Subgroups — #16952330

Order 4352.

338 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#63214	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#63214	C₂	1	2	yes	maximal
3	C₂×#11762585	C₂	1	1	yes	maximal, characteristic
4	#14526862	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#63214	C₂²	1	2	yes	—
7	C₂²×#63214	C₂²	1	1	yes	characteristic
8	C₂²×#63214	C₂²	1	1	yes	characteristic
9	C₄⋊#63214	—	2	2	no	—
10	C₄⋊#63214	—	2	4	no	—
11	#11762585	—	2	2	no	—
12	C₂×C₁₇×C₄⋊C₈	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₁₇	C₄	1	1	yes	characteristic
16	C₁₇×#336	C₄	1	2	yes	—
17	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₇	C₈	1	2	yes	—
21	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₁₇×C₄⋊C₈	—	2	2	no	—
24	C₄×C₈×C₁₇	—	2	2	no	—
25	C₄×C₈×C₁₇	—	2	2	no	—
26	C₄×C₈×C₁₇	D₄	1	2	yes	—
27	C₄×C₈×C₁₇	Q₈	1	2	yes	—
28	C₁₇×#95	C₈	1	1	yes	characteristic
29	C₁₇×#95	C₈	1	1	yes	characteristic
30	C₂×#63214	—	2	1	no	—
31	C₂×#63214	—	2	1	no	—
32	C₂×#63214	—	2	2	no	—
33	C₂×#63214	—	2	2	no	—
34	C₂×#63214	—	2	1	no	—
35	C₂×#63214	—	2	1	no	—
36	C₂×#63214	—	2	2	no	—
37	C₂×#63214	D₄	1	2	yes	—
38	C₂×#63214	Q₈	1	2	yes	—
39	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
40	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₇	—	2	1	no	—
45	C₂×C₈×C₁₇	—	2	1	no	—
46	C₂×C₈×C₁₇	—	2	2	no	—
47	C₂×C₈×C₁₇	—	2	2	no	—
48	C₂×C₈×C₁₇	—	2	2	no	—
49	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
55	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
56	C₄⋊C₄×C₁₇	—	2	4	no	—
57	C₄²×C₁₇	—	2	2	no	—
58	C₄²×C₁₇	—	2	2	no	—
59	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
60	C₄²×C₁₇	M₁₆	1	2	yes	—
61	#63214	—	2	2	no	—
62	#63214	—	2	2	no	—
63	#63214	—	4	2	no	—
64	#13160	—	17	1	no	maximal, Sylow
65	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₇	—	2	2	no	—
67	C₂×C₄×C₁₇	—	2	2	no	—
68	C₂×C₄×C₁₇	—	2	2	no	—
69	C₂×C₄×C₁₇	—	2	2	no	—
70	C₂×C₄×C₁₇	—	2	2	no	—
71	C₂×C₄×C₁₇	—	2	2	no	—
72	C₂×C₄×C₁₇	—	2	2	no	—
73	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
74	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
75	C₂×C₄×C₁₇	M₃₂	1	4	yes	—
76	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
81	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
82	C₈×C₁₇	—	2	2	no	—
83	C₈×C₁₇	—	4	2	no	—
84	C₈×C₁₇	#121	1	1	yes	characteristic
85	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
86	C₈×C₁₇	#125	1	1	yes	characteristic
87	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
88	C₂×C₄⋊C₁₆	—	17	2	no	—
89	C₂×C₄⋊C₁₆	—	17	1	no	—
90	C₂×#346	—	17	1	no	—
91	C₈×C₄⋊C₄	D₁₇	1	1	yes	characteristic
92	#1374	—	17	2	no	—
93	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
94	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
95	C₂²×C₁₇	C₈×Q₈	1	1	yes	characteristic
96	C₂²×C₁₇	C₂×M₃₂	1	2	yes	—
97	C₂²×C₁₇	#504	1	2	yes	—
98	C₄×C₁₇	—	2	2	no	—
99	C₄×C₁₇	—	2	2	no	—
100	C₄×C₁₇	—	2	4	no	—
101	C₄×C₁₇	—	4	2	no	—
102	C₄×C₁₇	#389	1	1	yes	characteristic
103	C₄×C₁₇	#424	1	1	yes	characteristic
104	C₄×C₁₇	#436	1	1	yes	characteristic
105	C₄×C₁₇	#446	1	1	yes	characteristic
106	C₂²×C₁₆	—	17	1	no	—
107	C₂²×C₁₆	—	17	1	no	—
108	C₂²×C₁₆	—	17	2	no	—
109	C₂×C₄⋊C₈	Dic₁₇	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂×D₁₇	1	2	yes	—
111	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
112	C₄×C₄⋊C₄	Dic₁₇	1	1	yes	characteristic
113	C₄⋊C₁₆	—	34	4	no	—
114	C₄⋊C₁₆	—	34	2	no	—
115	#346	—	34	2	no	—
116	#336	Dic₁₇	1	2	yes	—
117	C₂×C₁₇	#1939	1	1	yes	characteristic
118	C₂×C₁₇	#2150	1	1	yes	characteristic
119	C₂×C₁₇	#2151	1	1	yes	characteristic
120	C₂×C₁₇	#2168	1	2	yes	—
121	C₂×C₁₇	#2183	1	2	yes	—
122	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
123	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
124	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
125	C₂×C₄⋊C₄	#3622	1	2	yes	—
126	C₂×C₄²	C₂×Dic₁₇	1	2	yes	—
127	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
128	C₂×C₁₆	—	17	2	no	—
129	C₂×C₁₆	—	17	2	no	—
130	C₂×C₁₆	—	34	1	no	—
131	C₂×C₁₆	—	34	1	no	—
132	C₂×C₁₆	—	34	1	no	—
133	C₂×C₁₆	—	34	1	no	—
134	C₂×C₁₆	—	34	2	no	—
135	C₂×C₁₆	—	34	2	no	—
136	C₂×C₁₆	—	34	2	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	—	2	2	no	—
140	C₄×C₈	Dic₃₄	1	2	yes	—
141	C₄×C₈	D₆₈	1	2	yes	—
142	#95	#3622	1	1	yes	characteristic
143	#95	#3622	1	1	yes	characteristic
144	C₁₇	#13160	1	1	yes	characteristic, complemented, Sylow
145	C₂²×C₄	C₂×#3622	1	2	yes	—
146	C₂²×C₄	C₂×#3622	1	1	yes	characteristic
147	C₂²×C₄	C₂×#3622	1	1	yes	characteristic
148	C₂²×C₄	C₂×#3622	1	2	yes	—
149	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
156	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
157	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
158	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
159	C₂×C₈	#63254	1	1	yes	characteristic
160	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
161	C₂×C₈	#63256	1	1	yes	characteristic
162	C₄⋊C₄	—	2	4	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	#63223	1	2	yes	—
166	C₄²	C₄⋊Dic₁₇	1	2	yes	—
167	C₁₆	—	34	2	no	—
168	C₁₆	—	34	2	no	—
169	C₁₆	—	68	2	no	—
170	C₂³	C₂²×#3622	1	1	yes	characteristic
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	C₄⋊#3622	1	2	yes	—
179	C₂×C₄	C₄⋊#3622	1	2	yes	—
180	C₂×C₄	#10587483	1	4	yes	—
181	C₂×C₄	C₂×#63223	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
183	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
184	C₂×C₄	Q₈×Dic₁₇	1	1	yes	characteristic
185	C₂×C₄	#10587634	1	1	yes	characteristic
186	C₂×C₄	#10587634	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	—	4	2	no	—
189	C₈	#10587574	1	1	yes	characteristic
190	C₈	#10587576	1	1	yes	characteristic
191	C₈	#10587581	1	1	yes	characteristic
192	C₈	#10587583	1	1	yes	characteristic
193	C₂²	C₂×C₄⋊#3622	1	1	yes	characteristic
194	C₂²	D₄×#3622	1	1	yes	characteristic
195	C₂²	Q₈×#3622	1	1	yes	characteristic
196	C₂²	C₂×#10587483	1	2	yes	—
197	C₂²	#11763226	1	2	yes	—
198	C₄	—	2	4	no	—
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	4	2	no	—
202	C₄	#11762708	1	1	yes	characteristic
203	C₄	#11762836	1	1	yes	characteristic
204	C₄	#11762887	1	1	yes	characteristic
205	C₄	#11762940	1	1	yes	characteristic
206	C₂	#14529503	1	1	yes	characteristic
207	C₂	#14530540	1	1	yes	characteristic
208	C₂	#14530543	1	1	yes	characteristic
209	C₂	#14530610	1	2	yes	—
210	C₂	#14530693	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₇
commutator, socle, prime radical	C₂³×C₁₇	C₂²×C₈
Fitting	C₈×C₄⋊C₄×C₁₇	C₂
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₈×C₄⋊C₄	D₁₇
hypocenter	C₁₇	#13160
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
144. C₁₇	#13160	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#63214	2
C₂×#11762585	2
#14526862	2
C₈×C₄⋊C₄×C₁₇	2
#13160	17