Subgroups — #17961659

Order 4352.

382 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763732	C₂	1	2	yes	maximal
2	C₂×#11763731	C₂	1	2	yes	maximal
3	#14545118	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587538	C₂²	1	4	yes	—
6	#11763732	—	2	4	no	—
7	#11763731	—	2	4	no	—
8	#11763730	—	2	4	no	—
9	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
14	C₄²×Dic₁₇	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₁₇	—	2	2	no	—
20	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
23	C₂²×#3622	C₂³	1	1	yes	characteristic
24	#10587538	—	2	4	no	—
25	#10587538	—	2	4	no	—
26	C₂×#63216	—	2	4	no	—
27	C₄×C₈×C₁₇	—	2	2	no	—
28	C₈×Dic₁₇	—	2	2	no	—
29	C₈×Dic₁₇	C₈	1	2	yes	—
30	C₈×Dic₁₇	C₈	1	2	yes	—
31	C₈×Dic₁₇	D₄	1	2	yes	—
32	C₈×Dic₁₇	D₄	1	2	yes	—
33	C₈×Dic₁₇	Q₈	1	2	yes	—
34	C₈×Dic₁₇	Q₈	1	2	yes	—
35	C₄×#3622	D₄	1	2	yes	—
36	C₄×#3622	Q₈	1	2	yes	—
37	C₂²×C₄×C₁₇	—	2	1	no	—
38	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
39	C₂²×Dic₁₇	—	2	1	no	—
40	C₂²×Dic₁₇	M₁₆	1	2	yes	—
41	C₂×C₈×C₁₇	—	2	1	no	—
42	C₂×C₈×C₁₇	—	2	1	no	—
43	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
47	C₄²×C₁₇	—	2	1	no	—
48	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
49	C₄×Dic₁₇	—	2	8	no	—
50	C₄×Dic₁₇	—	2	1	no	—
51	C₄×Dic₁₇	C₄⋊C₄	1	2	yes	—
52	C₄×Dic₁₇	C₄⋊C₄	1	2	yes	—
53	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
54	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
55	C₂×#3622	C₄⋊C₄	1	2	yes	—
56	C₂×#3622	C₂×C₈	1	2	yes	—
57	C₂×#3622	M₁₆	1	2	yes	—
58	C₂×#3622	C₂×D₄	1	1	yes	characteristic
59	C₂×#3622	C₂×D₄	1	1	yes	characteristic
60	C₂×#3622	C₂×D₄	1	1	yes	characteristic
61	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
62	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
63	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
64	#63216	—	4	4	no	—
65	#12900	—	17	1	no	maximal, Sylow
66	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
67	C₂×C₄×C₁₇	—	2	4	no	—
68	C₂×C₄×C₁₇	—	2	2	no	—
69	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
75	C₂×Dic₁₇	—	2	4	no	—
76	C₂×Dic₁₇	—	2	2	no	—
77	C₂×Dic₁₇	—	2	4	no	—
78	C₂×Dic₁₇	C₄⋊C₈	1	2	yes	—
79	C₂×Dic₁₇	C₄⋊C₈	1	2	yes	—
80	C₈×C₁₇	—	2	2	no	—
81	C₈×C₁₇	C₄⋊C₈	1	4	yes	—
82	#3622	C₄⋊C₈	1	4	yes	—
83	#3622	#127	1	1	yes	characteristic
84	#3622	C₄⋊Q₈	1	1	yes	characteristic
85	#3622	C₄⋊Q₈	1	1	yes	characteristic
86	#3622	C₄⋊Q₈	1	1	yes	characteristic
87	C₂×#364	—	17	2	no	—
88	C₂×#364	—	17	2	no	—
89	#2150	—	17	2	no	—
90	C₄²×C₈	—	17	1	no	—
91	C₂²×C₁₇	—	2	2	no	—
92	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
93	C₂²×C₁₇	#423	1	1	yes	characteristic
94	C₂²×C₁₇	#423	1	1	yes	characteristic
95	C₄×C₁₇	—	2	4	no	—
96	C₄×C₁₇	#384	1	1	yes	characteristic
97	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
98	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
99	C₄×C₁₇	#423	1	1	yes	characteristic
100	Dic₁₇	—	2	4	no	—
101	Dic₁₇	—	2	2	no	—
102	Dic₁₇	#364	1	4	yes	complemented
103	C₂×M₃₂	—	17	4	no	—
104	C₂×C₄×C₈	—	17	1	no	—
105	C₂×C₄×C₈	—	17	1	no	—
106	C₂×C₄×C₈	—	17	1	no	—
107	C₂×C₄×C₈	—	17	1	no	—
108	C₂×C₄×C₈	—	17	1	no	—
109	C₂×C₄×C₈	#595	1	1	yes	characteristic
110	C₄³	—	17	1	no	—
111	#364	—	34	4	no	—
112	#364	—	34	4	no	—
113	C₄⋊C₁₆	—	34	4	no	—
114	C₂×C₁₇	—	2	1	no	—
115	C₂×C₁₇	—	2	1	no	—
116	C₂×C₁₇	#1842	1	1	yes	characteristic
117	C₂×C₁₇	C₂×#364	1	2	yes	—
118	C₂²×C₈	—	17	1	no	—
119	C₂²×C₈	—	17	1	no	—
120	C₂²×C₈	C₂×#595	1	1	yes	characteristic
121	C₂²×C₈	C₂×#595	1	1	yes	characteristic
122	C₂×C₄²	—	17	1	no	—
123	C₂×C₄²	—	17	1	no	—
124	C₂×C₄²	—	34	2	no	—
125	C₂×C₄²	C₂×#595	1	1	yes	characteristic
126	M₃₂	—	34	4	no	—
127	M₃₂	—	34	4	no	—
128	C₂×C₁₆	—	34	4	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	17	2	no	—
131	C₄×C₈	—	17	2	no	—
132	C₄×C₈	—	17	2	no	—
133	C₄×C₈	—	17	2	no	—
134	C₄×C₈	—	17	2	no	—
135	C₄×C₈	—	17	2	no	—
136	C₄×C₈	—	17	2	no	—
137	C₄×C₈	—	17	2	no	—
138	C₄×C₈	—	34	2	no	—
139	C₁₇	#12900	1	1	yes	characteristic, complemented, Sylow
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	17	2	no	—
142	C₂²×C₄	—	34	1	no	—
143	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	17	2	no	—
147	C₂×C₈	—	17	2	no	—
148	C₂×C₈	—	17	1	no	—
149	C₂×C₈	—	17	1	no	—
150	C₂×C₈	—	17	2	no	—
151	C₂×C₈	—	17	1	no	—
152	C₂×C₈	—	17	1	no	—
153	C₂×C₈	—	17	1	no	—
154	C₂×C₈	—	17	1	no	—
155	C₂×C₈	#63242	1	2	yes	—
156	C₂×C₈	C₄⋊#595	1	2	yes	—
157	C₂×C₈	C₄⋊#595	1	2	yes	—
158	C₂×C₈	#63247	1	2	yes	—
159	C₄²	—	2	1	no	—
160	C₄²	—	17	1	no	—
161	C₄²	—	17	1	no	—
162	C₄²	—	17	2	no	—
163	C₄²	—	17	2	no	—
164	C₄²	—	34	8	no	—
165	C₄²	—	34	1	no	—
166	C₄²	C₄⋊#595	1	2	yes	—
167	C₁₆	—	68	4	no	—
168	C₂³	D₁₇⋊M₁₆	1	1	yes	characteristic
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	17	2	no	—
172	C₂×C₄	—	17	2	no	—
173	C₂×C₄	—	34	4	no	—
174	C₂×C₄	—	34	4	no	—
175	C₂×C₄	—	34	2	no	—
176	C₂×C₄	C₂×#63242	1	1	yes	characteristic
177	C₂×C₄	D₁₇⋊M₁₆	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
181	C₂×C₄	C₂×#63247	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	—	17	1	no	—
184	C₈	—	17	1	no	—
185	C₈	—	17	1	no	—
186	C₈	—	17	1	no	—
187	C₈	—	17	4	no	—
188	C₈	#10587544	1	4	yes	—
189	C₂²	—	2	2	no	—
190	C₂²	#11763810	1	1	yes	characteristic
191	C₂²	#11763812	1	1	yes	characteristic
192	C₂²	#11763814	1	1	yes	characteristic
193	C₄	—	2	4	no	—
194	C₄	—	17	4	no	—
195	C₄	—	34	4	no	—
196	C₄	—	34	2	no	—
197	C₄	#11763755	1	1	yes	characteristic
198	C₄	#11763810	1	1	yes	characteristic
199	C₄	C₂×#10587544	1	1	yes	characteristic
200	C₄	#11763812	1	1	yes	characteristic
201	C₂	—	2	1	no	—
202	C₂	—	2	1	no	—
203	C₂	#14544717	1	1	yes	characteristic
204	C₂	#14545123	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#595
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle	C₂²×C₁₇	C₂×C₄⋊C₈
prime radical	C₂³×C₁₇	C₂×M₁₆
cosocle	C₂²×#3622	C₂³
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#12900
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
102. Dic₁₇	#364	semidirect
139. C₁₇	#12900	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763732	2
C₂×#11763731	2
#14545118	2
C₄×C₈×Dic₁₇	2
#12900	17