Subgroups — #17982461

Order 4352.

438 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545970	C₂	1	2	yes	maximal
2	C₂×#11763902	C₂	1	2	yes	maximal
3	#14545855	C₂	1	2	yes	maximal
4	C₄×C₈×#595	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587665	C₂²	1	4	yes	—
6	#11763912	C₄	1	4	yes	complemented
7	C₄⋊#63241	C₄	1	4	yes	complemented
8	#11763902	—	2	4	no	—
9	C₄×#63242	C₂²	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	C₄	1	1	yes	characteristic
15	C₄×C₈×D₁₇	C₄	1	1	yes	characteristic
16	C₂×C₄×#595	C₈	1	2	yes	—
17	C₂×C₄×#595	C₈	1	2	yes	—
18	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
20	C₂×#63242	C₂×C₄	1	1	yes	characteristic
21	C₂×#63242	C₂³	1	1	yes	characteristic
22	C₂×#63241	C₂×C₄	1	8	yes	—
23	#10587665	—	2	4	no	—
24	#10587665	—	2	4	no	—
25	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
27	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
28	C₄×#3624	C₂×C₄	1	2	yes	—
29	C₄×#3624	C₂×C₄	1	2	yes	—
30	C₈×#595	—	2	2	no	—
31	C₈×#595	—	2	2	no	—
32	C₈×#595	—	2	2	no	—
33	C₈×#595	D₄	1	2	yes	—
34	C₈×#595	Q₈	1	2	yes	—
35	C₄×C₈×C₁₇	C₈	1	1	yes	characteristic
36	C₈×Dic₁₇	C₈	1	1	yes	characteristic
37	C₈×Dic₁₇	C₈	1	1	yes	characteristic
38	C₄×#3622	C₈	1	1	yes	characteristic
39	C₂²×#595	C₂×C₈	1	2	yes	—
40	C₂²×#595	C₂×C₈	1	2	yes	—
41	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×#3624	C₄²	1	2	yes	—
45	C₂×#3624	C₂²×C₄	1	2	yes	—
46	C₄×#595	—	2	4	no	—
47	C₄×#595	—	2	1	no	—
48	C₄×#595	—	2	1	no	—
49	C₄×#595	C₁₆	1	8	yes	complemented
50	C₄×#595	C₄⋊C₄	1	2	yes	—
51	C₄×#595	M₁₆	1	2	yes	—
52	#63242	—	2	1	no	—
53	#63242	—	2	1	no	—
54	#63242	C₂×D₄	1	1	yes	characteristic
55	#63242	C₂×Q₈	1	1	yes	characteristic
56	#63242	#41	1	2	yes	—
57	#63241	—	2	8	no	—
58	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
59	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
60	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
61	C₄×Dic₁₇	C₄²	1	1	yes	characteristic
62	C₄×Dic₁₇	C₄²	1	1	yes	characteristic
63	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
64	C₂×#3622	C₂×C₈	1	1	yes	characteristic
65	C₂×#3622	C₂×C₈	1	1	yes	characteristic
66	C₈×D₁₇	—	2	1	no	—
67	C₈×D₁₇	—	2	1	no	—
68	C₈×D₁₇	C₄⋊C₄	1	2	yes	—
69	C₈×D₁₇	M₁₆	1	2	yes	—
70	C₄×C₄⋊C₁₆	—	17	1	no	maximal, Sylow
71	C₂²×D₁₇	C₂²×C₈	1	1	yes	characteristic
72	C₂×#595	—	2	2	no	—
73	C₂×#595	—	2	2	no	—
74	C₂×#595	C₄⋊C₈	1	4	yes	—
75	C₂×#595	C₂×C₁₆	1	4	yes	—
76	C₂×#595	C₂×C₁₆	1	4	yes	—
77	C₂×#595	M₃₂	1	4	yes	—
78	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
81	C₂×Dic₁₇	C₄×C₈	1	1	yes	characteristic
82	C₂×Dic₁₇	C₄×C₈	1	1	yes	characteristic
83	C₂×Dic₁₇	C₂×C₄²	1	1	yes	characteristic
84	C₄×D₁₇	—	2	2	no	—
85	C₄×D₁₇	C₂×C₁₆	1	4	yes	—
86	C₄×D₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₄×D₁₇	#117	1	1	yes	characteristic
88	C₄×D₁₇	C₂×M₁₆	1	1	yes	characteristic
89	C₄×D₁₇	C₂×M₁₆	1	1	yes	characteristic
90	#3624	—	2	2	no	—
91	#3624	C₄×D₄	1	2	yes	—
92	#3624	C₄×Q₈	1	2	yes	—
93	C₈×C₁₇	—	2	1	no	—
94	C₈×C₁₇	C₄⋊C₈	1	2	yes	—
95	#3622	—	2	1	no	—
96	#3622	C₄⋊C₈	1	2	yes	—
97	C₂×C₄⋊C₁₆	—	17	2	no	—
98	C₂×C₄×C₁₆	—	17	2	no	—
99	C₄²×C₈	—	17	1	no	—
100	#1374	—	17	2	no	—
101	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
102	C₂×D₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
103	C₂×D₁₇	#431	1	1	yes	characteristic
104	C₂×D₁₇	C₂²×C₁₆	1	2	yes	—
105	C₂×D₁₇	C₂×M₃₂	1	2	yes	—
106	#595	—	2	4	no	—
107	#595	C₄⋊C₁₆	1	8	yes	complemented
108	C₄×C₁₇	—	2	1	no	—
109	C₄×C₁₇	C₄×C₁₆	1	2	yes	complemented
110	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
111	C₄×C₁₇	#431	1	1	yes	characteristic
112	Dic₁₇	—	2	1	no	—
113	Dic₁₇	C₄×C₁₆	1	2	yes	complemented
114	Dic₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
115	Dic₁₇	C₄×M₁₆	1	1	yes	characteristic
116	C₂²×C₁₆	—	17	4	no	—
117	C₂×C₄×C₈	—	17	1	no	—
118	C₂×C₄×C₈	—	17	1	no	—
119	C₂×C₄×C₈	—	17	1	no	—
120	C₂×C₄×C₈	—	17	1	no	—
121	C₂×C₄×C₈	—	17	1	no	—
122	C₂×C₄×C₈	—	17	1	no	—
123	C₄³	—	17	1	no	—
124	C₄⋊C₁₆	—	17	4	no	—
125	C₄⋊C₁₆	—	17	4	no	—
126	C₄×C₁₆	—	34	4	no	—
127	C₂×C₁₇	C₄×C₄⋊C₈	1	1	yes	characteristic
128	C₂×C₁₇	C₂×C₄×C₁₆	1	1	yes	characteristic
129	C₂×C₁₇	C₄×M₃₂	1	1	yes	characteristic
130	D₁₇	C₂×C₄⋊C₁₆	1	2	yes	—
131	D₁₇	#2162	1	2	yes	—
132	C₂²×C₈	—	17	1	no	—
133	C₂²×C₈	—	17	1	no	—
134	C₂²×C₈	—	17	1	no	—
135	C₂²×C₈	—	17	1	no	—
136	C₂×C₄²	—	17	2	no	—
137	C₂×C₄²	—	17	1	no	—
138	C₂×C₄²	—	17	1	no	—
139	C₂×C₄²	—	17	2	no	—
140	C₂×C₄²	—	17	1	no	—
141	C₂×C₁₆	—	17	8	no	—
142	C₂×C₁₆	—	34	4	no	—
143	C₂×C₁₆	—	34	4	no	—
144	C₄×C₈	—	17	2	no	—
145	C₄×C₈	—	17	2	no	—
146	C₄×C₈	—	17	2	no	—
147	C₄×C₈	—	17	2	no	—
148	C₄×C₈	—	17	1	no	—
149	C₄×C₈	—	17	1	no	—
150	C₄×C₈	—	17	1	no	—
151	C₄×C₈	—	34	2	no	—
152	C₄×C₈	—	34	2	no	—
153	C₄×C₈	—	34	2	no	—
154	C₄×C₈	#3633	1	1	yes	characteristic
155	C₁₇	C₄×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
156	C₂²×C₄	—	17	2	no	—
157	C₂²×C₄	—	17	1	no	—
158	C₂²×C₄	—	17	1	no	—
159	C₂²×C₄	—	17	2	no	—
160	C₂²×C₄	—	17	1	no	—
161	C₂×C₈	—	17	1	no	—
162	C₂×C₈	—	17	1	no	—
163	C₂×C₈	—	17	2	no	—
164	C₂×C₈	—	17	2	no	—
165	C₂×C₈	—	17	1	no	—
166	C₂×C₈	—	17	2	no	—
167	C₂×C₈	—	17	2	no	—
168	C₂×C₈	—	17	2	no	—
169	C₂×C₈	—	17	1	no	—
170	C₂×C₈	—	34	1	no	—
171	C₂×C₈	—	34	1	no	—
172	C₂×C₈	—	34	1	no	—
173	C₂×C₈	—	34	1	no	—
174	C₂×C₈	C₂×#3633	1	1	yes	characteristic
175	C₂×C₈	C₂×#3633	1	1	yes	characteristic
176	C₄²	—	17	8	no	—
177	C₄²	—	17	2	no	—
178	C₄²	—	17	2	no	—
179	C₄²	—	17	1	no	—
180	C₄²	—	17	1	no	—
181	C₄²	—	17	1	no	—
182	C₄²	—	34	1	no	—
183	C₄²	—	34	1	no	—
184	C₄²	—	34	4	no	—
185	C₄²	C₂×#3633	1	1	yes	characteristic
186	C₁₆	—	34	8	no	—
187	C₂³	—	17	1	no	—
188	C₂×C₄	—	17	4	no	—
189	C₂×C₄	—	17	4	no	—
190	C₂×C₄	—	17	4	no	—
191	C₂×C₄	—	17	1	no	—
192	C₂×C₄	—	17	1	no	—
193	C₂×C₄	—	17	4	no	—
194	C₂×C₄	—	17	1	no	—
195	C₂×C₄	—	17	1	no	—
196	C₂×C₄	—	17	1	no	—
197	C₂×C₄	—	17	1	no	—
198	C₂×C₄	—	17	4	no	—
199	C₂×C₄	—	17	1	no	—
200	C₂×C₄	—	34	2	no	—
201	C₂×C₄	—	34	2	no	—
202	C₂×C₄	—	34	2	no	—
203	C₂×C₄	C₄×#3633	1	1	yes	characteristic
204	C₂×C₄	C₄×#3633	1	1	yes	characteristic
205	C₂×C₄	C₂²×#3633	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	17	2	no	—
208	C₈	—	17	2	no	—
209	C₈	—	17	2	no	—
210	C₈	—	34	2	no	—
211	C₈	—	34	1	no	—
212	C₈	C₄⋊#3633	1	2	yes	—
213	C₂²	—	17	1	no	—
214	C₂²	—	17	1	no	—
215	C₂²	—	17	2	no	—
216	C₂²	—	17	2	no	—
217	C₂²	C₂×C₄×#3633	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	17	8	no	—
220	C₄	—	17	1	no	—
221	C₄	—	17	1	no	—
222	C₄	—	17	2	no	—
223	C₄	—	34	4	no	—
224	C₄	—	34	1	no	—
225	C₄	#11763902	1	2	yes	complemented
226	C₄	C₂×C₄⋊#3633	1	1	yes	characteristic
227	C₄	#11763922	1	1	yes	characteristic
228	C₂	—	17	2	no	—
229	C₂	—	17	2	no	—
230	C₂	C₄×C₄⋊#3633	1	1	yes	characteristic
231	C₂	C₂×#11763902	1	1	yes	characteristic
232	C₂	#595⋊M₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×#3633
commutator	C₂×C₁₇	C₂×C₄×C₁₆
Frattini	C₂×C₄	C₂²×#3633
Fitting	C₄×C₈×C₁₇	C₈
socle	C₂²×C₁₇	C₂×C₄×C₈
prime radical	C₂²×D₁₇	C₂²×C₈
cosocle	C₂×#63242	C₂³
hypercenter	C₄×C₈	#3633
hypocenter	C₁₇	C₄×C₄⋊C₁₆
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #11763912	C₄	semidirect
7. C₄⋊#63241	C₄	semidirect
49. C₄×#595	C₁₆	semidirect
107. #595	C₄⋊C₁₆	semidirect
109. C₄×C₁₇	C₄×C₁₆	semidirect
113. Dic₁₇	C₄×C₁₆	semidirect
155. C₁₇	C₄×C₄⋊C₁₆	semidirect
225. C₄	#11763902	semidirect

Maximal subgroups

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

subgroup	index
#14545970	2
C₂×#11763902	2
#14545855	2
C₄×C₈×#595	2
C₄×C₄⋊C₁₆	17