Subgroups — #14537716

Order 2176.

364 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763362	C₂	1	1	yes	maximal, characteristic
2	#11762937	C₂	1	4	yes	maximal, complemented
3	C₈×Dic₃₄	C₂	1	4	yes	maximal, complemented
4	#11762837	C₂	1	1	yes	maximal, characteristic
5	#11762836	C₂	1	1	yes	maximal, characteristic
6	#11762832	C₂	1	1	yes	maximal, characteristic
7	#11762764	C₂	1	1	yes	maximal, characteristic
8	C₂×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
9	C₁₇×#405	C₂	1	1	yes	maximal, characteristic
10	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
12	C₁₇×#117	C₂²	1	1	yes	characteristic
13	#10587612	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
15	C₂×#63223	C₂²	1	1	yes	characteristic
16	C₂²×#3622	C₂²	1	1	yes	characteristic
17	#10587576	C₄	1	2	yes	—
18	#10587574	C₄	1	2	yes	—
19	#10587564	C₄	1	4	yes	—
20	C₄×Dic₃₄	C₂²	1	4	yes	—
21	C₁₇×#97	C₂²	1	2	yes	—
22	C₄×C₈×C₁₇	C₂²	1	2	yes	—
23	#10587486	C₂²	1	2	yes	—
24	#10587485	C₂²	1	4	yes	—
25	#10587485	C₂²	1	4	yes	—
26	C₈×Dic₁₇	C₂²	1	2	yes	—
27	C₈×Dic₁₇	C₂²	1	2	yes	—
28	C₈×Dic₁₇	C₂²	1	2	yes	—
29	C₄⋊#3622	C₂²	1	4	yes	—
30	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
31	C₂²×Dic₁₇	C₂×C₄	1	2	yes	—
32	C₂×Dic₃₄	C₂×C₄	1	4	yes	—
33	M₁₆×C₁₇	—	2	2	no	—
34	C₂×C₈×C₁₇	Q₈	1	4	yes	complemented
35	C₂×C₈×C₁₇	C₂³	1	2	yes	—
36	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
39	C₂²⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
40	C₄²×C₁₇	C₂³	1	2	yes	—
41	C₂²⋊Dic₁₇	C₂×C₄	1	2	yes	—
42	C₄⋊Dic₁₇	C₂×C₄	1	4	yes	—
43	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
44	#63225	C₂×C₄	1	4	yes	—
45	#63225	C₂×C₄	1	4	yes	—
46	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
47	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
48	C₄×Dic₁₇	C₂³	1	2	yes	—
49	C₄×Dic₁₇	C₂³	1	2	yes	—
50	#63223	—	2	2	no	—
51	C₂×#3622	—	2	2	no	—
52	C₂×#3622	C₂³	1	2	yes	—
53	C₂×#3622	C₂³	1	2	yes	—
54	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₇	C₂²×C₄	1	4	yes	—
56	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₁₇	#41	1	2	yes	—
59	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
60	C₂×Dic₁₇	—	2	4	no	—
61	C₂×Dic₁₇	C₂²×C₄	1	4	yes	—
62	C₂×Dic₁₇	C₂²×C₄	1	2	yes	—
63	C₂×Dic₁₇	C₂²×C₄	1	2	yes	—
64	Dic₃₄	—	2	8	no	—
65	C₈×C₁₇	—	2	2	no	—
66	C₈×C₁₇	C₂×Q₈	1	2	yes	—
67	C₈×C₁₇	C₂×Q₈	1	2	yes	—
68	#3622	—	2	2	no	—
69	#3622	—	2	2	no	—
70	#2960	—	17	1	no	maximal, Sylow
71	C₂²×C₁₇	—	2	2	no	—
72	C₂²×C₁₇	C₄×Q₈	1	2	yes	—
73	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
74	C₄×C₁₇	—	2	4	no	—
75	C₄×C₁₇	C₄×Q₈	1	2	yes	—
76	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
77	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
78	Dic₁₇	—	2	4	no	—
79	Dic₁₇	—	2	2	no	—
80	Dic₁₇	#131	1	4	yes	—
81	#533	—	17	1	no	—
82	#446	—	17	4	no	—
83	C₈×Q₈	—	17	4	no	—
84	#424	—	17	1	no	—
85	#424	—	17	1	no	—
86	#423	—	17	1	no	—
87	#405	—	17	1	no	—
88	#405	D₁₇	1	1	yes	characteristic
89	C₂×C₄×C₈	—	17	1	no	—
90	C₂×C₁₇	—	2	2	no	—
91	C₂×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
92	C₂×C₁₇	C₂×#131	1	2	yes	—
93	C₂×M₁₆	—	17	1	no	—
94	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
95	C₂²×C₈	—	17	1	no	—
96	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
97	C₄⋊Q₈	—	17	2	no	—
98	#125	—	17	2	no	—
99	C₂²⋊Q₈	—	17	4	no	—
100	C₄×Q₈	—	17	4	no	—
101	#117	—	17	1	no	—
102	#117	C₂×D₁₇	1	1	yes	characteristic
103	C₂×C₄²	—	17	1	no	—
104	C₄⋊C₈	—	17	4	no	—
105	C₄⋊C₈	—	17	4	no	—
106	C₄⋊C₈	—	17	4	no	—
107	#97	—	17	2	no	—
108	#97	C₂×D₁₇	1	2	yes	—
109	C₄×C₈	—	17	2	no	—
110	C₄×C₈	—	17	2	no	—
111	C₄×C₈	—	17	2	no	—
112	C₄×C₈	C₂×D₁₇	1	2	yes	—
113	C₁₇	#2960	1	1	yes	characteristic, complemented, Sylow
114	C₂×Q₈	—	17	4	no	—
115	C₂²×C₄	—	17	2	no	—
116	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
117	M₁₆	—	2	2	no	—
118	M₁₆	—	34	2	no	—
119	C₂×C₈	—	17	2	no	—
120	C₂×C₈	—	17	2	no	—
121	C₂×C₈	—	34	2	no	—
122	C₂×C₈	Dic₃₄	1	4	yes	complemented
123	C₂×C₈	C₂²×D₁₇	1	2	yes	—
124	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
125	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
126	C₄⋊C₄	—	17	2	no	—
127	C₄⋊C₄	—	17	4	no	—
128	C₄⋊C₄	—	17	4	no	—
129	C₄⋊C₄	—	17	4	no	—
130	C₄⋊C₄	C₄×D₁₇	1	2	yes	—
131	C₂²⋊C₄	—	17	2	no	—
132	C₂²⋊C₄	C₄×D₁₇	1	2	yes	—
133	C₄²	—	17	1	no	—
134	C₄²	—	17	1	no	—
135	C₄²	—	17	2	no	—
136	C₄²	—	17	2	no	—
137	C₄²	C₂²×D₁₇	1	2	yes	—
138	C₂³	C₂×C₄×D₁₇	1	1	yes	characteristic
139	Q₈	—	34	8	no	—
140	C₂×C₄	—	17	2	no	—
141	C₂×C₄	—	17	2	no	—
142	C₂×C₄	—	17	4	no	—
143	C₂×C₄	—	34	4	no	—
144	C₂×C₄	C₂×Dic₃₄	1	2	yes	—
145	C₂×C₄	C₂×C₄×D₁₇	1	4	yes	—
146	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
147	C₂×C₄	#63252	1	2	yes	—
148	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	34	2	no	—
151	C₈	—	34	2	no	—
152	C₈	C₂×Dic₃₄	1	2	yes	—
153	C₈	C₂×Dic₃₄	1	2	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	C₄×Dic₃₄	1	2	yes	—
156	C₂²	C₂²×C₄×D₁₇	1	1	yes	characteristic
157	C₄	—	2	4	no	—
158	C₄	—	17	4	no	—
159	C₄	—	34	2	no	—
160	C₄	—	34	4	no	—
161	C₄	C₄×Dic₃₄	1	2	yes	—
162	C₄	C₂²×Dic₃₄	1	1	yes	characteristic
163	C₄	C₂×#63252	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	C₂×C₄×Dic₃₄	1	1	yes	characteristic
166	C₂	#131×D₁₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₇
commutator, socle	C₂²×C₁₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#405	C₂
prime radical	C₂³×C₁₇	C₂²×C₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	#405	D₁₇
hypocenter	C₁₇	#2960
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #11762937	C₂	semidirect
3. C₈×Dic₃₄	C₂	semidirect
34. C₂×C₈×C₁₇	Q₈	semidirect
113. C₁₇	#2960	semidirect
122. C₂×C₈	Dic₃₄	semidirect

Maximal subgroups

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

subgroup	index
#11763362	2
#11762937	2
C₈×Dic₃₄	2
#11762837	2
#11762836	2
#11762832	2
#11762764	2
C₂×C₈×Dic₁₇	2
C₁₇×#405	2
#2960	17