Subgroups — #17952262

Order 4352.

502 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	C₄×C₄⋊#3624	C₂	1	2	yes	maximal
2	C₂×#11763650	C₂	1	2	yes	maximal
3	#14543606	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#3624	C₄	1	4	yes	—
6	C₂×C₄×#3624	C₂²	1	4	yes	—
7	#11763675	C₄	1	4	yes	—
8	#11763650	—	2	4	no	—
9	C₂×C₄×#3622	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
14	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
16	C₂²×#3624	C₂×C₄	1	8	yes	—
17	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
22	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
23	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
24	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
25	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
26	C₄⋊#3624	—	2	8	no	—
27	C₄×#3624	—	2	4	no	—
28	C₄×#3624	—	2	4	no	—
29	C₄×C₈×C₁₇	C₈	1	4	yes	complemented
30	C₈×Dic₁₇	—	2	2	no	—
31	C₈×Dic₁₇	—	2	2	no	—
32	C₈×Dic₁₇	—	2	2	no	—
33	C₈×Dic₁₇	D₄	1	2	yes	—
34	C₈×Dic₁₇	Q₈	1	2	yes	—
35	C₄×#3622	C₈	1	4	yes	complemented
36	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
38	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
39	C₂²×Dic₁₇	C₄²	1	2	yes	—
40	C₂²×Dic₁₇	C₂²×C₄	1	2	yes	—
41	C₂×#3624	—	2	8	no	—
42	C₂×#3624	—	2	8	no	—
43	C₂×#3624	—	2	8	no	—
44	C₂×C₈×C₁₇	—	2	1	no	—
45	C₂×C₈×C₁₇	—	2	1	no	—
46	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
50	C₄²×C₁₇	C₂×C₈	1	2	yes	—
51	C₄²×C₁₇	C₂×C₈	1	2	yes	—
52	C₄×Dic₁₇	—	2	1	no	—
53	C₄×Dic₁₇	—	2	1	no	—
54	C₄×Dic₁₇	—	2	4	no	—
55	C₄×Dic₁₇	M₁₆	1	8	yes	—
56	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
57	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
58	C₄×Dic₁₇	#41	1	2	yes	—
59	C₂×#3622	—	2	1	no	—
60	C₂×#3622	—	2	1	no	—
61	C₂×#3622	C₄⋊C₄	1	2	yes	—
62	C₂×#3622	C₂×C₈	1	2	yes	—
63	C₂×#3622	C₂×C₈	1	2	yes	—
64	C₂×#3622	M₁₆	1	2	yes	—
65	#8260	—	17	1	no	maximal, Sylow
66	C₂³×C₁₇	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₁₇	—	2	2	no	—
68	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
69	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
70	C₂×C₄×C₁₇	#97	1	4	yes	—
71	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
77	C₂×Dic₁₇	—	2	2	no	—
78	C₂×Dic₁₇	—	2	2	no	—
79	C₂×Dic₁₇	#97	1	4	yes	—
80	C₂×Dic₁₇	C₄×D₄	1	2	yes	—
81	C₂×Dic₁₇	C₄×Q₈	1	2	yes	—
82	C₂×Dic₁₇	C₂×M₁₆	1	4	yes	—
83	C₂×Dic₁₇	#131	1	4	yes	—
84	#3624	—	4	8	no	—
85	C₈×C₁₇	—	2	2	no	—
86	C₈×C₁₇	C₄⋊C₈	1	4	yes	complemented
87	#3622	—	2	2	no	—
88	#3622	C₄⋊C₈	1	4	yes	complemented
89	C₄×C₄⋊C₈	—	17	2	no	—
90	C₄²×C₈	—	17	1	no	—
91	C₂×#322	—	17	2	no	—
92	#1273	—	17	2	no	—
93	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
95	C₂²×C₁₇	C₂×#97	1	2	yes	—
96	C₂²×C₁₇	C₄×M₁₆	1	1	yes	characteristic
97	C₂²×C₁₇	#405	1	2	yes	—
98	C₄×C₁₇	—	2	2	no	—
99	C₄×C₁₇	#322	1	4	yes	complemented
100	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
101	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₄×C₁₇	#431	1	1	yes	characteristic
103	C₄×C₁₇	#431	1	1	yes	characteristic
104	Dic₁₇	—	2	4	no	—
105	Dic₁₇	#436	1	4	yes	—
106	Dic₁₇	#446	1	4	yes	—
107	C₂×C₄⋊C₈	—	17	4	no	—
108	C₂×C₄×C₈	—	17	1	no	—
109	C₂×C₄×C₈	—	17	1	no	—
110	C₂×C₄×C₈	—	17	4	no	—
111	C₂×C₄×C₈	—	17	1	no	—
112	C₂×C₄×C₈	—	17	1	no	—
113	C₂×C₄×C₈	—	17	1	no	—
114	C₂×C₄×C₈	#595	1	1	yes	characteristic
115	C₄³	—	17	1	no	—
116	#336	—	17	4	no	—
117	#322	—	34	4	no	—
118	C₂×C₁₇	C₂×#322	1	1	yes	characteristic
119	C₂×C₁₇	#1450	1	1	yes	characteristic
120	C₂×C₁₇	#1451	1	2	yes	—
121	C₂×C₁₇	C₄×C₄⋊C₈	1	1	yes	characteristic
122	C₂×C₁₇	#1770	1	2	yes	—
123	C₂²×C₈	—	17	1	no	—
124	C₂²×C₈	—	17	1	no	—
125	C₂²×C₈	—	17	8	no	—
126	C₂²×C₈	C₂×#595	1	1	yes	characteristic
127	C₂²×C₈	C₂×#595	1	1	yes	characteristic
128	C₂×C₄²	—	17	2	no	—
129	C₂×C₄²	—	17	1	no	—
130	C₂×C₄²	—	17	2	no	—
131	C₂×C₄²	—	17	1	no	—
132	C₂×C₄²	C₂×#595	1	1	yes	characteristic
133	C₄⋊C₈	—	34	8	no	—
134	C₄×C₈	—	17	2	no	—
135	C₄×C₈	—	17	2	no	—
136	C₄×C₈	—	17	4	no	—
137	C₄×C₈	—	34	2	no	—
138	C₄×C₈	—	34	4	no	—
139	C₄×C₈	—	34	4	no	—
140	C₄×C₈	—	34	2	no	—
141	C₄×C₈	—	34	2	no	—
142	C₄×C₈	#3624	1	4	yes	complemented
143	C₁₇	#8260	1	1	yes	characteristic, complemented, Sylow
144	C₂²×C₄	—	17	2	no	—
145	C₂²×C₄	—	17	2	no	—
146	C₂²×C₄	C₄×#595	1	1	yes	characteristic
147	C₂²×C₄	C₄×#595	1	1	yes	characteristic
148	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	17	2	no	—
152	C₂×C₈	—	17	2	no	—
153	C₂×C₈	—	17	2	no	—
154	C₂×C₈	—	17	2	no	—
155	C₂×C₈	—	34	1	no	—
156	C₂×C₈	—	34	1	no	—
157	C₂×C₈	—	34	8	no	—
158	C₂×C₈	—	34	8	no	—
159	C₂×C₈	—	34	8	no	—
160	C₂×C₈	#63243	1	2	yes	—
161	C₂×C₈	C₄⋊#595	1	2	yes	—
162	C₂×C₈	C₂×#3624	1	2	yes	—
163	C₂×C₈	C₂×#3624	1	2	yes	—
164	C₄²	—	17	1	no	—
165	C₄²	—	17	1	no	—
166	C₄²	—	17	2	no	—
167	C₄²	—	17	8	no	—
168	C₄²	—	34	4	no	—
169	C₄²	—	34	1	no	—
170	C₄²	—	34	1	no	—
171	C₄²	#63242	1	2	yes	—
172	C₄²	C₂×#3624	1	2	yes	—
173	C₂³	C₂×C₄×#595	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	17	2	no	—
176	C₂×C₄	—	17	2	no	—
177	C₂×C₄	—	17	4	no	—
178	C₂×C₄	—	17	4	no	—
179	C₂×C₄	—	17	4	no	—
180	C₂×C₄	—	34	2	no	—
181	C₂×C₄	—	34	2	no	—
182	C₂×C₄	#10587532	1	4	yes	—
183	C₂×C₄	C₄×#3624	1	2	yes	—
184	C₂×C₄	C₄×#3624	1	2	yes	—
185	C₂×C₄	C₂×#63242	1	1	yes	characteristic
186	C₂×C₄	C₂×#63243	1	1	yes	characteristic
187	C₂×C₄	D₁₇⋊M₁₆	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
189	C₂×C₄	#10587677	1	1	yes	characteristic
190	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
191	C₈	—	2	2	no	—
192	C₈	—	17	4	no	—
193	C₈	—	34	2	no	—
194	C₈	—	68	8	no	—
195	C₈	C₄⋊#3624	1	4	yes	complemented
196	C₂²	C₄×C₄⋊#595	1	1	yes	characteristic
197	C₂²	C₂×C₄×#3624	1	1	yes	characteristic
198	C₂²	C₂×#10587532	1	2	yes	—
199	C₂²	C₄×#63243	1	1	yes	characteristic
200	C₂²	#11763772	1	2	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	17	4	no	—
203	C₄	—	17	4	no	—
204	C₄	—	34	4	no	—
205	C₄	#11763650	1	4	yes	complemented
206	C₄	C₂×C₄⋊#3624	1	1	yes	characteristic
207	C₄	#11763810	1	1	yes	characteristic
208	C₄	#11763825	1	1	yes	characteristic
209	C₄	#11763828	1	1	yes	characteristic
210	C₂	C₂×#11763650	1	1	yes	characteristic
211	C₂	#14543987	1	1	yes	characteristic
212	C₂	#14543994	1	2	yes	—
213	C₂	C₄×C₄⋊#3624	1	1	yes	characteristic
214	C₂	#14544578	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#595
commutator	C₂²×C₁₇	C₂×C₄×C₈
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂×C₄²
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#8260
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
29. C₄×C₈×C₁₇	C₈	semidirect
35. C₄×#3622	C₈	semidirect
86. C₈×C₁₇	C₄⋊C₈	semidirect
88. #3622	C₄⋊C₈	semidirect
99. C₄×C₁₇	#322	semidirect
142. C₄×C₈	#3624	semidirect
143. C₁₇	#8260	semidirect
195. C₈	C₄⋊#3624	semidirect
205. C₄	#11763650	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#3624	2
C₂×#11763650	2
#14543606	2
C₄×C₈×Dic₁₇	2
#8260	17