no	—
94	#1378	—	17	2	no	—
95	C₂²×C₁₇	—	4	1	no	—
96	C₂²×C₁₇	#424	1	1	yes	characteristic
97	C₂²×C₁₇	#436	1	1	yes	characteristic
98	C₂²×C₁₇	C₈×Q₈	1	1	yes	characteristic
99	C₄×C₁₇	—	2	2	no	—
100	C₄×C₁₇	—	4	1	no	—
101	C₄×C₁₇	#389	1	1	yes	characteristic
102	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
103	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
104	C₄×C₁₇	#446	1	1	yes	characteristic
105	Dic₁₇	—	8	2	no	—
106	C₂×M₃₂	—	17	2	no	—
107	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
108	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
109	C₂×#108	—	17	1	no	—
110	#427	—	17	1	no	—
111	#405	—	17	2	no	—
112	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
113	#364	—	34	4	no	—
114	#364	—	34	1	no	—
115	#364	—	34	1	no	—
116	#346	—	2	1	no	—
117	C₄×C₁₆	—	2	1	no	—
118	#339	—	17	2	no	—
119	C₂×C₁₇	—	2	1	no	—
120	C₂×C₁₇	—	2	1	no	—
121	C₂×C₁₇	—	2	1	no	—
122	C₂×C₁₇	#1939	1	1	yes	characteristic
123	C₂×M₁₆	—	17	2	no	—
124	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
126	#117	—	17	2	no	—
127	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
128	M₃₂	—	2	1	no	—
129	M₃₂	—	2	1	no	—
130	M₃₂	—	34	2	no	—
131	M₃₂	—	34	2	no	—
132	M₃₂	Dic₃₄	1	2	yes	—
133	M₃₂	D₆₈	1	2	yes	—
134	C₂×C₁₆	—	2	1	no	—
135	C₂×C₁₆	—	2	1	no	—
136	C₂×C₁₆	—	34	2	no	—
137	#108	—	17	2	no	—
138	#108	—	17	2	no	—
139	#107	—	34	1	no	—
140	#106	—	34	1	no	—
141	#97	—	34	2	no	—
142	C₄×C₈	—	34	2	no	—
143	C₄×C₈	Dic₃₄	1	1	yes	characteristic
144	C₄×C₈	Dic₃₄	1	1	yes	characteristic
145	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
146	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
147	C₁₇	#13164	1	1	yes	characteristic, complemented, Sylow
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
150	M₁₆	—	17	4	no	—
151	M₁₆	—	34	2	no	—
152	C₂×C₈	—	34	2	no	—
153	C₂×C₈	C₈×D₁₇	1	1	yes	characteristic
154	C₂×C₈	C₈×D₁₇	1	1	yes	characteristic
155	C₂×C₈	#63218	1	1	yes	characteristic
156	C₂×C₈	#63218	1	1	yes	characteristic
157	C₂×C₈	#63225	1	1	yes	characteristic
158	C₂×C₈	#63225	1	1	yes	characteristic
159	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
160	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
161	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
162	C₂×C₈	#63252	1	1	yes	characteristic
163	C₂×C₈	#63252	1	1	yes	characteristic
164	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
165	C₄⋊C₄	—	34	2	no	—
166	C₂²⋊C₄	—	34	2	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	34	2	no	—
169	C₄²	#63225	1	1	yes	characteristic
170	C₄²	#63225	1	1	yes	characteristic
171	C₁₆	—	2	2	no	—
172	C₁₆	—	2	2	no	—
173	C₁₆	—	68	2	no	—
174	C₂³	#10587587	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	68	2	no	—
179	C₂×C₄	C₄×Dic₃₄	1	1	yes	characteristic
180	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
181	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
182	C₂×C₄	C₂×#63218	1	1	yes	characteristic
183	C₂×C₄	#10587587	1	1	yes	characteristic
184	C₂×C₄	C₂×#63225	1	1	yes	characteristic
185	C₈	—	34	4	no	—
186	C₈	#10587485	1	1	yes	characteristic
187	C₈	#10587485	1	1	yes	characteristic
188	C₈	#10587485	1	1	yes	characteristic
189	C₈	#10587485	1	1	yes	characteristic
190	C₈	#10587555	1	1	yes	characteristic
191	C₈	#10587556	1	1	yes	characteristic
192	C₈	#10587610	1	1	yes	characteristic
193	C₈	#10587616	1	1	yes	characteristic
194	C₂²	—	4	1	no	—
195	C₂²	#11762835	1	1	yes	characteristic
196	C₂²	#11762888	1	1	yes	characteristic
197	C₂²	C₈×Dic₃₄	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	4	1	no	—
200	C₄	—	136	2	no	—
201	C₄	#11762707	1	1	yes	characteristic
202	C₄	C₂×#10587485	1	1	yes	characteristic
203	C₄	C₈×D₆₈	1	1	yes	characteristic
204	C₄	#11762939	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	#14529502	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#10587616
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₄×C₁₇×M₃₂	C₂
socle	C₂×C₁₇	#1939
prime radical	C₂³×C₁₇	#131
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₄×M₃₂	D₁₇
hypocenter	C₁₇	#13164
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
147. C₁₇	#13164	semidirect

Maximal subgroups

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

subgroup	index
#14530549	2
C₂×#11762630	2
#14528663	2
#14526871	2
C₄×C₁₇×M₃₂	2
#13164	17