Subgroups — #17978662

Order 4352.

490 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545201	C₂	1	8	yes	maximal, complemented
2	#14545121	C₂	1	2	yes	maximal
3	#14545116	C₂	1	1	yes	maximal, characteristic
4	#14544995	C₂	1	2	yes	maximal
5	C₂×#11763697	C₂	1	1	yes	maximal, characteristic
6	#14537714	C₂	1	1	yes	maximal, characteristic
7	C₂×#10587538	C₂²	1	2	yes	—
8	C₂²×#63216	C₂²	1	1	yes	characteristic
9	C₂²×#63216	C₂²	1	1	yes	characteristic
10	#11763730	C₂²	1	8	yes	—
11	C₄⋊#63216	C₂²	1	4	yes	—
12	#11763698	C₂²	1	4	yes	—
13	#11763697	C₂²	1	4	yes	—
14	#11763695	C₂²	1	4	yes	—
15	#11763366	C₄	1	1	yes	characteristic
16	#11762936	C₄	1	4	yes	—
17	#11762933	C₂²	1	4	yes	—
18	#11762837	C₄	1	2	yes	—
19	#11762767	C₂²	1	2	yes	—
20	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
21	C₁₇×#423	C₄	1	1	yes	characteristic
22	C₂×M₁₆×C₁₇	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
24	#10587612	C₂×C₄	1	2	yes	—
25	#10587610	C₈	1	4	yes	—
26	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
27	C₂²×#3622	C₂³	1	1	yes	characteristic
28	#10587556	C₈	1	2	yes	—
29	#10587555	C₈	1	2	yes	—
30	C₄×Dic₃₄	C₂×C₄	1	4	yes	—
31	#10587538	—	2	4	no	—
32	C₂×#63216	—	2	2	no	—
33	C₂×#63216	Q₈	1	4	yes	complemented
34	C₂×#63216	C₂³	1	4	yes	—
35	C₂×#63216	C₂³	1	2	yes	—
36	C₂×#63216	C₂³	1	2	yes	—
37	C₁₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
38	#10587486	C₂×C₄	1	4	yes	—
39	#10587485	C₂×C₄	1	8	yes	—
40	C₈×Dic₁₇	C₂³	1	4	yes	—
41	C₄×#3622	C₂×C₄	1	2	yes	—
42	C₄×#3622	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₃₄	C₂×C₈	1	4	yes	—
46	M₁₆×C₁₇	—	2	4	no	—
47	C₂×C₈×C₁₇	C₂²×C₄	1	4	yes	—
48	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
50	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
51	C₂²⋊Dic₁₇	C₂×C₈	1	4	yes	—
52	C₄⋊Dic₁₇	C₂×C₈	1	4	yes	—
53	C₄⋊Dic₁₇	C₂×C₈	1	4	yes	—
54	#63225	C₂×C₈	1	8	yes	—
55	C₄×Dic₁₇	C₂²×C₄	1	4	yes	—
56	C₂×#3622	—	2	2	no	—
57	C₂×#3622	C₂²×C₄	1	2	yes	—
58	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
59	C₂×#3622	C₂×Q₈	1	2	yes	—
60	C₂×#3622	#41	1	2	yes	—
61	C₂×#3622	C₂⁴	1	1	yes	characteristic
62	#63216	—	2	2	no	—
63	#63216	—	2	4	no	—
64	#63216	C₂×Q₈	1	4	yes	—
65	#33475	—	17	1	no	maximal, Sylow
66	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₁₇	—	4	2	no	—
68	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
69	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
70	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
71	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₁₇	#131	1	2	yes	—
73	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
74	C₂×Dic₁₇	C₂²×C₈	1	8	yes	—
75	Dic₃₄	—	4	4	no	—
76	C₈×C₁₇	—	2	4	no	—
77	#3622	—	2	2	no	—
78	#3622	C₄×Q₈	1	2	yes	—
79	#3622	C₂²×Q₈	1	1	yes	characteristic
80	#3622	C₂×#41	1	1	yes	characteristic
81	#2960	—	17	1	no	—
82	#2183	—	17	8	no	—
83	#2151	—	17	2	no	—
84	#2150	—	17	1	no	—
85	#2108	—	17	2	no	—
86	C₂×#346	—	17	1	no	—
87	C₂²×C₁₇	—	2	2	no	—
88	C₂²×C₁₇	C₈×Q₈	1	2	yes	—
89	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
90	C₄×C₁₇	—	2	2	no	—
91	C₄×C₁₇	—	2	2	no	—
92	C₄×C₁₇	C₈×Q₈	1	2	yes	—
93	C₄×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
94	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
95	Dic₁₇	—	2	8	no	—
96	#533	—	17	1	no	—
97	C₂×M₃₂	—	17	2	no	—
98	C₂²×C₁₆	—	17	1	no	—
99	C₂²×C₁₆	—	17	1	no	—
100	#446	—	17	4	no	—
101	C₈×Q₈	—	17	4	no	—
102	#424	—	17	2	no	—
103	#423	#595	1	1	yes	characteristic
104	#405	—	17	2	no	—
105	C₂×C₄×C₈	—	17	1	no	—
106	C₄⋊C₁₆	—	17	4	no	—
107	C₄⋊C₁₆	—	17	8	no	—
108	#346	—	17	4	no	—
109	#346	—	17	4	no	—
110	C₄×C₁₆	—	17	4	no	—
111	C₂×C₁₇	—	2	2	no	—
112	C₂×C₁₇	C₂×C₈×Q₈	1	1	yes	characteristic
113	C₂×C₁₇	#3407	1	2	yes	—
114	C₂×M₁₆	C₂×#595	1	2	yes	—
115	C₂²×C₈	—	17	1	no	—
116	C₂²×C₈	—	17	1	no	—
117	C₄⋊Q₈	—	17	2	no	—
118	#125	—	17	2	no	—
119	C₂²⋊Q₈	—	17	4	no	—
120	C₄×Q₈	—	17	4	no	—
121	#117	—	17	2	no	—
122	C₂×C₄²	C₂×#595	1	1	yes	characteristic
123	M₃₂	—	34	4	no	—
124	C₂×C₁₆	—	17	2	no	—
125	C₂×C₁₆	—	17	4	no	—
126	C₂×C₁₆	—	17	4	no	—
127	C₂×C₁₆	—	17	2	no	—
128	C₂×C₁₆	—	34	2	no	—
129	C₄⋊C₈	—	17	8	no	—
130	C₄⋊C₈	C₂×#595	1	4	yes	—
131	#97	—	17	4	no	—
132	C₄×C₈	—	17	4	no	—
133	C₄×C₈	—	17	2	no	—
134	C₄×C₈	—	17	2	no	—
135	C₁₇	#33475	1	1	yes	characteristic, complemented, Sylow
136	C₂×Q₈	—	17	4	no	—
137	C₂²×C₄	#63242	1	2	yes	—
138	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
139	M₁₆	—	2	4	no	—
140	C₂×C₈	—	17	2	no	—
141	C₂×C₈	—	17	2	no	—
142	C₂×C₈	—	17	1	no	—
143	C₂×C₈	—	17	1	no	—
144	C₂×C₈	—	17	2	no	—
145	C₂×C₈	—	34	2	no	—
146	C₂×C₈	C₂²×#595	1	4	yes	—
147	C₄⋊C₄	—	17	4	no	—
148	C₄⋊C₄	—	17	8	no	—
149	C₄⋊C₄	—	17	4	no	—
150	C₂²⋊C₄	—	17	4	no	—
151	C₄²	—	17	4	no	—
152	C₄²	#63242	1	1	yes	characteristic
153	C₄²	#63242	1	1	yes	characteristic
154	C₄²	C₂²×#595	1	2	yes	—
155	C₁₆	—	17	4	no	—
156	C₁₆	—	34	2	no	—
157	C₁₆	—	34	4	no	—
158	C₂³	C₂×#63242	1	1	yes	characteristic
159	Q₈	—	68	4	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	17	8	no	—
162	C₂×C₄	C₂×#63242	1	2	yes	—
163	C₂×C₄	C₂×#63242	1	2	yes	—
164	C₂×C₄	C₂×#63242	1	1	yes	characteristic
165	C₂×C₄	#10587678	1	2	yes	—
166	C₂×C₄	Q₈×#595	1	2	yes	—
167	C₂×C₄	C₂³×#595	1	1	yes	characteristic
168	C₈	—	2	4	no	—
169	C₈	—	17	1	no	—
170	C₈	—	17	1	no	—
171	C₈	—	17	2	no	—
172	C₈	—	34	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	#11763858	1	2	yes	—
175	C₂²	C₂²×#63242	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	34	8	no	—
179	C₄	#11763858	1	2	yes	—
180	C₄	C₂×Q₈×#595	1	1	yes	characteristic
181	C₄	C₂×#10587678	1	1	yes	characteristic
182	C₂	—	2	2	no	—
183	C₂	C₂×#11763858	1	1	yes	characteristic
184	C₂	#14545753	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₈
Fitting	C₁₇×#423	C₄
prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	#423	#595
hypocenter	C₁₇	#33475
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
1. #14545201	C₂	semidirect
33. C₂×#63216	Q₈	semidirect
135. C₁₇	#33475	semidirect

Maximal subgroups

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

subgroup	index
#14545201	2
#14545121	2
#14545116	2
#14544995	2
C₂×#11763697	2
#14537714	2
#33475	17