Subgroups — #16945868

Order 4352.

270 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14530540	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#3622	C₂	1	1	yes	maximal, characteristic
3	#14526847	C₂	1	4	yes	maximal, complemented
4	C₁₇×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587483	—	2	1	no	—
6	C₂×C₄×#3622	C₄	1	1	yes	characteristic
7	C₂×C₄×#3622	C₄	1	1	yes	characteristic
8	C₄⋊#63214	—	2	1	no	—
9	C₄⋊#63214	C₂²	1	2	yes	—
10	C₈×#3622	C₄	1	2	yes	—
11	C₈×#3622	C₂²	1	2	yes	—
12	C₂×C₁₇×M₃₂	—	2	1	no	—
13	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×C₄⋊C₁₆	—	2	1	no	—
15	C₁₇×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×#3622	—	2	1	no	—
20	C₂²×#3622	C₈	1	2	yes	—
21	C₁₇×M₃₂	—	4	2	no	—
22	C₂×C₁₆×C₁₇	—	2	2	no	—
23	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
25	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
26	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
27	#10587483	—	4	2	no	—
28	C₂×#63214	—	2	2	no	—
29	C₄×#3622	—	2	1	no	—
30	C₄×#3622	C₈	1	1	yes	characteristic
31	C₄×#3622	C₈	1	1	yes	characteristic
32	C₄×#3622	C₂×C₄	1	2	yes	—
33	C₄×#3622	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₁₇	—	2	2	no	—
38	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
41	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
42	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
46	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
47	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×#3622	—	2	4	no	—
49	C₂×#3622	—	2	2	no	—
50	C₂×#3622	—	2	4	no	—
51	C₂×#3622	C₂×C₈	1	2	yes	—
52	C₂×#3622	C₂×C₈	1	2	yes	—
53	C₁₆×C₁₇	—	4	2	no	—
54	#63214	—	4	2	no	—
55	#11996	—	17	1	no	maximal, Sylow
56	C₂³×C₁₇	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₇	—	2	2	no	—
58	C₂×C₄×C₁₇	—	2	2	no	—
59	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₁₇	#106	1	2	yes	—
63	C₂×C₄×C₁₇	#108	1	2	yes	—
64	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
67	C₈×C₁₇	—	2	1	no	—
68	C₈×C₁₇	—	2	1	no	—
69	C₈×C₁₇	#106	1	2	yes	—
70	C₈×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
71	C₈×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
72	#3622	—	2	4	no	—
73	#3622	—	2	2	no	—
74	#3622	M₃₂	1	4	yes	complemented
75	#2150	—	17	1	no	—
76	#2150	D₁₇	1	1	yes	characteristic
77	C₂×C₈²	—	17	1	no	—
78	#1367	—	17	4	no	—
79	C₂²×C₁₇	—	2	1	no	—
80	C₂²×C₁₇	—	2	1	no	—
81	C₂²×C₁₇	#334	1	2	yes	—
82	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
83	C₄×C₁₇	—	2	1	no	—
84	C₄×C₁₇	—	2	1	no	—
85	C₄×C₁₇	#334	1	2	yes	—
86	C₄×C₁₇	#364	1	2	yes	—
87	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
88	C₄×C₁₇	C₂×#108	1	1	yes	characteristic
89	C₄×C₁₇	C₂×M₃₂	1	2	yes	—
90	C₂×M₃₂	—	2	1	no	—
91	C₂×M₃₂	—	34	1	no	—
92	C₂×C₄×C₈	—	17	1	no	—
93	C₂×C₄×C₈	—	17	1	no	—
94	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
95	C₄⋊C₁₆	—	2	1	no	—
96	C₄⋊C₁₆	—	17	2	no	—
97	C₄⋊C₁₆	—	34	1	no	—
98	C₄⋊C₁₆	C₂×D₁₇	1	2	yes	—
99	C₈²	—	17	2	no	—
100	C₈²	—	17	2	no	—
101	C₂×C₁₇	—	2	2	no	—
102	C₂×C₁₇	C₂×#334	1	1	yes	characteristic
103	C₂×C₁₇	#2150	1	1	yes	characteristic
104	C₂×C₁₇	C₂×#364	1	1	yes	characteristic
105	C₂²×C₈	—	17	2	no	—
106	C₂²×C₈	—	34	1	no	—
107	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
108	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
109	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
110	M₃₂	—	4	2	no	—
111	M₃₂	—	68	2	no	—
112	C₂×C₁₆	—	2	2	no	—
113	C₂×C₁₆	—	34	2	no	—
114	C₄×C₈	—	17	2	no	—
115	C₄×C₈	—	17	2	no	—
116	C₄×C₈	—	17	1	no	—
117	C₄×C₈	—	17	1	no	—
118	C₄×C₈	—	34	1	no	—
119	C₄×C₈	Dic₃₄	1	1	yes	characteristic
120	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
121	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
122	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
123	C₁₇	#11996	1	1	yes	characteristic, complemented, Sylow
124	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
125	C₂²×C₄	#63218	1	1	yes	characteristic
126	C₂²×C₄	#63225	1	1	yes	characteristic
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	17	2	no	—
129	C₂×C₈	—	17	2	no	—
130	C₂×C₈	—	34	4	no	—
131	C₂×C₈	—	34	4	no	—
132	C₂×C₈	—	34	2	no	—
133	C₂×C₈	#63225	1	1	yes	characteristic
134	C₂×C₈	#63225	1	1	yes	characteristic
135	C₂×C₈	#63229	1	2	yes	—
136	C₂×C₈	#63230	1	2	yes	—
137	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
138	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
139	C₄²	C₈×D₁₇	1	1	yes	characteristic
140	C₄²	#63218	1	1	yes	characteristic
141	C₄²	#63225	1	1	yes	characteristic
142	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
143	C₁₆	—	4	2	no	—
144	C₁₆	—	68	2	no	—
145	C₂³	#10587485	1	1	yes	characteristic
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	#10587479	1	2	yes	—
149	C₂×C₄	#10587485	1	1	yes	characteristic
150	C₂×C₄	#10587485	1	1	yes	characteristic
151	C₂×C₄	#10587485	1	1	yes	characteristic
152	C₂×C₄	#10587493	1	2	yes	—
153	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
154	C₂×C₄	C₂×#63218	1	1	yes	characteristic
155	C₂×C₄	C₂×#63225	1	1	yes	characteristic
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	17	4	no	—
159	C₈	—	34	4	no	—
160	C₈	—	34	2	no	—
161	C₈	#10587479	1	2	yes	—
162	C₈	C₂×#63229	1	1	yes	characteristic
163	C₈	C₂×#63230	1	1	yes	characteristic
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	#11762559	1	2	yes	—
167	C₂²	C₂×#10587485	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	#11762559	1	2	yes	—
171	C₄	#11762629	1	2	yes	—
172	C₄	C₂×#10587479	1	1	yes	characteristic
173	C₄	C₂×#10587493	1	1	yes	characteristic
174	C₄	M₃₂×D₁₇	1	2	yes	—
175	C₂	—	2	2	no	—
176	C₂	C₂×#11762559	1	1	yes	characteristic
177	C₂	#14530539	1	1	yes	characteristic
178	C₂	C₂×#11762629	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#2150	C₂
socle	C₂²×C₁₇	C₂×C₄⋊C₈
prime radical	C₂³×C₁₇	C₄⋊C₈
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	#2150	D₁₇
hypocenter	C₁₇	#11996
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
3. #14526847	C₂	semidirect
74. #3622	M₃₂	semidirect
123. C₁₇	#11996	semidirect

Maximal subgroups

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

subgroup	index
#14530540	2
C₂×C₈×#3622	2
#14526847	2
C₁₇×#2150	2
#11996	17