Subgroups — #17952272

Order 4352.

494 conjugacy classes of subgroups.

The subgroup lattice has 236 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	1	yes	maximal, characteristic
3	C₂×C₈×#3624	C₂	1	1	yes	maximal, characteristic
4	#14543606	C₂	1	2	yes	maximal
5	#97×Dic₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#3624	C₄	1	4	yes	—
7	C₂×C₄×#3624	C₂²	1	2	yes	—
8	C₂×C₄×#3624	C₂²	1	2	yes	—
9	#11763675	C₄	1	4	yes	—
10	#11763650	—	2	2	no	—
11	C₈×#3624	—	2	2	no	—
12	C₂×#10587474	C₄	1	1	yes	characteristic
13	C₂×#10587486	C₄	1	1	yes	characteristic
14	C₂×#10587486	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
17	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
18	C₂×C₁₇×#97	C₄	1	1	yes	characteristic
19	C₂²×#3624	C₂×C₄	1	4	yes	—
20	C₂²×#3624	C₂×C₄	1	4	yes	—
21	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
26	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
27	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
28	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
29	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
30	C₄⋊#3624	—	2	8	no	—
31	C₄×#3624	—	2	2	no	—
32	C₄×#3624	—	2	2	no	—
33	C₄×#3624	—	2	2	no	—
34	C₄×#3624	—	2	2	no	—
35	C₁₇×#97	C₈	1	4	yes	complemented
36	#10587486	—	2	2	no	—
37	#10587486	—	2	2	no	—
38	C₈×Dic₁₇	—	2	2	no	—
39	C₈×Dic₁₇	D₄	1	2	yes	—
40	C₈×Dic₁₇	Q₈	1	2	yes	—
41	#10587474	C₈	1	4	yes	complemented
42	C₂²×C₄×C₁₇	C₄²	1	2	yes	—
43	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₁₇	C₄²	1	1	yes	characteristic
45	C₂²×Dic₁₇	C₄²	1	1	yes	characteristic
46	C₂²×Dic₁₇	C₂²×C₄	1	2	yes	—
47	C₂×#3624	—	2	4	no	—
48	C₂×#3624	—	2	4	no	—
49	C₂×#3624	—	2	4	no	—
50	C₂×#3624	—	2	4	no	—
51	C₂×#3624	—	2	4	no	—
52	C₂×#3624	M₁₆	1	8	yes	—
53	C₂×C₈×C₁₇	—	2	1	no	—
54	C₂×C₈×C₁₇	—	2	1	no	—
55	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
57	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
59	C₄²×C₁₇	C₂×C₈	1	2	yes	—
60	C₄²×C₁₇	C₂×C₈	1	2	yes	—
61	C₄×Dic₁₇	—	2	1	no	—
62	C₄×Dic₁₇	—	2	1	no	—
63	C₄×Dic₁₇	—	2	4	no	—
64	C₄×Dic₁₇	—	2	4	no	—
65	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
66	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
67	C₄×Dic₁₇	#41	1	2	yes	—
68	C₂×#3622	—	2	1	no	—
69	C₂×#3622	—	2	1	no	—
70	C₂×#3622	C₄⋊C₄	1	2	yes	—
71	C₂×#3622	C₂×C₈	1	2	yes	—
72	C₂×#3622	C₂×C₈	1	2	yes	—
73	C₂×#3622	M₁₆	1	2	yes	—
74	#8262	—	17	1	no	maximal, Sylow
75	C₂³×C₁₇	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₁₇	—	2	2	no	—
77	C₂×C₄×C₁₇	—	2	2	no	—
78	C₂×C₄×C₁₇	C₄×C₈	1	4	yes	—
79	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
85	C₂×Dic₁₇	—	2	2	no	—
86	C₂×Dic₁₇	—	2	1	no	—
87	C₂×Dic₁₇	—	2	1	no	—
88	C₂×Dic₁₇	—	2	2	no	—
89	C₂×Dic₁₇	C₄×D₄	1	2	yes	—
90	C₂×Dic₁₇	C₄×Q₈	1	2	yes	—
91	C₂×Dic₁₇	C₂×M₁₆	1	4	yes	—
92	C₂×Dic₁₇	#131	1	4	yes	—
93	#3624	—	2	8	no	—
94	#3624	—	4	4	no	—
95	C₈×C₁₇	—	2	2	no	—
96	C₈×C₁₇	C₄⋊C₈	1	4	yes	complemented
97	#3622	—	2	2	no	—
98	#3622	C₄⋊C₈	1	4	yes	complemented
99	C₄×C₄⋊C₈	—	17	2	no	—
100	C₄×#97	—	17	1	no	—
101	C₂×#322	—	17	1	no	—
102	C₂×C₈²	—	17	1	no	—
103	#1273	—	17	2	no	—
104	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
106	C₂²×C₁₇	C₄×M₁₆	1	2	yes	—
107	C₂²×C₁₇	C₄×M₁₆	1	1	yes	characteristic
108	C₂²×C₁₇	#405	1	2	yes	—
109	C₄×C₁₇	—	2	4	no	—
110	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
111	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₄×C₁₇	#431	1	1	yes	characteristic
113	C₄×C₁₇	#431	1	1	yes	characteristic
114	Dic₁₇	—	4	1	no	—
115	Dic₁₇	—	4	1	no	—
116	Dic₁₇	#435	1	4	yes	—
117	Dic₁₇	#446	1	4	yes	—
118	C₂×C₄⋊C₈	—	17	4	no	—
119	C₂×#97	—	17	1	no	—
120	C₂×#97	—	17	1	no	—
121	C₂×#97	—	17	1	no	—
122	C₂×#97	#595	1	1	yes	characteristic
123	C₂×C₄×C₈	—	17	1	no	—
124	C₂×C₄×C₈	—	17	2	no	—
125	C₂×C₄×C₈	—	17	2	no	—
126	C₂×C₄×C₈	—	17	1	no	—
127	C₄³	—	17	1	no	—
128	#336	—	17	4	no	—
129	#322	—	34	2	no	—
130	C₈²	—	34	2	no	—
131	C₂×C₁₇	C₈×M₁₆	1	2	yes	—
132	C₂×C₁₇	#1453	1	1	yes	characteristic
133	C₂×C₁₇	#1455	1	1	yes	characteristic
134	C₂×C₁₇	C₄×C₄⋊C₈	1	1	yes	characteristic
135	C₂×C₁₇	#1771	1	2	yes	—
136	C₂²×C₈	—	17	1	no	—
137	C₂²×C₈	—	17	4	no	—
138	C₂²×C₈	—	17	1	no	—
139	C₂²×C₈	—	17	4	no	—
140	C₂²×C₈	C₂×#595	1	1	yes	characteristic
141	C₂²×C₈	C₂×#595	1	1	yes	characteristic
142	C₂×C₄²	—	17	1	no	—
143	C₂×C₄²	—	17	2	no	—
144	C₂×C₄²	—	17	2	no	—
145	C₂×C₄²	—	17	1	no	—
146	C₂×C₄²	C₂×#595	1	1	yes	characteristic
147	C₄⋊C₈	—	34	8	no	—
148	#97	—	17	4	no	—
149	#97	—	34	2	no	—
150	#97	—	34	2	no	—
151	#97	#3624	1	4	yes	complemented
152	C₄×C₈	—	17	2	no	—
153	C₄×C₈	—	17	2	no	—
154	C₄×C₈	—	34	2	no	—
155	C₄×C₈	—	34	2	no	—
156	C₄×C₈	—	34	2	no	—
157	C₄×C₈	—	34	2	no	—
158	C₄×C₈	—	34	2	no	—
159	C₁₇	#8262	1	1	yes	characteristic, complemented, Sylow
160	C₂²×C₄	—	17	2	no	—
161	C₂²×C₄	—	17	1	no	—
162	C₂²×C₄	—	17	1	no	—
163	C₂²×C₄	C₄×#595	1	2	yes	—
164	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	17	2	no	—
168	C₂×C₈	—	17	2	no	—
169	C₂×C₈	—	17	2	no	—
170	C₂×C₈	—	17	8	no	—
171	C₂×C₈	—	17	2	no	—
172	C₂×C₈	—	34	4	no	—
173	C₂×C₈	—	34	4	no	—
174	C₂×C₈	—	34	1	no	—
175	C₂×C₈	—	34	1	no	—
176	C₂×C₈	—	34	4	no	—
177	C₂×C₈	—	34	4	no	—
178	C₂×C₈	—	34	4	no	—
179	C₂×C₈	#63243	1	2	yes	—
180	C₂×C₈	C₄⋊#595	1	2	yes	—
181	C₂×C₈	C₂×#3624	1	2	yes	—
182	C₂×C₈	C₂×#3624	1	2	yes	—
183	C₄²	—	17	1	no	—
184	C₄²	—	17	1	no	—
185	C₄²	—	17	2	no	—
186	C₄²	—	34	4	no	—
187	C₄²	—	34	4	no	—
188	C₄²	—	34	1	no	—
189	C₄²	—	34	1	no	—
190	C₄²	#63242	1	2	yes	—
191	C₄²	C₂×#3624	1	2	yes	—
192	C₂³	C₂×C₄×#595	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	17	2	no	—
196	C₂×C₄	—	17	2	no	—
197	C₂×C₄	—	17	4	no	—
198	C₂×C₄	—	17	4	no	—
199	C₂×C₄	—	34	2	no	—
200	C₂×C₄	—	34	1	no	—
201	C₂×C₄	—	34	1	no	—
202	C₂×C₄	—	34	2	no	—
203	C₂×C₄	C₄×#3624	1	4	yes	—
204	C₂×C₄	C₂×#63242	1	1	yes	characteristic
205	C₂×C₄	C₂×#63243	1	1	yes	characteristic
206	C₂×C₄	D₁₇⋊M₁₆	1	1	yes	characteristic
207	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
208	C₂×C₄	#10587677	1	1	yes	characteristic
209	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	17	4	no	—
212	C₈	—	34	8	no	—
213	C₈	—	34	2	no	—
214	C₈	—	68	4	no	—
215	C₈	C₄⋊#3624	1	4	yes	complemented
216	C₂²	C₄×C₄⋊#595	1	1	yes	characteristic
217	C₂²	C₂×C₄×#3624	1	1	yes	characteristic
218	C₂²	M₁₆×#595	1	2	yes	—
219	C₂²	C₄×#63243	1	1	yes	characteristic
220	C₂²	#11763775	1	2	yes	—
221	C₄	—	2	4	no	—
222	C₄	—	17	4	no	—
223	C₄	—	17	4	no	—
224	C₄	—	68	1	no	—
225	C₄	—	68	1	no	—
226	C₄	C₂×C₄⋊#3624	1	1	yes	characteristic
227	C₄	#11763810	1	1	yes	characteristic
228	C₄	#11763825	1	1	yes	characteristic
229	C₄	#11763828	1	1	yes	characteristic
230	C₂	M₁₆×#3624	1	2	yes	—
231	C₂	#14544004	1	1	yes	characteristic
232	C₂	#14544008	1	1	yes	characteristic
233	C₂	C₄×C₄⋊#3624	1	1	yes	characteristic
234	C₂	#14544582	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#595
commutator	C₂²×C₁₇	C₂×C₄×C₈
Fitting	C₂×C₁₇×#97	C₄
socle, prime radical	C₂³×C₁₇	C₂×C₄²
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	C₂×#97	#595
hypocenter	C₁₇	#8262
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
35. C₁₇×#97	C₈	semidirect
41. #10587474	C₈	semidirect
96. C₈×C₁₇	C₄⋊C₈	semidirect
98. #3622	C₄⋊C₈	semidirect
151. #97	#3624	semidirect
159. C₁₇	#8262	semidirect
215. C₈	C₄⋊#3624	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
C₂×C₈×#3624	2
#14543606	2
#97×Dic₁₇	2
#8262	17