Subgroups — #17981623

Order 4352.

502 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#3633	C₂	1	2	yes	maximal
2	C₂×#11763893	C₂	1	2	yes	maximal
3	(C₂×#3633)⋊C₈	C₂	1	2	yes	maximal
4	C₄×C₈×#595	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#3633	C₄	1	4	yes	—
6	C₂×C₄×#3633	C₂²	1	4	yes	—
7	(C₂×#3633)⋊C₄	C₄	1	4	yes	—
8	#11763893	—	2	4	no	—
9	C₄×#63242	C₄	1	1	yes	characteristic
10	C₂×C₈×#595	C₄	1	1	yes	characteristic
11	C₂×C₈×#595	C₄	1	1	yes	characteristic
12	C₂×C₈×#595	C₂²	1	1	yes	characteristic
13	C₂×C₈×#595	C₂²	1	1	yes	characteristic
14	C₄²×#595	C₂²	1	1	yes	characteristic
15	C₄×C₈×D₁₇	C₄	1	1	yes	characteristic
16	C₂²×#3633	C₂×C₄	1	8	yes	—
17	C₂×C₄×#595	C₂×C₄	1	2	yes	—
18	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×#595	C₂×C₄	1	2	yes	—
20	C₂×C₄×#595	C₂³	1	1	yes	characteristic
21	C₂×#63242	C₂×C₄	1	1	yes	characteristic
22	C₂×#63242	C₂×C₄	1	1	yes	characteristic
23	C₄⋊#3633	—	2	8	no	—
24	C₄×#3633	—	2	4	no	—
25	C₄×#3633	—	2	4	no	—
26	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
28	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
29	C₄×#3624	C₈	1	2	yes	complemented
30	C₄×#3624	C₈	1	2	yes	complemented
31	C₈×#595	—	2	2	no	—
32	C₈×#595	—	2	2	no	—
33	C₈×#595	—	2	2	no	—
34	C₈×#595	D₄	1	2	yes	—
35	C₈×#595	Q₈	1	2	yes	—
36	C₄×C₈×C₁₇	C₈	1	1	yes	characteristic, complemented
37	C₈×Dic₁₇	C₈	1	1	yes	characteristic, complemented
38	C₈×Dic₁₇	C₈	1	1	yes	characteristic, complemented
39	C₄×#3622	C₈	1	1	yes	characteristic, complemented
40	C₂²×#595	C₄²	1	2	yes	—
41	C₂²×#595	C₂²×C₄	1	2	yes	—
42	C₂×#3633	—	2	8	no	—
43	C₂×#3633	—	2	8	no	—
44	C₂×#3633	—	2	8	no	—
45	C₂×C₄×D₁₇	C₄²	1	1	yes	characteristic
46	C₂×C₄×D₁₇	C₄²	1	1	yes	characteristic
47	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×#3624	C₂×C₈	1	2	yes	—
49	C₂×#3624	C₂×C₈	1	2	yes	—
50	C₄×#595	—	2	4	no	—
51	C₄×#595	—	2	1	no	—
52	C₄×#595	—	2	1	no	—
53	C₄×#595	M₁₆	1	8	yes	—
54	C₄×#595	C₂×D₄	1	1	yes	characteristic
55	C₄×#595	C₂×Q₈	1	1	yes	characteristic
56	C₄×#595	#41	1	2	yes	—
57	#63242	—	2	1	no	—
58	#63242	—	2	1	no	—
59	#63242	C₄⋊C₄	1	2	yes	—
60	#63242	M₁₆	1	2	yes	—
61	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
62	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
63	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
64	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
65	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
66	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
67	C₂×#3622	C₂×C₈	1	1	yes	characteristic
68	C₂×#3622	C₂×C₈	1	1	yes	characteristic
69	C₈×D₁₇	—	2	1	no	—
70	C₈×D₁₇	—	2	1	no	—
71	C₈×D₁₇	C₄⋊C₄	1	2	yes	—
72	C₈×D₁₇	M₁₆	1	2	yes	—
73	#8260	—	17	1	no	maximal, Sylow
74	C₂²×D₁₇	C₂×C₄²	1	1	yes	characteristic
75	C₂×#595	—	2	2	no	—
76	C₂×#595	—	2	2	no	—
77	C₂×#595	#97	1	4	yes	—
78	C₂×#595	C₄×D₄	1	2	yes	—
79	C₂×#595	C₄×Q₈	1	2	yes	—
80	C₂×#595	C₂×M₁₆	1	4	yes	—
81	C₂×#595	#131	1	4	yes	—
82	#3633	—	4	8	no	—
83	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
84	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
86	C₂×Dic₁₇	C₄×C₈	1	1	yes	characteristic
87	C₂×Dic₁₇	C₄×C₈	1	1	yes	characteristic
88	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
89	C₄×D₁₇	—	2	2	no	—
90	C₄×D₁₇	#97	1	4	yes	—
91	C₄×D₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₄×D₁₇	#117	1	1	yes	characteristic
93	C₄×D₁₇	C₂×M₁₆	1	1	yes	characteristic
94	C₄×D₁₇	C₂×M₁₆	1	1	yes	characteristic
95	#3624	—	2	2	no	—
96	#3624	C₄⋊C₈	1	4	yes	complemented
97	C₈×C₁₇	—	2	1	no	—
98	C₈×C₁₇	C₄⋊C₈	1	2	yes	complemented
99	#3622	—	2	1	no	—
100	#3622	C₄⋊C₈	1	2	yes	complemented
101	C₄×C₄⋊C₈	—	17	2	no	—
102	C₄²×C₈	—	17	1	no	—
103	C₂×#322	—	17	2	no	—
104	#1273	—	17	2	no	—
105	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
106	C₂×D₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
107	C₂×D₁₇	C₂×#97	1	2	yes	—
108	C₂×D₁₇	C₄×M₁₆	1	1	yes	characteristic
109	C₂×D₁₇	#405	1	2	yes	—
110	#595	—	2	4	no	—
111	#595	#436	1	4	yes	—
112	#595	#446	1	4	yes	—
113	C₄×C₁₇	—	2	1	no	—
114	C₄×C₁₇	#322	1	2	yes	complemented
115	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
116	C₄×C₁₇	#431	1	1	yes	characteristic
117	Dic₁₇	—	2	1	no	—
118	Dic₁₇	#322	1	2	yes	complemented
119	Dic₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
120	Dic₁₇	#431	1	1	yes	characteristic
121	C₂×C₄⋊C₈	—	17	4	no	—
122	C₂×C₄×C₈	—	17	1	no	—
123	C₂×C₄×C₈	—	17	1	no	—
124	C₂×C₄×C₈	—	17	1	no	—
125	C₂×C₄×C₈	—	17	1	no	—
126	C₂×C₄×C₈	—	17	1	no	—
127	C₂×C₄×C₈	—	17	1	no	—
128	C₂×C₄×C₈	—	17	4	no	—
129	C₄³	—	17	1	no	—
130	#336	—	17	4	no	—
131	#322	—	34	4	no	—
132	C₂×C₁₇	C₂×#322	1	1	yes	characteristic
133	C₂×C₁₇	#1450	1	1	yes	characteristic
134	C₂×C₁₇	C₄×C₄⋊C₈	1	1	yes	characteristic
135	D₁₇	#1451	1	2	yes	—
136	D₁₇	#1770	1	2	yes	—
137	C₂²×C₈	—	17	1	no	—
138	C₂²×C₈	—	17	1	no	—
139	C₂²×C₈	—	17	1	no	—
140	C₂²×C₈	—	17	1	no	—
141	C₂²×C₈	—	17	8	no	—
142	C₂×C₄²	—	17	2	no	—
143	C₂×C₄²	—	17	1	no	—
144	C₂×C₄²	—	17	1	no	—
145	C₂×C₄²	—	17	2	no	—
146	C₂×C₄²	—	17	1	no	—
147	C₄⋊C₈	—	34	8	no	—
148	C₄×C₈	—	17	2	no	—
149	C₄×C₈	—	17	2	no	—
150	C₄×C₈	—	17	2	no	—
151	C₄×C₈	—	17	2	no	—
152	C₄×C₈	—	17	1	no	—
153	C₄×C₈	—	17	1	no	—
154	C₄×C₈	—	17	1	no	—
155	C₄×C₈	—	34	4	no	—
156	C₄×C₈	—	34	4	no	—
157	C₄×C₈	—	34	2	no	—
158	C₄×C₈	—	34	2	no	—
159	C₄×C₈	—	34	2	no	—
160	C₄×C₈	#3633	1	1	yes	characteristic, complemented
161	C₁₇	#8260	1	1	yes	characteristic, complemented, Sylow
162	C₂²×C₄	—	17	2	no	—
163	C₂²×C₄	—	17	1	no	—
164	C₂²×C₄	—	17	1	no	—
165	C₂²×C₄	—	17	2	no	—
166	C₂²×C₄	—	17	1	no	—
167	C₂×C₈	—	17	2	no	—
168	C₂×C₈	—	17	2	no	—
169	C₂×C₈	—	17	2	no	—
170	C₂×C₈	—	17	1	no	—
171	C₂×C₈	—	17	2	no	—
172	C₂×C₈	—	17	2	no	—
173	C₂×C₈	—	17	2	no	—
174	C₂×C₈	—	17	1	no	—
175	C₂×C₈	—	34	1	no	—
176	C₂×C₈	—	34	1	no	—
177	C₂×C₈	—	34	1	no	—
178	C₂×C₈	—	34	1	no	—
179	C₂×C₈	—	34	8	no	—
180	C₂×C₈	—	34	8	no	—
181	C₂×C₈	—	34	8	no	—
182	C₂×C₈	C₂×#3633	1	1	yes	characteristic
183	C₂×C₈	C₂×#3633	1	1	yes	characteristic
184	C₄²	—	17	8	no	—
185	C₄²	—	17	1	no	—
186	C₄²	—	17	1	no	—
187	C₄²	—	17	2	no	—
188	C₄²	—	17	1	no	—
189	C₄²	—	17	1	no	—
190	C₄²	—	17	1	no	—
191	C₄²	—	34	1	no	—
192	C₄²	—	34	1	no	—
193	C₄²	—	34	4	no	—
194	C₄²	C₂×#3633	1	1	yes	characteristic
195	C₂³	—	17	1	no	—
196	C₂×C₄	—	17	2	no	—
197	C₂×C₄	—	17	4	no	—
198	C₂×C₄	—	17	2	no	—
199	C₂×C₄	—	17	4	no	—
200	C₂×C₄	—	17	1	no	—
201	C₂×C₄	—	17	1	no	—
202	C₂×C₄	—	17	4	no	—
203	C₂×C₄	—	17	1	no	—
204	C₂×C₄	—	17	1	no	—
205	C₂×C₄	—	17	1	no	—
206	C₂×C₄	—	17	1	no	—
207	C₂×C₄	—	17	4	no	—
208	C₂×C₄	—	17	1	no	—
209	C₂×C₄	—	34	2	no	—
210	C₂×C₄	—	34	2	no	—
211	C₂×C₄	—	34	2	no	—
212	C₂×C₄	C₄×#3633	1	1	yes	characteristic
213	C₂×C₄	C₄×#3633	1	1	yes	characteristic
214	C₂×C₄	C₂²×#3633	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	17	4	no	—
217	C₈	—	17	2	no	—
218	C₈	—	34	2	no	—
219	C₈	—	34	1	no	—
220	C₈	—	68	8	no	—
221	C₈	C₄⋊#3633	1	2	yes	complemented
222	C₂²	—	17	1	no	—
223	C₂²	—	17	1	no	—
224	C₂²	—	17	2	no	—
225	C₂²	—	17	2	no	—
226	C₂²	C₂×C₄×#3633	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	17	4	no	—
229	C₄	—	17	4	no	—
230	C₄	—	17	1	no	—
231	C₄	—	17	1	no	—
232	C₄	—	17	2	no	—
233	C₄	—	34	4	no	—
234	C₄	—	34	1	no	—
235	C₄	#11763893	1	2	yes	complemented
236	C₄	C₂×C₄⋊#3633	1	1	yes	characteristic
237	C₄	#11763922	1	1	yes	characteristic
238	C₂	—	17	2	no	—
239	C₂	—	17	2	no	—
240	C₂	C₂×#11763893	1	1	yes	characteristic
241	C₂	#14545893	1	1	yes	characteristic
242	C₂	C₄×C₄⋊#3633	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#3633
commutator, socle	C₂²×C₁₇	C₂×C₄×C₈
Fitting	C₄×C₈×C₁₇	C₈
prime radical	C₂²×D₁₇	C₂×C₄²
cosocle	C₂×C₄×#595	C₂³
hypercenter	C₄×C₈	#3633
hypocenter	C₁₇	#8260
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
29. C₄×#3624	C₈	semidirect
30. C₄×#3624	C₈	semidirect
36. C₄×C₈×C₁₇	C₈	semidirect
37. C₈×Dic₁₇	C₈	semidirect
38. C₈×Dic₁₇	C₈	semidirect
39. C₄×#3622	C₈	semidirect
96. #3624	C₄⋊C₈	semidirect
98. C₈×C₁₇	C₄⋊C₈	semidirect
100. #3622	C₄⋊C₈	semidirect
114. C₄×C₁₇	#322	semidirect
118. Dic₁₇	#322	semidirect
160. C₄×C₈	#3633	semidirect
161. C₁₇	#8260	semidirect
221. C₈	C₄⋊#3633	semidirect
235. C₄	#11763893	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#3633	2
C₂×#11763893	2
(C₂×#3633)⋊C₈	2
C₄×C₈×#595	2
#8260	17