Subgroups — #17189682

Order 4352.

578 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	#14539298	C₂	1	1	yes	maximal, characteristic
2	#14531029	C₂	1	4	yes	maximal, complemented
3	#14531011	C₂	1	4	yes	maximal, complemented
4	#14530504	C₂	1	4	yes	maximal
5	C₄×#10587483	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₇×#494	C₂	1	1	yes	maximal, characteristic
7	#11763366	—	2	1	no	—
8	C₂×#10587483	C₂²	1	2	yes	—
9	#11763214	C₂²	1	2	yes	—
10	#11763211	—	2	1	no	—
11	#11763211	C₂²	1	2	yes	—
12	#11763207	—	2	1	no	—
13	#11762847	—	2	1	no	—
14	#11762847	C₂²	1	2	yes	—
15	#11762613	C₂²	1	16	yes	—
16	#11762585	C₂²	1	2	yes	—
17	C₄×#63214	C₂²	1	2	yes	—
18	C₂²×Q₁₆×C₁₇	—	2	1	no	—
19	C₂²×Q₁₆×C₁₇	C₂²	1	2	yes	—
20	C₂×C₁₇×C₄⋊Q₈	—	2	1	no	—
21	C₁₇×#494	—	2	1	no	—
22	C₁₇×#494	—	2	1	no	—
23	C₁₇×#494	C₂²	1	2	yes	—
24	C₁₇×#494	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
26	C₂²×Q₈×C₁₇	—	2	2	no	—
27	C₂×Q₁₆×C₁₇	—	2	2	no	—
28	C₂×Q₁₆×C₁₇	—	2	2	no	—
29	C₂×Q₁₆×C₁₇	—	2	2	no	—
30	C₂×Q₁₆×C₁₇	—	2	8	no	—
31	C₂×Q₁₆×C₁₇	—	4	4	no	—
32	C₂×Q₁₆×C₁₇	C₂³	1	4	yes	—
33	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
35	C₁₇×C₄⋊Q₈	—	2	2	no	—
36	C₁₇×C₄⋊Q₈	—	2	2	no	—
37	C₁₇×C₄⋊Q₈	—	4	2	no	—
38	C₂×C₄⋊C₄×C₁₇	—	4	2	no	—
39	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
40	#10587612	—	2	2	no	—
41	#10587610	—	4	2	no	—
42	#10587556	—	2	2	no	—
43	#10587555	—	2	2	no	—
44	C₄×Dic₃₄	—	4	2	no	—
45	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
49	#10587489	—	2	2	no	—
50	#10587489	C₂³	1	4	yes	—
51	#10587488	—	2	2	no	—
52	#10587488	—	2	2	no	—
53	#10587483	—	2	4	no	—
54	C₂×#63214	C₂³	1	4	yes	—
55	C₂×Q₈×C₁₇	—	2	4	no	—
56	C₂×Q₈×C₁₇	—	4	2	no	—
57	C₂×Q₈×C₁₇	—	4	4	no	—
58	C₂×Q₈×C₁₇	—	4	4	no	—
59	C₂²×C₄×C₁₇	—	4	2	no	—
60	C₂²×C₄×C₁₇	D₈	1	2	yes	—
61	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
62	C₂×Dic₃₄	—	4	2	no	—
63	Q₁₆×C₁₇	—	2	8	no	—
64	Q₁₆×C₁₇	—	4	4	no	—
65	Q₁₆×C₁₇	—	4	4	no	—
66	Q₁₆×C₁₇	—	4	4	no	—
67	C₂×C₈×C₁₇	SD₁₆	1	4	yes	—
68	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₇	#41	1	4	yes	—
72	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
73	C₄⋊C₄×C₁₇	—	4	4	no	—
74	C₄⋊C₄×C₁₇	—	4	4	no	—
75	C₄²×C₁₇	D₈	1	2	yes	—
76	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
78	C₂²⋊Dic₁₇	—	4	2	no	—
79	C₄⋊Dic₁₇	—	2	4	no	—
80	C₄⋊Dic₁₇	—	4	2	no	—
81	#63225	—	4	4	no	—
82	C₄×Dic₁₇	—	4	2	no	—
83	#63214	—	2	4	no	—
84	#33705	—	17	1	no	maximal, Sylow
85	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
86	Q₈×C₁₇	—	4	4	no	—
87	Q₈×C₁₇	—	4	4	no	—
88	Q₈×C₁₇	—	8	4	no	—
89	C₂×C₄×C₁₇	—	2	4	no	—
90	C₂×C₄×C₁₇	—	4	4	no	—
91	C₂×C₄×C₁₇	—	8	4	no	—
92	C₂×C₄×C₁₇	#124	1	2	yes	—
93	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
97	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
98	C₂×Dic₁₇	—	4	4	no	—
99	Dic₃₄	—	8	4	no	—
100	C₈×C₁₇	#124	1	2	yes	—
101	C₈×C₁₇	C₂×SD₁₆	1	2	yes	—
102	C₈×C₁₇	C₂×SD₁₆	1	2	yes	—
103	C₈×C₁₇	C₂×#41	1	2	yes	—
104	#3160	—	17	1	no	—
105	C₂×#494	D₁₇	1	1	yes	characteristic
106	#2245	—	17	4	no	—
107	#2241	—	17	4	no	—
108	#2141	—	17	4	no	—
109	C₄×M₃₂	—	17	1	no	—
110	C₂²×C₁₇	—	2	2	no	—
111	C₂²×C₁₇	#486	1	2	yes	—
112	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
113	C₄×C₁₇	—	2	2	no	—
114	C₄×C₁₇	—	8	4	no	—
115	C₄×C₁₇	#486	1	2	yes	—
116	C₄×C₁₇	#510	1	4	yes	—
117	C₄×C₁₇	C₂×#124	1	1	yes	characteristic
118	C₄×C₁₇	C₂²×SD₁₆	1	1	yes	characteristic
119	Dic₁₇	—	8	4	no	—
120	C₂²×Q₁₆	—	2	1	no	—
121	C₂²×Q₁₆	C₂×D₁₇	1	2	yes	—
122	#533	—	34	1	no	—
123	C₂×C₄⋊Q₈	—	2	1	no	—
124	C₂×M₃₂	—	17	2	no	—
125	#500	—	17	2	no	—
126	#499	—	17	2	no	—
127	#499	—	34	1	no	—
128	#498	—	34	1	no	—
129	#494	—	2	1	no	—
130	#494	—	2	1	no	—
131	#494	C₂×D₁₇	1	2	yes	—
132	#494	C₂×D₁₇	1	2	yes	—
133	#427	—	17	2	no	—
134	#427	—	34	1	no	—
135	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
136	#358	—	17	16	no	—
137	#346	—	17	2	no	—
138	C₄×C₁₆	—	17	2	no	—
139	C₂×C₁₇	—	2	2	no	—
140	C₂×C₁₇	C₂×#486	1	1	yes	characteristic
141	C₂×C₁₇	C₂×#510	1	2	yes	—
142	C₂²×Q₈	—	2	2	no	—
143	C₂×Q₁₆	—	2	8	no	—
144	C₂×Q₁₆	—	2	2	no	—
145	C₂×Q₁₆	—	2	2	no	—
146	C₂×Q₁₆	—	2	2	no	—
147	C₂×Q₁₆	—	4	4	no	—
148	C₂×Q₁₆	C₂²×D₁₇	1	4	yes	—
149	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
151	C₄⋊Q₈	—	2	2	no	—
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	—	4	2	no	—
154	C₄⋊Q₈	—	34	2	no	—
155	#125	—	34	2	no	—
156	C₂²⋊Q₈	—	68	2	no	—
157	C₄×Q₈	—	68	2	no	—
158	#117	—	34	2	no	—
159	C₂×C₄⋊C₄	—	4	2	no	—
160	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
161	M₃₂	—	34	4	no	—
162	C₂×C₁₆	—	17	4	no	—
163	#107	—	17	4	no	—
164	#107	—	34	2	no	—
165	#106	—	34	2	no	—
166	#106	—	34	2	no	—
167	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
168	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
169	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
170	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
171	C₁₇	#33705	1	1	yes	characteristic, complemented, Sylow
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	4	no	—
175	C₂×Q₈	—	4	4	no	—
176	C₂×Q₈	—	68	2	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	#63228	1	2	yes	—
179	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
180	Q₁₆	—	2	8	no	—
181	Q₁₆	—	4	4	no	—
182	Q₁₆	—	4	4	no	—
183	Q₁₆	—	4	4	no	—
184	C₂×C₈	#63229	1	4	yes	—
185	C₂×C₈	#63254	1	4	yes	—
186	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
187	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
188	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
189	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
190	C₄⋊C₄	—	4	4	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄⋊C₄	—	34	4	no	—
193	C₄⋊C₄	—	68	4	no	—
194	C₄⋊C₄	—	68	2	no	—
195	C₂²⋊C₄	—	68	2	no	—
196	C₄²	—	68	2	no	—
197	C₄²	#63228	1	2	yes	—
198	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
199	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
200	C₁₆	—	34	4	no	—
201	C₂³	C₂×#63228	1	1	yes	characteristic
202	Q₈	—	4	4	no	—
203	Q₈	—	4	4	no	—
204	Q₈	—	8	4	no	—
205	Q₈	—	136	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	8	4	no	—
209	C₂×C₄	—	68	4	no	—
210	C₂×C₄	C₂×#63228	1	2	yes	—
211	C₂×C₄	C₂×#63228	1	2	yes	—
212	C₂×C₄	C₂×#63228	1	1	yes	characteristic
213	C₂×C₄	C₂×#63229	1	2	yes	—
214	C₂×C₄	#10587622	1	2	yes	—
215	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
216	C₈	C₂×#63229	1	2	yes	—
217	C₈	C₂×#63229	1	2	yes	—
218	C₈	#10587622	1	2	yes	—
219	C₈	C₂×#63254	1	2	yes	—
220	C₂²	—	2	2	no	—
221	C₂²	#11763155	1	2	yes	—
222	C₂²	C₂²×#63228	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	8	4	no	—
225	C₄	—	136	4	no	—
226	C₄	#11763155	1	2	yes	—
227	C₄	#11763250	1	4	yes	—
228	C₄	C₂×#10587622	1	1	yes	characteristic
229	C₄	C₂²×#63229	1	1	yes	characteristic
230	C₂	—	2	2	no	—
231	C₂	C₂×#11763155	1	1	yes	characteristic
232	C₂	C₂×#11763250	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#63228
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₁₇×#494	C₂
socle	C₂²×C₁₇	C₂²×D₈
prime radical	C₂³×C₁₇	C₂×D₈
hypercenter	C₂×#494	D₁₇
hypocenter	C₁₇	#33705
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
2. #14531029	C₂	semidirect
3. #14531011	C₂	semidirect
171. C₁₇	#33705	semidirect

Maximal subgroups

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

subgroup	index
#14539298	2
#14531029	2
#14531011	2
#14530504	2
C₄×#10587483	2
C₂×C₁₇×#494	2
#33705	17