Subgroups — #16950771

Order 4352.

314 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762639	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762638	C₂	1	2	yes	maximal
3	C₂×C₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic
4	#14528954	C₂	1	2	yes	maximal
5	C₂×C₁₇×#368	C₂	1	1	yes	maximal, characteristic
6	C₂²×#63214	C₂²	1	1	yes	characteristic
7	C₂×#10587490	C₂²	1	2	yes	—
8	C₂×#10587493	—	2	2	no	—
9	#11762832	—	2	2	no	—
10	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
11	#11762639	—	2	2	no	—
12	#11762638	—	2	4	no	—
13	C₁₆×Dic₁₇	C₄	1	2	yes	—
14	C₁₆×Dic₁₇	C₄	1	2	yes	—
15	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
16	C₂×C₁₇×#108	C₂²	1	2	yes	—
17	C₁₇×#368	—	2	2	no	—
18	C₂×M₁₆×C₁₇	—	2	2	no	—
19	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
20	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
21	C₂×#63223	—	2	2	no	—
22	C₂²×#3622	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
25	C₂×C₁₆×C₁₇	Q₈	1	2	yes	—
26	C₁₇×#108	—	2	2	no	—
27	C₁₇×#108	—	2	2	no	—
28	#10587493	—	4	4	no	—
29	#10587490	—	2	2	no	—
30	#10587490	—	2	2	no	—
31	#10587485	—	4	4	no	—
32	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
33	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
34	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
35	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
36	C₂×#63214	C₂×C₄	1	2	yes	—
37	C₂×#63214	D₄	1	2	yes	—
38	C₂×#63214	Q₈	1	2	yes	—
39	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₁₇	Q₁₆	1	2	yes	—
41	M₁₆×C₁₇	—	4	2	no	—
42	M₁₆×C₁₇	—	4	2	no	—
43	C₂×C₈×C₁₇	—	4	2	no	—
44	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
50	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
51	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
52	C₄×Dic₁₇	D₈	1	2	yes	—
53	#63223	—	4	2	no	—
54	#63223	—	4	2	no	—
55	C₂×#3622	—	4	2	no	—
56	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
57	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
58	C₂×#3622	D₈	1	2	yes	—
59	C₂×#3622	Q₁₆	1	2	yes	—
60	C₁₆×C₁₇	C₄⋊C₄	1	4	yes	—
61	#63214	C₄⋊C₄	1	4	yes	—
62	#12909	—	17	1	no	maximal, Sylow
63	C₂³×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
64	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×Dic₁₇	—	2	4	no	—
71	C₂×Dic₁₇	#107	1	2	yes	—
72	C₂×Dic₁₇	#107	1	2	yes	—
73	C₈×C₁₇	—	8	2	no	—
74	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
77	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
78	#3622	—	8	2	no	—
79	#3622	#107	1	2	yes	—
80	#3622	#107	1	2	yes	—
81	C₂×#368	—	17	1	no	—
82	C₂×#368	—	17	2	no	—
83	C₂×#368	D₁₇	1	1	yes	characteristic
84	C₂×C₄×C₁₆	—	17	1	no	—
85	#1848	—	17	2	no	—
86	C₂²×C₁₇	—	2	2	no	—
87	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
88	C₂²×C₁₇	#494	1	1	yes	characteristic
89	C₂²×C₁₇	#500	1	1	yes	characteristic
90	C₄×C₁₇	#384	1	1	yes	characteristic
91	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
92	C₄×C₁₇	#493	1	1	yes	characteristic
93	C₄×C₁₇	#500	1	1	yes	characteristic
94	Dic₁₇	—	2	2	no	—
95	Dic₁₇	#368	1	4	yes	complemented
96	C₂²×C₁₆	—	17	1	no	—
97	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
98	C₂×#108	—	17	2	no	—
99	C₂×#108	—	34	2	no	—
100	C₂×#108	C₂×D₁₇	1	2	yes	—
101	#423	—	34	2	no	—
102	C₂×C₄×C₈	—	17	1	no	—
103	#368	—	2	2	no	—
104	#368	—	34	4	no	—
105	#368	—	34	2	no	—
106	C₄×C₁₆	—	17	2	no	—
107	C₄×C₁₆	—	17	2	no	—
108	C₂×C₁₇	—	2	1	no	—
109	C₂×C₁₇	—	2	1	no	—
110	C₂×C₁₇	#1845	1	1	yes	characteristic
111	C₂×C₁₇	C₂×#368	1	2	yes	—
112	C₂×M₁₆	—	2	2	no	—
113	C₂×M₁₆	—	34	2	no	—
114	C₂²×C₈	—	17	1	no	—
115	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
116	C₂×C₄²	—	17	1	no	—
117	C₂×C₁₆	—	17	2	no	—
118	C₂×C₁₆	—	17	2	no	—
119	C₂×C₁₆	—	17	2	no	—
120	C₂×C₁₆	Dic₃₄	1	2	yes	—
121	C₂×C₁₆	C₄×D₁₇	1	2	yes	—
122	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
123	#108	—	2	2	no	—
124	#108	—	2	2	no	—
125	#108	—	34	2	no	—
126	#108	—	34	2	no	—
127	#108	—	68	4	no	—
128	C₄⋊C₈	—	68	4	no	—
129	C₄×C₈	—	17	1	no	—
130	C₄×C₈	—	17	1	no	—
131	C₄×C₈	—	17	1	no	—
132	C₄×C₈	—	17	1	no	—
133	C₁₇	#12909	1	1	yes	characteristic, complemented, Sylow
134	C₂²×C₄	—	17	2	no	—
135	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
136	M₁₆	—	4	2	no	—
137	M₁₆	—	4	2	no	—
138	M₁₆	—	68	2	no	—
139	M₁₆	—	68	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	17	1	no	—
142	C₂×C₈	—	17	1	no	—
143	C₂×C₈	—	17	2	no	—
144	C₂×C₈	—	17	2	no	—
145	C₂×C₈	—	68	2	no	—
146	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
147	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
148	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
149	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
150	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
151	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
152	C₄²	—	17	1	no	—
153	C₄²	—	17	1	no	—
154	C₄²	—	17	2	no	—
155	C₁₆	—	17	4	no	—
156	C₁₆	#63225	1	4	yes	—
157	C₂³	Q₁₆×D₁₇	1	1	yes	characteristic
158	C₂×C₄	—	17	2	no	—
159	C₂×C₄	—	17	2	no	—
160	C₂×C₄	—	34	4	no	—
161	C₂×C₄	#10587574	1	1	yes	characteristic
162	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
163	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
164	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
165	C₂×C₄	Q₁₆×D₁₇	1	1	yes	characteristic
166	C₂×C₄	#10587626	1	1	yes	characteristic
167	C₈	—	8	2	no	—
168	C₈	—	17	2	no	—
169	C₈	—	17	2	no	—
170	C₈	—	136	2	no	—
171	C₈	#10587574	1	1	yes	characteristic
172	C₈	C₄⋊C₄×D₁₇	1	1	yes	characteristic
173	C₈	C₂×#63225	1	1	yes	characteristic
174	C₈	#10587630	1	1	yes	characteristic
175	C₂²	—	2	2	no	—
176	C₂²	#107×D₁₇	1	1	yes	characteristic
177	C₂²	#11763187	1	1	yes	characteristic
178	C₂²	#11763213	1	1	yes	characteristic
179	C₄	—	17	4	no	—
180	C₄	—	34	2	no	—
181	C₄	#11762678	1	1	yes	characteristic
182	C₄	#107×D₁₇	1	1	yes	characteristic
183	C₄	#11763184	1	1	yes	characteristic
184	C₄	#11763213	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	#14528942	1	1	yes	characteristic
188	C₂	D₁₇×#368	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₁₇
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#368	C₂
socle	C₂²×C₁₇	C₂×#107
prime radical	C₂³×C₁₇	C₂×Q₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×#368	D₁₇
hypocenter	C₁₇	#12909
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
95. Dic₁₇	#368	semidirect
133. C₁₇	#12909	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762639	2
C₂×#11762638	2
C₂×C₁₆×Dic₁₇	2
#14528954	2
C₂×C₁₇×#368	2
#12909	17