Subgroups — C₁₇×#2336 #14525342

Order 2176.

428 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇×#125	C₂	1	1	yes	maximal, characteristic
2	C₁₇×#389	C₂	1	2	yes	maximal
3	C₁₇×#387	C₂	1	2	yes	maximal
4	C₁₇×#387	C₂	1	2	yes	maximal
5	C₁₇×#387	C₂	1	2	yes	maximal
6	C₁₇×#387	C₂	1	2	yes	maximal
7	C₁₇×#383	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₄×C₁₇	C₂	1	2	yes	maximal
9	C₄×C₄⋊C₄×C₁₇	C₂	1	1	yes	maximal, characteristic
10	C₁₇×#125	C₄	1	8	yes	complemented
11	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₇	C₂²	1	2	yes	—
19	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₁₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
23	C₁₇×#95	C₂²	1	2	yes	—
24	C₁₇×#95	C₂²	1	2	yes	—
25	C₁₇×#95	C₂²	1	2	yes	—
26	C₁₇×#95	C₂²	1	1	yes	characteristic
27	C₁₇×#95	C₂²	1	2	yes	—
28	C₁₇×#95	C₂²	1	1	yes	characteristic
29	C₁₇×#95	C₂²	1	2	yes	—
30	C₁₇×#95	C₂²	1	2	yes	—
31	C₁₇×#95	C₂²	1	2	yes	—
32	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
33	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
34	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
35	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
41	C₄⋊C₄×C₁₇	—	2	4	no	—
42	C₄⋊C₄×C₁₇	—	2	4	no	—
43	C₄⋊C₄×C₁₇	—	4	2	no	—
44	C₄⋊C₄×C₁₇	C₂×C₄	1	8	yes	—
45	C₄⋊C₄×C₁₇	C₂×C₄	1	4	yes	—
46	C₄⋊C₄×C₁₇	C₂×C₄	1	4	yes	—
47	C₄⋊C₄×C₁₇	C₂×C₄	1	8	yes	—
48	C₄²×C₁₇	—	2	4	no	—
49	C₄²×C₁₇	—	4	1	no	—
50	C₄²×C₁₇	—	4	2	no	—
51	C₄²×C₁₇	—	4	1	no	—
52	C₄²×C₁₇	C₂×C₄	1	4	yes	—
53	C₂³×C₁₇	C₂⁴	1	1	yes	characteristic
54	C₂×C₄×C₁₇	—	2	2	no	—
55	C₂×C₄×C₁₇	—	2	2	no	—
56	C₂×C₄×C₁₇	—	2	2	no	—
57	C₂×C₄×C₁₇	—	2	1	no	—
58	C₂×C₄×C₁₇	—	2	2	no	—
59	C₂×C₄×C₁₇	—	2	2	no	—
60	C₂×C₄×C₁₇	—	2	2	no	—
61	C₂×C₄×C₁₇	—	2	1	no	—
62	C₂×C₄×C₁₇	—	2	2	no	—
63	C₂×C₄×C₁₇	—	2	2	no	—
64	C₂×C₄×C₁₇	—	2	2	no	—
65	C₂×C₄×C₁₇	—	2	2	no	—
66	C₂×C₄×C₁₇	—	2	2	no	—
67	C₂×C₄×C₁₇	—	2	2	no	—
68	C₂×C₄×C₁₇	—	2	2	no	—
69	C₂×C₄×C₁₇	—	2	2	no	—
70	C₂×C₄×C₁₇	—	2	2	no	—
71	C₂×C₄×C₁₇	—	2	2	no	—
72	C₂×C₄×C₁₇	—	2	2	no	—
73	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₁₇	C₂²×C₄	1	4	yes	—
77	C₂×C₄×C₁₇	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₁₇	#41	1	4	yes	—
79	#2336	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
81	C₂²×C₁₇	C₂×#41	1	2	yes	—
82	C₂²×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
83	C₂²×C₁₇	2₋¹⁺⁴	1	2	yes	—
84	C₂²×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
85	C₄×C₁₇	—	2	4	no	—
86	C₄×C₁₇	—	2	2	no	—
87	C₄×C₁₇	—	2	2	no	—
88	C₄×C₁₇	—	2	2	no	—
89	C₄×C₁₇	—	2	4	no	—
90	C₄×C₁₇	—	2	4	no	—
91	C₄×C₁₇	—	4	2	no	—
92	C₄×C₁₇	—	4	2	no	—
93	C₄×C₁₇	—	4	2	no	—
94	C₂×#125	C₂×C₁₇	1	1	yes	characteristic
95	#389	C₂×C₁₇	1	2	yes	—
96	#387	C₂×C₁₇	1	2	yes	—
97	#387	C₂×C₁₇	1	2	yes	—
98	#387	C₂×C₁₇	1	2	yes	—
99	#387	C₂×C₁₇	1	2	yes	—
100	#383	C₂×C₁₇	1	1	yes	characteristic
101	C₄×C₄⋊C₄	C₂×C₁₇	1	2	yes	—
102	C₄×C₄⋊C₄	C₂×C₁₇	1	1	yes	characteristic
103	C₂×C₁₇	C₄×#41	1	1	yes	characteristic
104	C₂×C₁₇	#519	1	1	yes	characteristic
105	C₂×C₁₇	#520	1	1	yes	characteristic
106	C₂×C₁₇	#539	1	2	yes	—
107	C₂×C₁₇	#541	1	2	yes	—
108	#125	C₄×C₁₇	1	8	yes	complemented
109	C₂×C₄⋊C₄	C₂²×C₁₇	1	2	yes	—
110	C₂×C₄⋊C₄	C₂²×C₁₇	1	2	yes	—
111	C₂×C₄⋊C₄	C₂²×C₁₇	1	2	yes	—
112	C₂×C₄⋊C₄	C₂²×C₁₇	1	2	yes	—
113	C₂×C₄⋊C₄	C₂²×C₁₇	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂²×C₁₇	1	1	yes	characteristic
115	C₂×C₄⋊C₄	C₂²×C₁₇	1	2	yes	—
116	C₂×C₄²	C₂²×C₁₇	1	2	yes	—
117	C₂×C₄²	C₂²×C₁₇	1	1	yes	characteristic
118	C₂×C₄²	C₂²×C₁₇	1	1	yes	characteristic
119	C₂×C₄²	C₂²×C₁₇	1	2	yes	—
120	C₂×C₄²	C₂²×C₁₇	1	1	yes	characteristic
121	#95	C₂²×C₁₇	1	2	yes	—
122	#95	C₂²×C₁₇	1	2	yes	—
123	#95	C₂²×C₁₇	1	2	yes	—
124	#95	C₂²×C₁₇	1	1	yes	characteristic
125	#95	C₂²×C₁₇	1	2	yes	—
126	#95	C₂²×C₁₇	1	1	yes	characteristic
127	#95	C₂²×C₁₇	1	2	yes	—
128	#95	C₂²×C₁₇	1	2	yes	—
129	#95	C₂²×C₁₇	1	2	yes	—
130	C₁₇	#2336	1	1	yes	characteristic, direct product factor, Sylow
131	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
132	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
133	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
134	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
135	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
136	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
137	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
138	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
139	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
140	C₄⋊C₄	—	2	4	no	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	C₂×C₄×C₁₇	1	8	yes	—
144	C₄⋊C₄	C₂×C₄×C₁₇	1	4	yes	—
145	C₄⋊C₄	C₂×C₄×C₁₇	1	4	yes	—
146	C₄⋊C₄	C₂×C₄×C₁₇	1	8	yes	—
147	C₄²	—	2	4	no	—
148	C₄²	—	4	1	no	—
149	C₄²	—	4	2	no	—
150	C₄²	—	4	1	no	—
151	C₄²	C₂×C₄×C₁₇	1	4	yes	—
152	C₂³	C₂⁴×C₁₇	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	C₂²×C₄×C₁₇	1	2	yes	—
173	C₂×C₄	C₂²×C₄×C₁₇	1	2	yes	—
174	C₂×C₄	C₂²×C₄×C₁₇	1	2	yes	—
175	C₂×C₄	C₂²×C₄×C₁₇	1	4	yes	—
176	C₂×C₄	C₂²×C₄×C₁₇	1	4	yes	—
177	C₂×C₄	#41×C₁₇	1	4	yes	—
178	C₂²	C₂³×C₄×C₁₇	1	1	yes	characteristic
179	C₂²	C₂×#41×C₁₇	1	2	yes	—
180	C₂²	C₁₇×H(5,𝔽₂)	1	1	yes	characteristic
181	C₂²	C₁₇×2₋¹⁺⁴	1	2	yes	—
182	C₂²	C₁₇×2₋¹⁺⁴	1	1	yes	characteristic
183	C₄	—	2	4	no	—
184	C₄	—	2	2	no	—
185	C₄	—	2	2	no	—
186	C₄	—	2	2	no	—
187	C₄	—	2	4	no	—
188	C₄	—	2	4	no	—
189	C₄	—	4	2	no	—
190	C₄	—	4	2	no	—
191	C₄	—	4	2	no	—
192	C₂	C₄×#41×C₁₇	1	1	yes	characteristic
193	C₂	C₁₇×#519	1	1	yes	characteristic
194	C₂	C₁₇×#520	1	1	yes	characteristic
195	C₂	C₁₇×#539	1	2	yes	—
196	C₂	C₁₇×#541	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₁₇
Frattini, cosocle	C₂³	C₂⁴×C₁₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₁₇×#125	C₄	semidirect
79. #2336	C₁₇	direct (N × H)
108. #125	C₄×C₁₇	semidirect
130. C₁₇	#2336	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₇×#125	2
C₁₇×#389	2
C₁₇×#387	2
C₁₇×#383	2
C₄×C₄⋊C₄×C₁₇	2
#2336	17

Subgroups — C17×#2336 #14525342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#2336 #14525342