Subgroups — #16927122

Order 4352.

516 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528936	C₂	1	2	yes	maximal
2	C₂×#11762558	C₂	1	2	yes	maximal
3	#14526667	C₂	1	2	yes	maximal
4	C₄×C₁₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#3622	—	2	2	no	—
6	C₂×C₄⋊#3622	—	2	2	no	—
7	C₂×C₄⋊#3622	C₂²	1	4	yes	—
8	#11762558	—	2	4	no	—
9	C₂×C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₇	C₄	1	2	yes	—
11	C₁₇×#391	C₄	1	2	yes	—
12	C₁₇×#389	—	2	2	no	—
13	C₁₇×#384	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₁₇	—	2	2	no	—
15	C₄³×C₁₇	C₂²	1	1	yes	characteristic
16	C₂²×Q₈×C₁₇	C₂×C₄	1	2	yes	—
17	C₁₇×C₄⋊Q₈	—	2	2	no	—
18	C₁₇×C₄⋊Q₈	D₄	1	2	yes	—
19	C₁₇×C₄⋊Q₈	Q₈	1	2	yes	—
20	C₄×Q₈×C₁₇	—	4	4	no	—
21	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
22	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
23	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
24	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₁₇	—	2	2	no	—
26	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₇	D₄	1	2	yes	—
29	C₂×C₄²×C₁₇	D₄	1	2	yes	—
30	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
31	C₂²×#3622	—	2	4	no	—
32	C₁₇×#95	—	2	4	no	—
33	C₄⋊#3622	—	2	4	no	—
34	C₄⋊#3622	—	2	4	no	—
35	C₄⋊#3622	—	4	4	no	—
36	C₄⋊#3622	—	4	4	no	—
37	C₂×Q₈×C₁₇	—	2	2	no	—
38	C₂×Q₈×C₁₇	—	4	2	no	—
39	C₂×Q₈×C₁₇	—	4	2	no	—
40	C₂×Q₈×C₁₇	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₁₇	—	2	2	no	—
42	C₂²×C₄×C₁₇	—	2	2	no	—
43	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
48	C₄⋊C₄×C₁₇	—	2	4	no	—
49	C₄⋊C₄×C₁₇	—	2	4	no	—
50	C₄⋊C₄×C₁₇	—	4	2	no	—
51	C₄⋊C₄×C₁₇	—	4	4	no	—
52	C₄⋊C₄×C₁₇	—	4	2	no	—
53	C₄⋊C₄×C₁₇	—	4	2	no	—
54	C₄²×C₁₇	—	2	2	no	—
55	C₄²×C₁₇	—	2	2	no	—
56	C₄²×C₁₇	—	2	2	no	—
57	C₄²×C₁₇	—	4	1	no	—
58	C₄²×C₁₇	—	4	4	no	—
59	C₄²×C₁₇	SD₁₆	1	4	yes	—
60	C₄²×C₁₇	Q₁₆	1	4	yes	—
61	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₁₇	#41	1	2	yes	—
64	C₂×#3622	—	4	4	no	—
65	C₂×#3622	—	4	4	no	—
66	C₂×#3622	—	4	4	no	—
67	#8587	—	17	1	no	maximal, Sylow
68	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	Q₈×C₁₇	—	4	4	no	—
70	Q₈×C₁₇	—	4	4	no	—
71	C₂×C₄×C₁₇	—	2	1	no	—
72	C₂×C₄×C₁₇	—	2	2	no	—
73	C₂×C₄×C₁₇	—	2	1	no	—
74	C₂×C₄×C₁₇	—	2	2	no	—
75	C₂×C₄×C₁₇	—	2	4	no	—
76	C₂×C₄×C₁₇	—	2	1	no	—
77	C₂×C₄×C₁₇	—	2	2	no	—
78	C₂×C₄×C₁₇	—	4	2	no	—
79	C₂×C₄×C₁₇	—	4	2	no	—
80	C₂×C₄×C₁₇	—	4	2	no	—
81	C₂×C₄×C₁₇	—	4	4	no	—
82	C₂×C₄×C₁₇	#100	1	2	yes	—
83	C₂×C₄×C₁₇	#103	1	4	yes	—
84	C₂×C₄×C₁₇	#103	1	4	yes	—
85	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
87	C₂×C₄×C₁₇	#121	1	2	yes	—
88	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
89	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
90	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
91	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
92	C₂×C₄×C₁₇	#137	1	2	yes	—
93	#3622	—	8	4	no	—
94	C₄×C₄⋊Q₈	D₁₇	1	1	yes	characteristic
95	#1842	—	17	2	no	—
96	C₂×#333	—	17	2	no	—
97	#1296	—	17	2	no	—
98	C₂²×C₁₇	#380	1	1	yes	characteristic
99	C₂²×C₁₇	C₂×#100	1	1	yes	characteristic
100	C₂²×C₁₇	#413	1	1	yes	characteristic
101	C₂²×C₁₇	C₂×#103	1	2	yes	—
102	C₂²×C₁₇	#417	1	2	yes	—
103	C₄×C₁₇	—	2	1	no	—
104	C₄×C₁₇	—	2	1	no	—
105	C₄×C₁₇	—	2	4	no	—
106	C₄×C₁₇	—	4	4	no	—
107	C₄×C₁₇	—	4	1	no	—
108	C₄×C₁₇	—	8	2	no	—
109	C₄×C₁₇	#333	1	4	yes	complemented
110	C₄×C₁₇	#460	1	2	yes	—
111	C₄×C₁₇	#462	1	2	yes	—
112	C₄×C₁₇	#475	1	2	yes	—
113	C₄×C₁₇	#477	1	2	yes	—
114	C₂×C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
115	C₂×C₄×Q₈	Dic₁₇	1	2	yes	—
116	C₂×C₄⋊C₈	—	17	4	no	—
117	C₂×C₄⋊C₈	—	34	2	no	—
118	C₂×C₄⋊C₈	—	34	2	no	—
119	#391	Dic₁₇	1	2	yes	—
120	#389	—	2	2	no	—
121	#384	C₂×D₁₇	1	1	yes	characteristic
122	C₄×C₄⋊C₄	—	2	2	no	—
123	C₄³	C₂×D₁₇	1	1	yes	characteristic
124	#333	—	34	4	no	—
125	C₂×C₁₇	C₂×#333	1	1	yes	characteristic
126	C₂×C₁₇	#1550	1	2	yes	—
127	C₂×C₁₇	#1560	1	1	yes	characteristic
128	C₂×C₁₇	#1800	1	1	yes	characteristic
129	C₂×C₁₇	#1803	1	2	yes	—
130	C₂²×Q₈	C₂×Dic₁₇	1	2	yes	—
131	C₂²×C₈	—	34	4	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	Dic₃₄	1	2	yes	—
134	C₄⋊Q₈	D₆₈	1	2	yes	—
135	C₄×Q₈	—	4	4	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
140	C₂×C₄²	—	2	2	no	—
141	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
142	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
143	C₂×C₄²	C₂²⋊D₁₇	1	2	yes	—
144	C₂×C₄²	C₂²⋊D₁₇	1	2	yes	—
145	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
146	C₄⋊C₈	—	34	4	no	—
147	C₄⋊C₈	—	34	4	no	—
148	C₄⋊C₈	—	68	4	no	—
149	C₄⋊C₈	—	68	4	no	—
150	#95	—	2	4	no	—
151	C₁₇	#8587	1	1	yes	characteristic, complemented, Sylow
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	4	2	no	—
155	C₂×Q₈	C₄⋊Dic₁₇	1	4	yes	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
159	C₂²×C₄	C₂²⋊Dic₁₇	1	2	yes	—
160	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
161	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
162	C₂²×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
163	C₂×C₈	—	68	4	no	—
164	C₂×C₈	—	68	4	no	—
165	C₂×C₈	—	68	4	no	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	4	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	4	1	no	—
176	C₄²	—	4	4	no	—
177	C₄²	#63230	1	4	yes	—
178	C₄²	#63231	1	4	yes	—
179	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
180	C₄²	C₂×D₆₈	1	1	yes	characteristic
181	C₄²	#63252	1	2	yes	—
182	C₂³	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
183	Q₈	—	4	4	no	—
184	Q₈	—	4	4	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	#10587504	1	2	yes	—
197	C₂×C₄	#10587506	1	4	yes	—
198	C₂×C₄	#10587506	1	4	yes	—
199	C₂×C₄	#10587610	1	2	yes	—
200	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
201	C₂×C₄	#10587612	1	1	yes	characteristic
202	C₂×C₄	#10587616	1	2	yes	—
203	C₂×C₄	C₂×#63230	1	2	yes	—
204	C₂×C₄	C₂×#63231	1	2	yes	—
205	C₂×C₄	#10587635	1	2	yes	—
206	C₂×C₄	#10587637	1	2	yes	—
207	C₈	—	136	4	no	—
208	C₂²	#11762669	1	1	yes	characteristic
209	C₂²	C₂×#10587504	1	1	yes	characteristic
210	C₂²	#11762793	1	1	yes	characteristic
211	C₂²	C₂×#10587506	1	2	yes	—
212	C₂²	#11762810	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	4	no	—
216	C₄	—	4	4	no	—
217	C₄	—	4	1	no	—
218	C₄	—	8	2	no	—
219	C₄	#11762558	1	4	yes	complemented
220	C₄	#11763009	1	2	yes	—
221	C₄	#11763019	1	2	yes	—
222	C₄	#11763094	1	2	yes	—
223	C₄	#11763104	1	2	yes	—
224	C₂	C₂×#11762558	1	1	yes	characteristic
225	C₂	#14527474	1	2	yes	—
226	C₂	#14527519	1	1	yes	characteristic
227	C₂	#14528776	1	1	yes	characteristic
228	C₂	#14528789	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₇
commutator	C₂²×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₄×C₁₇×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₁₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₄×C₄⋊Q₈	D₁₇
hypocenter	C₁₇	#8587
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
109. C₄×C₁₇	#333	semidirect
151. C₁₇	#8587	semidirect
219. C₄	#11762558	semidirect

Maximal subgroups

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

subgroup	index
#14528936	2
C₂×#11762558	2
#14526667	2
C₄×C₁₇×C₄⋊Q₈	2
#8587	17