Subgroups — #16952344

Order 4352.

318 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762629	C₂	1	2	yes	maximal
2	#14530540	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊Dic₁₇	C₂	1	1	yes	maximal, characteristic
4	#14526865	C₂	1	2	yes	maximal
5	C₄×C₁₇×M₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587483	C₂²	1	2	yes	—
7	C₂×C₄⋊#3622	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₁₇	C₂²	1	2	yes	—
9	C₄×C₄⋊Dic₁₇	C₄	1	1	yes	characteristic
10	#11762629	—	2	4	no	—
11	C₄⋊#63214	—	2	1	no	—
12	C₄⋊#63214	—	2	1	no	—
13	#11762563	C₄	1	2	yes	—
14	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×#346	—	2	1	no	—
18	C₄×C₁₆×C₁₇	—	2	1	no	—
19	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₇	—	2	1	no	—
23	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
24	C₂²×#3622	C₂×C₄	1	2	yes	—
25	C₁₇×M₃₂	—	2	1	no	—
26	C₁₇×M₃₂	—	2	1	no	—
27	C₁₇×M₃₂	D₄	1	2	yes	—
28	C₁₇×M₃₂	Q₈	1	2	yes	—
29	C₂×C₁₆×C₁₇	—	2	1	no	—
30	C₂×C₁₆×C₁₇	—	2	1	no	—
31	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
32	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
33	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
34	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
35	#10587502	—	2	1	no	—
36	C₈×Dic₁₇	—	2	2	no	—
37	C₈×Dic₁₇	—	2	2	no	—
38	#10587483	—	2	2	no	—
39	#10587483	—	2	2	no	—
40	C₂×#63214	—	2	2	no	—
41	C₄⋊#3622	C₈	1	2	yes	—
42	C₄⋊#3622	C₈	1	2	yes	—
43	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₁₇	—	2	2	no	—
47	C₂×C₈×C₁₇	—	2	1	no	—
48	C₂×C₈×C₁₇	—	2	1	no	—
49	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
56	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
57	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
59	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
60	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
61	C₄⋊Dic₁₇	—	2	4	no	—
62	C₄×Dic₁₇	—	2	2	no	—
63	C₄×Dic₁₇	—	2	2	no	—
64	C₂×#3622	—	2	2	no	—
65	C₂×#3622	—	2	2	no	—
66	C₂×#3622	C₂×C₈	1	4	yes	—
67	C₁₆×C₁₇	—	2	2	no	—
68	C₁₆×C₁₇	—	2	2	no	—
69	#63214	—	4	2	no	—
70	#13163	—	17	1	no	maximal, Sylow
71	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₁₇	—	2	2	no	—
73	C₂×C₄×C₁₇	—	4	1	no	—
74	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
83	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
84	C₂×Dic₁₇	—	2	4	no	—
85	C₂×Dic₁₇	—	4	2	no	—
86	C₂×Dic₁₇	—	4	2	no	—
87	C₈×C₁₇	—	2	1	no	—
88	C₈×C₁₇	—	2	1	no	—
89	C₈×C₁₇	#121	1	1	yes	characteristic
90	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
91	C₈×C₁₇	#125	1	1	yes	characteristic
92	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
93	#3622	—	2	4	no	—
94	C₂×#364	—	17	2	no	—
95	#2150	—	17	1	no	—
96	C₄×M₃₂	D₁₇	1	1	yes	characteristic
97	C₈×C₄⋊C₄	—	17	1	no	—
98	#1376	—	17	2	no	—
99	C₂²×C₁₇	—	2	1	no	—
100	C₂²×C₁₇	—	2	1	no	—
101	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₂²×C₁₇	#436	1	1	yes	characteristic
103	C₂²×C₁₇	#446	1	1	yes	characteristic
104	C₄×C₁₇	—	2	1	no	—
105	C₄×C₁₇	—	2	1	no	—
106	C₄×C₁₇	—	2	1	no	—
107	C₄×C₁₇	#364	1	2	yes	complemented
108	C₄×C₁₇	#389	1	1	yes	characteristic
109	C₄×C₁₇	#424	1	1	yes	characteristic
110	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
111	C₄×C₁₇	C₈×Q₈	1	1	yes	characteristic
112	Dic₁₇	—	4	4	no	—
113	C₂×M₃₂	—	17	2	no	—
114	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
115	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
116	C₂×C₄⋊C₈	—	17	1	no	—
117	C₂×C₄×C₈	—	17	2	no	—
118	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
119	C₄×C₄⋊C₄	—	17	1	no	—
120	#364	—	34	4	no	—
121	C₄⋊C₁₆	—	34	1	no	—
122	C₄⋊C₁₆	—	34	1	no	—
123	#346	—	2	1	no	—
124	C₄×C₁₆	—	2	1	no	—
125	#336	—	17	2	no	—
126	C₂×C₁₇	—	2	1	no	—
127	C₂×C₁₇	—	2	1	no	—
128	C₂×C₁₇	#1939	1	1	yes	characteristic
129	C₂×C₁₇	C₂×#364	1	1	yes	characteristic
130	C₂×C₁₇	#2153	1	1	yes	characteristic
131	C₂²×C₈	—	17	2	no	—
132	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
133	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
134	C₂×C₄⋊C₄	—	34	1	no	—
135	C₂×C₄²	—	17	2	no	—
136	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
137	M₃₂	—	2	1	no	—
138	M₃₂	—	2	1	no	—
139	M₃₂	—	34	2	no	—
140	M₃₂	—	34	2	no	—
141	M₃₂	Dic₃₄	1	2	yes	—
142	M₃₂	D₆₈	1	2	yes	—
143	C₂×C₁₆	—	2	1	no	—
144	C₂×C₁₆	—	2	1	no	—
145	C₂×C₁₆	—	34	2	no	—
146	C₄⋊C₈	—	17	2	no	—
147	C₄⋊C₈	—	17	2	no	—
148	C₄×C₈	—	34	2	no	—
149	C₄×C₈	—	34	2	no	—
150	C₄×C₈	Dic₃₄	1	1	yes	characteristic
151	C₄×C₈	Dic₃₄	1	1	yes	characteristic
152	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
153	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
154	#95	—	34	1	no	—
155	C₁₇	#13163	1	1	yes	characteristic, complemented, Sylow
156	C₂²×C₄	—	34	2	no	—
157	C₂²×C₄	#63218	1	1	yes	characteristic
158	C₂²×C₄	#63218	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	17	4	no	—
163	C₂×C₈	—	34	2	no	—
164	C₂×C₈	—	34	2	no	—
165	C₂×C₈	#63225	1	1	yes	characteristic
166	C₂×C₈	#63225	1	1	yes	characteristic
167	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
168	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
169	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
170	C₂×C₈	#63252	1	1	yes	characteristic
171	C₂×C₈	#63252	1	1	yes	characteristic
172	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
173	C₄⋊C₄	—	34	4	no	—
174	C₄²	—	34	2	no	—
175	C₄²	—	34	2	no	—
176	C₄²	C₈×D₁₇	1	1	yes	characteristic
177	C₄²	C₈×D₁₇	1	1	yes	characteristic
178	C₄²	#63225	1	1	yes	characteristic
179	C₄²	#63225	1	1	yes	characteristic
180	C₁₆	—	2	2	no	—
181	C₁₆	—	2	2	no	—
182	C₁₆	—	68	2	no	—
183	C₂³	C₂×#63218	1	1	yes	characteristic
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	34	4	no	—
187	C₂×C₄	—	68	2	no	—
188	C₂×C₄	—	68	2	no	—
189	C₂×C₄	#10587485	1	1	yes	characteristic
190	C₂×C₄	#10587485	1	1	yes	characteristic
191	C₂×C₄	#10587485	1	1	yes	characteristic
192	C₂×C₄	#10587485	1	1	yes	characteristic
193	C₂×C₄	C₄×Dic₃₄	1	1	yes	characteristic
194	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
195	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
196	C₂×C₄	#10587587	1	1	yes	characteristic
197	C₂×C₄	#10587587	1	1	yes	characteristic
198	C₂×C₄	C₂×#63225	1	1	yes	characteristic
199	C₈	—	2	1	no	—
200	C₈	—	2	1	no	—
201	C₈	—	34	4	no	—
202	C₈	#10587555	1	1	yes	characteristic
203	C₈	#10587556	1	1	yes	characteristic
204	C₈	#10587610	1	1	yes	characteristic
205	C₈	#10587616	1	1	yes	characteristic
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	C₂×#10587485	1	1	yes	characteristic
209	C₂²	#11762888	1	1	yes	characteristic
210	C₂²	#11762939	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	68	4	no	—
215	C₄	#11762629	1	2	yes	complemented
216	C₄	#11762707	1	1	yes	characteristic
217	C₄	#11762835	1	1	yes	characteristic
218	C₄	C₈×D₆₈	1	1	yes	characteristic
219	C₄	C₈×Dic₃₄	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#14529502	1	1	yes	characteristic
223	C₂	C₂×#11762629	1	1	yes	characteristic
224	C₂	#14530549	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₄×C₁₇×M₃₂	C₂
socle	C₂²×C₁₇	C₂×C₄⋊C₈
prime radical	C₂³×C₁₇	C₂×M₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₄×M₃₂	D₁₇
hypocenter	C₁₇	#13163
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
107. C₄×C₁₇	#364	semidirect
155. C₁₇	#13163	semidirect
215. C₄	#11762629	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762629	2
#14530540	2
C₈×C₄⋊Dic₁₇	2
#14526865	2
C₄×C₁₇×M₃₂	2
#13163	17