Subgroups — #17053230

Order 4352.

876 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	C₂×Q₈×#3622	C₂	1	2	yes	maximal
2	#14529503	C₂	1	6	yes	maximal
3	#14528901	C₂	1	3	yes	maximal
4	C₄×C₄⋊#3622	C₂	1	3	yes	maximal
5	C₄²×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×#3622	—	2	8	no	—
7	C₂×C₄⋊#3622	C₂²	1	6	yes	—
8	C₂×C₄⋊#3622	C₂²	1	6	yes	—
9	C₂×C₄×#3622	C₂²	1	6	yes	—
10	#11762564	C₂²	1	1	yes	characteristic
11	#11762564	C₂²	1	3	yes	—
12	#11762564	C₂²	1	6	yes	—
13	C₂×C₄×Q₈×C₁₇	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₁₇	C₄	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₁₇	C₄	1	3	yes	—
17	C₄×C₄⋊C₄×C₁₇	C₄	1	3	yes	—
18	C₄×C₄⋊C₄×C₁₇	C₂²	1	3	yes	—
19	C₄³×C₁₇	C₂²	1	3	yes	—
20	C₂²×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₄×Q₈×C₁₇	—	2	4	no	—
22	C₄×Q₈×C₁₇	C₈	1	8	yes	complemented
23	C₄×Q₈×C₁₇	D₄	1	4	yes	—
24	C₄×Q₈×C₁₇	Q₈	1	4	yes	—
25	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	3	yes	—
26	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	3	yes	—
27	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	3	yes	—
28	C₂×C₄²×C₁₇	C₂×C₄	1	3	yes	—
29	C₂×C₄²×C₁₇	C₂×C₄	1	3	yes	—
30	C₂×C₄²×C₁₇	C₂×C₄	1	3	yes	—
31	C₂×C₄²×C₁₇	C₂×C₄	1	3	yes	—
32	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₁₇	C₂³	1	3	yes	—
34	C₂×C₄²×C₁₇	C₂³	1	3	yes	—
35	C₂²×#3622	C₂³	1	6	yes	—
36	C₂²×#3622	C₂³	1	2	yes	—
37	C₁₇×#95	C₂×C₄	1	6	yes	—
38	C₄⋊#3622	—	2	12	no	—
39	C₄⋊#3622	—	2	12	no	—
40	C₄×#3622	—	2	12	no	—
41	C₂×Q₈×C₁₇	C₄⋊C₄	1	4	yes	—
42	C₂×Q₈×C₁₇	C₂×C₈	1	4	yes	—
43	C₂×Q₈×C₁₇	M₁₆	1	4	yes	—
44	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
45	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
46	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
47	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
50	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
51	C₄⋊C₄×C₁₇	—	2	6	no	—
52	C₄⋊C₄×C₁₇	C₄⋊C₄	1	12	yes	—
53	C₄⋊C₄×C₁₇	C₂×C₈	1	12	yes	—
54	C₄²×C₁₇	—	2	6	no	—
55	C₄²×C₁₇	—	2	6	no	—
56	C₄²×C₁₇	—	2	6	no	—
57	C₄²×C₁₇	—	2	2	no	—
58	C₄²×C₁₇	—	4	3	no	—
59	C₄²×C₁₇	C₂×C₈	1	12	yes	—
60	C₄²×C₁₇	C₂×D₄	1	6	yes	—
61	C₄²×C₁₇	C₂×Q₈	1	6	yes	—
62	C₂×#3622	—	2	2	no	—
63	C₂×#3622	—	2	2	no	—
64	C₂×#3622	—	2	6	no	—
65	C₂×#3622	—	2	6	no	—
66	C₂×#3622	—	2	6	no	—
67	C₂×#3622	#41	1	4	yes	—
68	#22734	—	17	1	no	maximal, Sylow
69	C₂³×C₁₇	C₂³×C₄	1	1	yes	characteristic
70	Q₈×C₁₇	C₄⋊C₈	1	16	yes	complemented
71	C₂×C₄×C₁₇	—	2	3	no	—
72	C₂×C₄×C₁₇	—	2	1	no	—
73	C₂×C₄×C₁₇	—	2	1	no	—
74	C₂×C₄×C₁₇	—	2	1	no	—
75	C₂×C₄×C₁₇	—	2	3	no	—
76	C₂×C₄×C₁₇	—	2	3	no	—
77	C₂×C₄×C₁₇	—	2	1	no	—
78	C₂×C₄×C₁₇	—	2	3	no	—
79	C₂×C₄×C₁₇	—	2	3	no	—
80	C₂×C₄×C₁₇	—	2	3	no	—
81	C₂×C₄×C₁₇	—	2	3	no	—
82	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
83	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
84	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
85	C₂×C₄×C₁₇	C₂²×C₈	1	6	yes	—
86	C₂×C₄×C₁₇	C₂²×C₈	1	6	yes	—
87	C₂×C₄×C₁₇	C₂×M₁₆	1	6	yes	—
88	C₂×C₄×C₁₇	#131	1	2	yes	—
89	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
91	C₂×C₄×C₁₇	C₂×#41	1	2	yes	—
92	C₂×C₄×C₁₇	2₋¹⁺⁴	1	2	yes	—
93	#3622	—	2	4	no	—
94	#3622	—	4	6	no	—
95	C₂×C₈×Q₈	—	17	2	no	—
96	C₄²×Q₈	D₁₇	1	1	yes	characteristic
97	#1939	—	17	6	no	—
98	#1831	—	17	3	no	—
99	C₄×C₄⋊C₈	—	17	3	no	—
100	C₂²×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₁₇	C₄×#41	1	1	yes	characteristic
102	C₂²×C₁₇	#519	1	1	yes	characteristic
103	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
104	C₂²×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
105	C₂²×C₁₇	C₂×#131	1	1	yes	characteristic
106	C₂²×C₁₇	#568	1	1	yes	characteristic
107	C₄×C₁₇	—	2	6	no	—
108	C₄×C₁₇	—	2	6	no	—
109	C₄×C₁₇	—	2	2	no	—
110	C₄×C₁₇	—	2	2	no	—
111	C₄×C₁₇	—	4	3	no	—
112	C₄×C₁₇	C₂×C₄⋊C₈	1	12	yes	—
113	C₄×C₁₇	#548	1	2	yes	—
114	C₄×C₁₇	#557	1	2	yes	—
115	C₂×C₄×Q₈	Dic₁₇	1	1	yes	characteristic
116	C₂×C₄×Q₈	Dic₁₇	1	1	yes	characteristic
117	C₂×C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
118	C₈×Q₈	—	34	8	no	—
119	C₂×C₄⋊C₈	—	17	6	no	—
120	C₂×C₄⋊C₈	—	17	6	no	—
121	C₂×C₄×C₈	—	17	6	no	—
122	C₄×C₄⋊C₄	Dic₁₇	1	3	yes	—
123	C₄×C₄⋊C₄	Dic₁₇	1	3	yes	—
124	C₄×C₄⋊C₄	C₂×D₁₇	1	3	yes	—
125	C₄³	C₂×D₁₇	1	3	yes	—
126	#336	—	17	3	no	—
127	#336	—	17	6	no	—
128	#336	—	17	1	no	—
129	C₂×C₁₇	#2351	1	1	yes	characteristic
130	C₂×C₁₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
131	C₂×C₁₇	#2906	1	1	yes	characteristic
132	C₂×C₁₇	C₈×#41	1	1	yes	characteristic
133	C₂×C₁₇	#2971	1	1	yes	characteristic
134	C₂×C₁₇	#2982	1	1	yes	characteristic
135	C₂×C₁₇	#2984	1	1	yes	characteristic
136	C₂²×Q₈	C₂×Dic₁₇	1	1	yes	characteristic
137	C₂²×C₈	—	17	2	no	—
138	C₂²×C₈	—	17	6	no	—
139	C₄×Q₈	—	2	4	no	—
140	C₄×Q₈	#3622	1	8	yes	complemented
141	C₄×Q₈	Dic₃₄	1	4	yes	—
142	C₄×Q₈	D₆₈	1	4	yes	—
143	C₂×C₄⋊C₄	C₂×Dic₁₇	1	3	yes	—
144	C₂×C₄⋊C₄	C₂×Dic₁₇	1	3	yes	—
145	C₂×C₄⋊C₄	C₂×Dic₁₇	1	3	yes	—
146	C₂×C₄²	C₂×Dic₁₇	1	3	yes	—
147	C₂×C₄²	C₂×Dic₁₇	1	3	yes	—
148	C₂×C₄²	C₂×Dic₁₇	1	3	yes	—
149	C₂×C₄²	C₂×Dic₁₇	1	3	yes	—
150	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
151	C₂×C₄²	C₂²×D₁₇	1	3	yes	—
152	C₂×C₄²	C₂²×D₁₇	1	3	yes	—
153	C₄⋊C₈	—	34	12	no	—
154	C₄⋊C₈	—	34	12	no	—
155	C₄×C₈	—	34	12	no	—
156	#95	C₂×Dic₁₇	1	6	yes	—
157	C₁₇	#22734	1	1	yes	characteristic, complemented, Sylow
158	C₂×Q₈	C₂×#3622	1	4	yes	—
159	C₂×Q₈	#63223	1	4	yes	—
160	C₂×Q₈	C₄⋊Dic₁₇	1	4	yes	—
161	C₂²×C₄	C₂²×Dic₁₇	1	3	yes	—
162	C₂²×C₄	C₂²×Dic₁₇	1	3	yes	—
163	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
164	C₂²×C₄	C₂²×Dic₁₇	1	3	yes	—
165	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
166	C₂²×C₄	C₂²×Dic₁₇	1	3	yes	—
167	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
168	C₂×C₈	—	17	4	no	—
169	C₂×C₈	—	34	2	no	—
170	C₂×C₈	—	34	2	no	—
171	C₂×C₈	—	34	6	no	—
172	C₂×C₈	—	34	6	no	—
173	C₂×C₈	—	34	6	no	—
174	C₄⋊C₄	—	2	6	no	—
175	C₄⋊C₄	C₂×#3622	1	12	yes	—
176	C₄⋊C₄	C₄⋊Dic₁₇	1	12	yes	—
177	C₄²	—	2	2	no	—
178	C₄²	—	2	6	no	—
179	C₄²	—	2	6	no	—
180	C₄²	—	2	6	no	—
181	C₄²	—	4	3	no	—
182	C₄²	C₂×#3622	1	12	yes	—
183	C₄²	C₂×Dic₃₄	1	6	yes	—
184	C₄²	C₂×D₆₈	1	6	yes	—
185	C₂³	C₂³×Dic₁₇	1	1	yes	characteristic
186	Q₈	C₄⋊#3622	1	16	yes	complemented
187	C₂×C₄	—	2	3	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	3	no	—
192	C₂×C₄	—	2	3	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	3	no	—
195	C₂×C₄	—	2	3	no	—
196	C₂×C₄	—	2	3	no	—
197	C₂×C₄	—	2	3	no	—
198	C₂×C₄	C₂²×#3622	1	2	yes	—
199	C₂×C₄	C₂²×#3622	1	6	yes	—
200	C₂×C₄	C₂²×#3622	1	6	yes	—
201	C₂×C₄	C₂×#63223	1	6	yes	—
202	C₂×C₄	C₂×C₄⋊Dic₁₇	1	6	yes	—
203	C₂×C₄	C₂×C₄⋊Dic₁₇	1	6	yes	—
204	C₂×C₄	#10587634	1	2	yes	—
205	C₂×C₄	C₂²×Dic₃₄	1	1	yes	characteristic
206	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
207	C₂×C₄	#10587694	1	2	yes	—
208	C₂×C₄	#41×D₁₇	1	2	yes	—
209	C₈	—	34	4	no	—
210	C₈	—	68	6	no	—
211	C₂²	C₂²×C₄⋊Dic₁₇	1	1	yes	characteristic
212	C₂²	#41×Dic₁₇	1	1	yes	characteristic
213	C₂²	#11763287	1	1	yes	characteristic
214	C₂²	C₂³×#3622	1	1	yes	characteristic
215	C₂²	C₂²×#63223	1	1	yes	characteristic
216	C₂²	C₂×#10587634	1	1	yes	characteristic
217	C₂²	#11763576	1	1	yes	characteristic
218	C₄	—	2	6	no	—
219	C₄	—	2	6	no	—
220	C₄	—	2	2	no	—
221	C₄	—	2	2	no	—
222	C₄	—	4	3	no	—
223	C₄	C₂×C₄⋊#3622	1	12	yes	—
224	C₄	#11763471	1	2	yes	—
225	C₄	#11763535	1	2	yes	—
226	C₂	#14531705	1	1	yes	characteristic
227	C₂	C₂²×C₄⋊#3622	1	1	yes	characteristic
228	C₂	#14537352	1	1	yes	characteristic
229	C₂	#41×#3622	1	1	yes	characteristic
230	C₂	#14537804	1	1	yes	characteristic
231	C₂	#14537865	1	1	yes	characteristic
232	C₂	#14537873	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₇
commutator	C₂²×C₁₇	C₂³×C₈
Fitting	C₄²×Q₈×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂³×C₄
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	C₄²×Q₈	D₁₇
hypocenter	C₁₇	#22734
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
22. C₄×Q₈×C₁₇	C₈	semidirect
70. Q₈×C₁₇	C₄⋊C₈	semidirect
140. C₄×Q₈	#3622	semidirect
157. C₁₇	#22734	semidirect
186. Q₈	C₄⋊#3622	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#3622	2
#14529503	2
#14528901	2
C₄×C₄⋊#3622	2
C₄²×Q₈×C₁₇	2
#22734	17