Subgroups — #16933028

Order 4352.

298 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14529559	C₂	1	2	yes	maximal
2	#14527543	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762562	C₂	1	1	yes	maximal, characteristic
4	#14526680	C₂	1	2	yes	maximal
5	C₈×M₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587493	—	2	2	no	—
7	#11762833	C₂²	1	2	yes	—
8	C₂×C₄⋊#3622	C₂²	1	2	yes	—
9	#11762573	—	2	2	no	—
10	#11762562	—	2	1	no	—
11	#11762562	—	2	1	no	—
12	#11762562	C₄	1	2	yes	—
13	#11762562	C₄	1	2	yes	—
14	C₁₇×#431	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×#322	—	2	1	no	—
18	C₈²×C₁₇	—	2	1	no	—
19	C₂×M₁₆×C₁₇	D₄	1	1	yes	characteristic
20	C₂×M₁₆×C₁₇	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
22	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
24	C₂×#63223	—	2	2	no	—
25	C₂²×#3622	—	2	2	no	—
26	C₁₇×C₄⋊C₈	—	2	1	no	—
27	C₁₇×C₂²⋊C₈	Q₈	1	2	yes	—
28	C₁₇×#97	—	2	1	no	—
29	C₄×C₈×C₁₇	—	2	1	no	—
30	C₄×C₈×C₁₇	—	2	1	no	—
31	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
34	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
35	#10587493	—	4	4	no	—
36	C₄⋊#3622	—	2	2	no	—
37	C₄⋊#3622	—	2	2	no	—
38	C₄⋊#3622	—	2	2	no	—
39	C₄⋊#3622	C₂×C₄	1	4	yes	—
40	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
43	M₁₆×C₁₇	—	2	1	no	—
44	M₁₆×C₁₇	—	2	1	no	—
45	M₁₆×C₁₇	D₈	1	2	yes	—
46	M₁₆×C₁₇	Q₁₆	1	2	yes	—
47	C₂×C₈×C₁₇	—	2	1	no	—
48	C₂×C₈×C₁₇	—	2	1	no	—
49	C₂×C₈×C₁₇	—	2	2	no	—
50	C₂×C₈×C₁₇	—	2	2	no	—
51	C₂×C₈×C₁₇	—	4	1	no	—
52	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
56	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₁₇	#41	1	1	yes	characteristic
59	C₄²×C₁₇	#41	1	1	yes	characteristic
60	#63223	—	4	2	no	—
61	#63223	—	4	2	no	—
62	C₂×#3622	—	2	4	no	—
63	C₂×#3622	—	4	2	no	—
64	C₂×#3622	—	4	2	no	—
65	C₂×#3622	—	4	2	no	—
66	#9624	—	17	1	no	maximal, Sylow
67	C₂³×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₁₇	—	2	1	no	—
69	C₂×C₄×C₁₇	—	2	1	no	—
70	C₂×C₄×C₁₇	—	4	1	no	—
71	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₁₇	#125	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
80	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
81	C₈×C₁₇	—	2	1	no	—
82	C₈×C₁₇	—	2	1	no	—
83	C₈×C₁₇	—	2	1	no	—
84	C₈×C₁₇	—	2	2	no	—
85	C₈×C₁₇	—	2	2	no	—
86	C₈×C₁₇	—	4	2	no	—
87	C₈×C₁₇	#108	1	2	yes	complemented
88	#3622	—	4	4	no	—
89	#3622	—	8	2	no	—
90	#1950	—	17	2	no	—
91	#1567	—	17	1	no	—
92	C₂×#335	—	17	1	no	—
93	C₈×M₁₆	D₁₇	1	1	yes	characteristic
94	#1301	—	17	2	no	—
95	C₂²×C₁₇	—	2	1	no	—
96	C₂²×C₁₇	—	2	1	no	—
97	C₂²×C₁₇	#389	1	1	yes	characteristic
98	C₂²×C₁₇	#499	1	1	yes	characteristic
99	C₂²×C₁₇	#500	1	1	yes	characteristic
100	C₄×C₁₇	—	2	1	no	—
101	C₄×C₁₇	—	2	1	no	—
102	C₄×C₁₇	—	2	1	no	—
103	C₄×C₁₇	C₂×#108	1	1	yes	characteristic
104	C₄×C₁₇	#430	1	1	yes	characteristic
105	C₄×C₁₇	C₄×D₈	1	1	yes	characteristic
106	C₄×C₁₇	C₄×Q₁₆	1	1	yes	characteristic
107	C₄×C₁₇	#466	1	1	yes	characteristic
108	C₄×C₁₇	#467	1	1	yes	characteristic
109	#431	C₂×D₁₇	1	1	yes	characteristic
110	C₂×#108	—	34	2	no	—
111	#423	—	17	2	no	—
112	C₂×C₄⋊C₈	—	17	2	no	—
113	C₄×M₁₆	C₂×D₁₇	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
115	#341	—	34	2	no	—
116	#335	—	17	2	no	—
117	#335	—	17	2	no	—
118	#335	—	34	1	no	—
119	#335	—	34	1	no	—
120	#322	—	2	1	no	—
121	C₈²	—	2	1	no	—
122	C₂×C₁₇	—	2	1	no	—
123	C₂×C₁₇	—	2	1	no	—
124	C₂×C₁₇	#1942	1	1	yes	characteristic
125	C₂×C₁₇	#1950	1	1	yes	characteristic
126	C₂×C₁₇	#1953	1	1	yes	characteristic
127	C₂×M₁₆	—	34	2	no	—
128	C₂×M₁₆	D₆₈	1	1	yes	characteristic
129	C₂×M₁₆	D₆₈	1	1	yes	characteristic
130	C₂²×C₈	—	34	2	no	—
131	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
132	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
133	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
134	#108	—	68	4	no	—
135	C₄⋊C₈	—	2	1	no	—
136	C₄⋊C₈	—	17	4	no	—
137	C₄⋊C₈	—	34	2	no	—
138	C₄⋊C₈	—	34	2	no	—
139	C₄⋊C₈	—	34	2	no	—
140	C₂²⋊C₈	Dic₃₄	1	2	yes	—
141	#97	—	2	1	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
145	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
146	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
148	C₁₇	#9624	1	1	yes	characteristic, complemented, Sylow
149	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
150	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
151	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
152	M₁₆	—	2	1	no	—
153	M₁₆	—	2	1	no	—
154	M₁₆	—	68	2	no	—
155	M₁₆	—	68	2	no	—
156	M₁₆	D₁₃₆	1	2	yes	—
157	M₁₆	Dic₆₈	1	2	yes	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	4	1	no	—
163	C₂×C₈	—	34	4	no	—
164	C₂×C₈	—	68	2	no	—
165	C₂×C₈	—	68	2	no	—
166	C₂×C₈	—	68	2	no	—
167	C₂×C₈	#63225	1	1	yes	characteristic
168	C₂×C₈	#63225	1	1	yes	characteristic
169	C₂×C₈	#63225	1	1	yes	characteristic
170	C₂×C₈	#63225	1	1	yes	characteristic
171	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
172	C₄²	#63252	1	1	yes	characteristic
173	C₄²	#63252	1	1	yes	characteristic
174	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
175	C₂³	#10587555	1	1	yes	characteristic
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	4	1	no	—
179	C₂×C₄	C₄×Dic₃₄	1	1	yes	characteristic
180	C₂×C₄	#10587556	1	1	yes	characteristic
181	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
182	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
183	C₂×C₄	#10587590	1	1	yes	characteristic
184	C₂×C₄	#10587590	1	1	yes	characteristic
185	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
186	C₂×C₄	C₂×#63225	1	1	yes	characteristic
187	C₂×C₄	#10587610	1	1	yes	characteristic
188	C₂×C₄	#10587616	1	1	yes	characteristic
189	C₈	—	2	2	no	—
190	C₈	—	2	2	no	—
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	2	1	no	—
194	C₈	—	4	2	no	—
195	C₈	—	68	4	no	—
196	C₈	—	136	2	no	—
197	C₈	#10587493	1	2	yes	complemented
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	#11762707	1	1	yes	characteristic
201	C₂²	#11763211	1	1	yes	characteristic
202	C₂²	#11763214	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	C₂×#10587493	1	1	yes	characteristic
207	C₄	#11762857	1	1	yes	characteristic
208	C₄	C₄×D₁₃₆	1	1	yes	characteristic
209	C₄	C₄×Dic₆₈	1	1	yes	characteristic
210	C₄	#11763045	1	1	yes	characteristic
211	C₄	#11763050	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	#14529517	1	1	yes	characteristic
215	C₂	#14529559	1	1	yes	characteristic
216	C₂	#14529574	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10587616
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₈×M₁₆×C₁₇	C₂
socle	C₂²×C₁₇	#389
prime radical	C₂³×C₁₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₈×M₁₆	D₁₇
hypocenter	C₁₇	#9624
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
87. C₈×C₁₇	#108	semidirect
148. C₁₇	#9624	semidirect
197. C₈	#10587493	semidirect

Maximal subgroups

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

subgroup	index
#14529559	2
#14527543	2
C₂×#11762562	2
#14526680	2
C₈×M₁₆×C₁₇	2
#9624	17