Subgroups — #16952884

Order 4352.

366 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	#14528663	C₂	1	1	yes	maximal, characteristic
3	#14526891	C₂	1	2	yes	maximal
4	#14526882	C₂	1	2	yes	maximal
5	C₂×C₁₇×#498	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587483	C₂²	1	2	yes	—
7	C₂×#10587490	C₂²	1	1	yes	characteristic
8	#11762847	C₂²	1	1	yes	characteristic
9	#11762764	—	2	1	no	—
10	#11762630	—	2	2	no	—
11	#11762569	—	2	1	no	—
12	C₂×C₁₇×C₄⋊Q₈	—	2	1	no	—
13	C₁₇×#498	—	2	2	no	—
14	C₁₇×#498	C₄	1	4	yes	complemented
15	C₂×C₁₇×#106	—	2	1	no	—
16	C₂×C₁₇×#106	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₂²×Q₈×C₁₇	—	4	1	no	—
19	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
21	C₁₇×C₄⋊Q₈	—	2	4	no	—
22	C₁₇×C₄⋊Q₈	—	4	2	no	—
23	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
24	C₂×C₄⋊C₄×C₁₇	—	4	1	no	—
25	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
26	#10587612	—	2	1	no	—
27	C₂×#63223	—	2	1	no	—
28	C₁₇×#106	—	2	4	no	—
29	C₁₇×#106	—	2	2	no	—
30	C₁₇×#106	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₁₇	D₄	1	2	yes	—
33	#10587490	—	2	1	no	—
34	#10587490	Q₈	1	2	yes	complemented
35	#10587489	Q₈	1	2	yes	complemented
36	#10587488	—	2	1	no	—
37	#10587486	—	4	1	no	—
38	C₈×Dic₁₇	—	4	1	no	—
39	#10587483	—	2	2	no	—
40	#10587483	—	2	2	no	—
41	C₂×#63214	—	2	2	no	—
42	C₂×Q₈×C₁₇	—	4	4	no	—
43	C₂×Q₈×C₁₇	—	8	4	no	—
44	C₂²×C₄×C₁₇	—	2	1	no	—
45	C₂²×C₄×C₁₇	—	4	2	no	—
46	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
48	C₂×C₈×C₁₇	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₁₇	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
51	C₂×C₈×C₁₇	C₂²×C₄	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	C₄⋊C₄×C₁₇	—	2	4	no	—
58	C₄⋊C₄×C₁₇	—	4	4	no	—
59	C₄⋊C₄×C₁₇	—	4	2	no	—
60	C₄²×C₁₇	—	2	1	no	—
61	C₄²×C₁₇	C₂²⋊C₄	1	2	yes	—
62	C₂²⋊Dic₁₇	—	4	1	no	—
63	C₄⋊Dic₁₇	—	2	2	no	—
64	C₄×Dic₁₇	—	4	1	no	—
65	#63223	—	2	2	no	—
66	#63223	—	4	1	no	—
67	C₂×#3622	—	4	1	no	—
68	#63214	—	4	2	no	—
69	#13272	—	17	1	no	maximal, Sylow
70	C₂³×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
71	Q₈×C₁₇	—	8	8	no	—
72	C₂×C₄×C₁₇	—	2	4	no	—
73	C₂×C₄×C₁₇	—	2	2	no	—
74	C₂×C₄×C₁₇	—	4	4	no	—
75	C₂×C₄×C₁₇	—	8	4	no	—
76	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
81	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂×Dic₁₇	—	4	2	no	—
83	C₈×C₁₇	#103	1	2	yes	—
84	C₈×C₁₇	#103	1	2	yes	—
85	C₈×C₁₇	C₄×Q₈	1	1	yes	characteristic
86	C₈×C₁₇	C₄×Q₈	1	1	yes	characteristic
87	C₈×C₁₇	#124	1	1	yes	characteristic
88	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
89	#3622	—	4	2	no	—
90	C₂×#498	D₁₇	1	1	yes	characteristic
91	C₂×#364	—	17	1	no	—
92	#1772	—	17	1	no	—
93	#1386	—	17	2	no	—
94	#1383	—	17	2	no	—
95	C₂²×C₁₇	—	2	2	no	—
96	C₂²×C₁₇	#418	1	1	yes	characteristic
97	C₂²×C₁₇	#476	1	1	yes	characteristic
98	C₂²×C₁₇	#477	1	1	yes	characteristic
99	C₄×C₁₇	—	2	4	no	—
100	C₄×C₁₇	—	8	4	no	—
101	C₄×C₁₇	#391	1	1	yes	characteristic
102	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
103	C₄×C₁₇	#476	1	1	yes	characteristic
104	C₄×C₁₇	#477	1	1	yes	characteristic
105	Dic₁₇	—	8	2	no	—
106	C₂×C₄⋊Q₈	—	2	1	no	—
107	C₂×M₃₂	—	17	2	no	—
108	#498	—	2	2	no	—
109	#498	Dic₁₇	1	4	yes	complemented
110	C₂×#108	—	17	1	no	—
111	#427	—	17	1	no	—
112	C₂×#106	—	2	1	no	—
113	C₂×#106	C₂×D₁₇	1	2	yes	—
114	#405	—	34	1	no	—
115	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
116	#364	—	34	2	no	—
117	#339	—	34	1	no	—
118	C₂×C₁₇	—	2	1	no	—
119	C₂×C₁₇	—	2	1	no	—
120	C₂×C₁₇	#1987	1	1	yes	characteristic
121	C₂×C₁₇	#2230	1	2	yes	—
122	C₂²×Q₈	—	4	1	no	—
123	C₂×M₁₆	—	34	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	#117	—	34	1	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	4	1	no	—
131	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
132	M₃₂	—	34	2	no	—
133	M₃₂	—	34	2	no	—
134	C₂×C₁₆	—	34	2	no	—
135	#108	—	17	2	no	—
136	#108	—	34	1	no	—
137	#107	—	17	2	no	—
138	#106	—	2	4	no	—
139	#106	—	2	2	no	—
140	#106	—	34	1	no	—
141	#106	C₂×Dic₁₇	1	4	yes	—
142	#97	—	68	1	no	—
143	C₄×C₈	—	68	1	no	—
144	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
145	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
146	C₁₇	#13272	1	1	yes	characteristic, complemented, Sylow
147	C₂×Q₈	—	4	4	no	—
148	C₂×Q₈	—	8	4	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
152	M₁₆	—	34	2	no	—
153	M₁₆	—	68	1	no	—
154	C₂×C₈	—	68	1	no	—
155	C₂×C₈	#63230	1	1	yes	characteristic
156	C₂×C₈	#63230	1	1	yes	characteristic
157	C₂×C₈	#63231	1	2	yes	—
158	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
159	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
160	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
161	C₂×C₈	#63256	1	1	yes	characteristic
162	C₂×C₈	#63256	1	1	yes	characteristic
163	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
164	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	34	2	no	—
169	C₂²⋊C₄	—	68	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	68	1	no	—
172	C₄²	C₂²⋊Dic₁₇	1	2	yes	—
173	C₁₆	—	68	2	no	—
174	C₂³	#10587618	1	1	yes	characteristic
175	Q₈	—	8	8	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	8	4	no	—
180	C₂×C₄	—	68	2	no	—
181	C₂×C₄	#10587618	1	1	yes	characteristic
182	C₂×C₄	C₂×#63230	1	1	yes	characteristic
183	C₂×C₄	C₂×#63231	1	1	yes	characteristic
184	C₂×C₄	#10587632	1	1	yes	characteristic
185	C₂×C₄	#10587632	1	1	yes	characteristic
186	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
187	C₈	—	68	2	no	—
188	C₈	#10587506	1	2	yes	—
189	C₈	#10587506	1	2	yes	—
190	C₈	#10587630	1	1	yes	characteristic
191	C₈	Q₈×Dic₁₇	1	1	yes	characteristic
192	C₈	Q₈×Dic₁₇	1	1	yes	characteristic
193	C₈	#10587633	1	1	yes	characteristic
194	C₂²	—	2	2	no	—
195	C₂²	#11762814	1	1	yes	characteristic
196	C₂²	#11763096	1	1	yes	characteristic
197	C₂²	#11763101	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	8	4	no	—
200	C₄	—	136	2	no	—
201	C₄	#11762716	1	1	yes	characteristic
202	C₄	C₂×#10587506	1	1	yes	characteristic
203	C₄	#11763096	1	1	yes	characteristic
204	C₄	#11763101	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	#14529733	1	1	yes	characteristic
208	C₂	#14530952	1	2	yes	—

	subgroup	G/H
center	C₂²	#11763096
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#498	C₂
socle	C₂²×C₁₇	#476
prime radical	C₂³×C₁₇	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×#498	D₁₇
hypocenter	C₁₇	#13272
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#498

Complements

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

N	H ≅ G/N	type
14. C₁₇×#498	C₄	semidirect
34. #10587490	Q₈	semidirect
35. #10587489	Q₈	semidirect
109. #498	Dic₁₇	semidirect
146. C₁₇	#13272	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
#14528663	2
#14526891	2
#14526882	2
C₂×C₁₇×#498	2
#13272	17