Subgroups — #16950791

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	C₂²×#10587483	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#63214	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#63214	C₂	1	1	yes	maximal, characteristic
4	#14526862	C₂	1	2	yes	maximal, complemented
5	C₈×C₂²⋊C₄×C₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587483	—	2	4	no	—
7	C₂×#10587483	—	2	2	no	—
8	C₂²×#63214	C₂²	1	1	yes	characteristic
9	C₂²×#63214	C₂²	1	1	yes	characteristic
10	C₂²×#63214	C₂²	1	2	yes	—
11	C₄⋊#63214	—	2	4	no	—
12	C₂²⋊#63214	—	2	2	no	—
13	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₂×C₁₇×C₂²⋊C₈	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₇	C₂²	1	2	yes	—
16	C₄×C₂²⋊C₄×C₁₇	C₄	1	1	yes	characteristic
17	C₁₇×#336	C₄	1	2	yes	—
18	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	—	2	2	no	—
20	C₂²×C₈×C₁₇	—	2	2	no	—
21	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₇	D₄	1	2	yes	—
24	C₂²×C₈×C₁₇	Q₈	1	2	yes	—
25	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₁₇	C₈	1	2	yes	—
27	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
28	C₁₇×C₂²⋊C₈	—	2	2	no	—
29	C₄×C₈×C₁₇	—	2	2	no	—
30	C₄×C₈×C₁₇	—	2	2	no	—
31	C₁₇×#95	C₈	1	1	yes	characteristic
32	C₁₇×#95	C₈	1	1	yes	characteristic
33	#10587483	—	2	8	no	—
34	C₂×#63214	—	2	2	no	—
35	C₂×#63214	—	2	2	no	—
36	C₂×#63214	—	2	1	no	—
37	C₂×#63214	—	2	1	no	—
38	C₂×#63214	—	2	2	no	—
39	C₂×#63214	—	2	2	no	—
40	C₂×#63214	D₄	1	4	yes	—
41	C₂⁴×C₁₇	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	—	2	2	no	—
43	C₂²×C₄×C₁₇	—	2	2	no	—
44	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₁₇	M₁₆	1	2	yes	—
49	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₇	—	2	2	no	—
51	C₂×C₈×C₁₇	—	2	2	no	—
52	C₂×C₈×C₁₇	—	2	4	no	—
53	C₂×C₈×C₁₇	—	2	2	no	—
54	C₂×C₈×C₁₇	—	2	2	no	—
55	C₂×C₈×C₁₇	—	2	2	no	—
56	C₂×C₈×C₁₇	—	2	2	no	—
57	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
58	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₁₇	#41	1	2	yes	—
62	C₂²⋊C₄×C₁₇	—	2	4	no	—
63	C₄²×C₁₇	—	2	2	no	—
64	C₄²×C₁₇	—	2	2	no	—
65	#63214	—	2	4	no	—
66	#63214	—	4	1	no	—
67	#63214	—	4	1	no	—
68	#12913	—	17	1	no	maximal, Sylow
69	C₂³×C₁₇	—	2	2	no	—
70	C₂³×C₁₇	—	2	2	no	—
71	C₂³×C₁₇	—	2	1	no	—
72	C₂³×C₁₇	—	2	1	no	—
73	C₂³×C₁₇	C₄⋊C₈	1	2	yes	—
74	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₇	—	2	2	no	—
76	C₂×C₄×C₁₇	—	2	2	no	—
77	C₂×C₄×C₁₇	—	2	2	no	—
78	C₂×C₄×C₁₇	—	2	2	no	—
79	C₂×C₄×C₁₇	—	2	2	no	—
80	C₂×C₄×C₁₇	—	2	4	no	—
81	C₂×C₄×C₁₇	—	2	2	no	—
82	C₂×C₄×C₁₇	—	2	2	no	—
83	C₂×C₄×C₁₇	—	2	2	no	—
84	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
85	C₂×C₄×C₁₇	M₃₂	1	4	yes	complemented
86	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₁₇	#131	1	2	yes	—
90	C₈×C₁₇	—	2	2	no	—
91	C₈×C₁₇	—	4	2	no	—
92	C₈×C₁₇	#120	1	1	yes	characteristic
93	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
94	C₈×C₁₇	#123	1	1	yes	characteristic
95	C₂²×M₃₂	—	17	1	no	—
96	C₂×C₄⋊C₁₆	—	17	2	no	—
97	C₂×C₂²⋊C₁₆	—	17	1	no	—
98	C₈×C₂²⋊C₄	D₁₇	1	1	yes	characteristic
99	#1374	—	17	2	no	—
100	C₂²×C₁₇	—	2	2	no	—
101	C₂²×C₁₇	—	2	2	no	—
102	C₂²×C₁₇	—	2	2	no	—
103	C₂²×C₁₇	—	4	2	no	—
104	C₂²×C₁₇	—	4	2	no	—
105	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
106	C₂²×C₁₇	C₈×D₄	1	2	yes	—
107	C₂²×C₁₇	C₂×M₃₂	1	2	yes	—
108	C₂²×C₁₇	#504	1	1	yes	characteristic
109	C₂²×C₁₇	#504	1	1	yes	characteristic
110	C₄×C₁₇	—	2	2	no	—
111	C₄×C₁₇	—	2	4	no	—
112	C₄×C₁₇	—	4	2	no	—
113	C₄×C₁₇	#385	1	1	yes	characteristic
114	C₄×C₁₇	#424	1	1	yes	characteristic
115	C₄×C₁₇	#435	1	2	yes	—
116	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
117	C₂×M₃₂	—	34	4	no	—
118	C₂×M₃₂	—	34	2	no	—
119	C₂²×C₁₆	—	17	2	no	—
120	C₂²×C₁₆	—	17	1	no	—
121	C₂²×C₁₆	—	17	1	no	—
122	C₂×C₂²⋊C₈	Dic₁₇	1	1	yes	characteristic
123	C₂×C₄×C₈	C₂×D₁₇	1	2	yes	—
124	C₄×C₂²⋊C₄	Dic₁₇	1	1	yes	characteristic
125	C₄⋊C₁₆	—	34	4	no	—
126	C₂²⋊C₁₆	—	34	2	no	—
127	#336	Dic₁₇	1	2	yes	—
128	C₂×C₁₇	—	4	2	no	—
129	C₂×C₁₇	#1850	1	1	yes	characteristic
130	C₂×C₁₇	#2150	1	1	yes	characteristic
131	C₂×C₁₇	#2151	1	1	yes	characteristic
132	C₂×C₁₇	#2168	1	2	yes	—
133	C₂×C₁₇	#2169	1	2	yes	—
134	C₂³×C₄	C₂×Dic₁₇	1	1	yes	characteristic
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	Dic₃₄	1	2	yes	—
138	C₂²×C₈	D₆₈	1	2	yes	—
139	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
140	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
141	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
142	C₂×C₂²⋊C₄	#3622	1	2	yes	—
143	C₂×C₄²	C₂×Dic₁₇	1	2	yes	—
144	M₃₂	—	34	8	no	—
145	C₂×C₁₆	—	17	4	no	—
146	C₂×C₁₆	—	34	2	no	—
147	C₂×C₁₆	—	34	2	no	—
148	C₂×C₁₆	—	34	1	no	—
149	C₂×C₁₆	—	34	1	no	—
150	C₂×C₁₆	—	34	2	no	—
151	C₂×C₁₆	—	34	2	no	—
152	C₂²⋊C₈	—	2	2	no	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	—	2	2	no	—
155	#95	#3622	1	1	yes	characteristic
156	#95	#3622	1	1	yes	characteristic
157	C₁₇	#12913	1	1	yes	characteristic, complemented, Sylow
158	C₂⁴	C₂×#3622	1	1	yes	characteristic
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	C₂×#3622	1	2	yes	—
162	C₂²×C₄	C₂×#3622	1	1	yes	characteristic
163	C₂²×C₄	C₂×#3622	1	2	yes	—
164	C₂²×C₄	#63223	1	2	yes	—
165	C₂²×C₄	C₄⋊Dic₁₇	1	2	yes	—
166	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
175	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
176	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
177	C₂×C₈	D₄×D₁₇	1	2	yes	—
178	C₂×C₈	#63254	1	2	yes	—
179	C₂²⋊C₄	—	2	4	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	2	2	no	—
182	C₁₆	—	34	4	no	—
183	C₁₆	—	68	1	no	—
184	C₁₆	—	68	1	no	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	1	no	—
187	C₂³	—	2	1	no	—
188	C₂³	—	2	2	no	—
189	C₂³	C₄⋊#3622	1	2	yes	—
190	C₂³	C₂²×#3622	1	1	yes	characteristic
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	C₄⋊#3622	1	2	yes	—
201	C₂×C₄	#10587483	1	4	yes	complemented
202	C₂×C₄	C₂×#63223	1	1	yes	characteristic
203	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
204	C₂×C₄	D₄×Dic₁₇	1	2	yes	—
205	C₂×C₄	#10587634	1	2	yes	—
206	C₈	—	2	2	no	—
207	C₈	—	4	2	no	—
208	C₈	#10587564	1	2	yes	—
209	C₈	#10587568	1	1	yes	characteristic
210	C₈	#10587572	1	1	yes	characteristic
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	C₂×C₄⋊#3622	1	1	yes	characteristic
217	C₂²	D₄×#3622	1	2	yes	—
218	C₂²	C₂×#10587483	1	2	yes	—
219	C₂²	#11763226	1	1	yes	characteristic
220	C₂²	#11763226	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	—	2	2	no	—
223	C₄	—	4	2	no	—
224	C₄	#11762684	1	1	yes	characteristic
225	C₄	#11762836	1	1	yes	characteristic
226	C₄	#11762881	1	2	yes	—
227	C₂	—	4	2	no	—
228	C₂	#14528964	1	1	yes	characteristic
229	C₂	#14530540	1	1	yes	characteristic
230	C₂	#14530543	1	1	yes	characteristic
231	C₂	#14530610	1	2	yes	—
232	C₂	#14530616	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₇
commutator, socle	C₂³×C₁₇	C₂²×C₈
Fitting	C₈×C₂²⋊C₄×C₁₇	C₂
prime radical	C₂⁴×C₁₇	C₂×C₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₈×C₂²⋊C₄	D₁₇
hypocenter	C₁₇	#12913
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₈×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊#63214	C₂	semidirect
4. #14526862	C₂	semidirect
85. C₂×C₄×C₁₇	M₃₂	semidirect
157. C₁₇	#12913	semidirect
201. C₂×C₄	#10587483	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#10587483	2
C₂×C₄⋊#63214	2
C₂×C₂²⋊#63214	2
#14526862	2
C₈×C₂²⋊C₄×C₁₇	2
#12913	17