Subgroups — C₁₇×#2680 #14525686

Order 2176.

568 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇×#123	C₂	1	2	yes	maximal
2	C₂×C₁₇×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
3	C₁₇×#398	C₂	1	2	yes	maximal, complemented
4	C₁₇×#397	C₂	1	2	yes	maximal, complemented
5	C₁₇×#395	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₇×#394	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#388	C₂	1	2	yes	maximal, complemented
8	C₁₇×#385	C₂	1	2	yes	maximal, complemented
9	C₁₇×#384	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₈×C₁₇	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₁₇	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₁₇	C₂²	1	2	yes	—
13	C₁₇×#123	—	2	8	no	—
14	C₁₇×C₂²⋊Q₈	—	2	8	no	—
15	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₁₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₁₇	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₁₇	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₁₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₁₇	C₂²	1	2	yes	—
27	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
28	C₁₇×#95	C₂²	1	2	yes	—
29	C₁₇×#95	C₂²	1	2	yes	—
30	C₁₇×#95	C₂²	1	2	yes	—
31	C₁₇×#95	C₂²	1	1	yes	characteristic
32	C₁₇×#95	C₂²	1	1	yes	characteristic
33	C₂⁴×C₁₇	C₂³	1	2	yes	—
34	C₂×Q₈×C₁₇	—	2	4	no	—
35	C₂×Q₈×C₁₇	—	4	1	no	—
36	C₂×D₄×C₁₇	—	2	4	no	—
37	C₂×D₄×C₁₇	—	4	1	no	—
38	C₂²×C₄×C₁₇	—	2	4	no	—
39	C₂²×C₄×C₁₇	—	2	4	no	—
40	C₂²×C₄×C₁₇	—	4	2	no	—
41	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
47	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
48	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
49	C₄⋊C₄×C₁₇	—	2	4	no	—
50	C₄⋊C₄×C₁₇	—	2	4	no	—
51	C₄⋊C₄×C₁₇	—	2	4	no	—
52	C₄⋊C₄×C₁₇	—	2	4	no	—
53	C₄⋊C₄×C₁₇	—	2	4	no	—
54	C₄⋊C₄×C₁₇	—	4	1	no	—
55	C₄⋊C₄×C₁₇	—	4	1	no	—
56	C₂²⋊C₄×C₁₇	—	2	4	no	—
57	C₂²⋊C₄×C₁₇	—	2	4	no	—
58	C₂²⋊C₄×C₁₇	—	2	4	no	—
59	C₂²⋊C₄×C₁₇	—	2	4	no	—
60	C₂²⋊C₄×C₁₇	D₄	1	8	yes	—
61	C₄²×C₁₇	—	2	2	no	—
62	C₂³×C₁₇	—	2	2	no	—
63	C₂³×C₁₇	—	2	2	no	—
64	C₂³×C₁₇	—	2	2	no	—
65	C₂³×C₁₇	—	2	2	no	—
66	C₂³×C₁₇	—	2	2	no	—
67	C₂³×C₁₇	—	2	2	no	—
68	C₂³×C₁₇	C₂×D₄	1	4	yes	—
69	C₂³×C₁₇	C₂⁴	1	1	yes	characteristic
70	Q₈×C₁₇	—	4	4	no	—
71	D₄×C₁₇	—	4	4	no	—
72	C₂×C₄×C₁₇	—	2	2	no	—
73	C₂×C₄×C₁₇	—	2	2	no	—
74	C₂×C₄×C₁₇	—	2	1	no	—
75	C₂×C₄×C₁₇	—	2	2	no	—
76	C₂×C₄×C₁₇	—	2	1	no	—
77	C₂×C₄×C₁₇	—	2	2	no	—
78	C₂×C₄×C₁₇	—	2	2	no	—
79	C₂×C₄×C₁₇	—	2	2	no	—
80	C₂×C₄×C₁₇	—	2	2	no	—
81	C₂×C₄×C₁₇	—	2	2	no	—
82	C₂×C₄×C₁₇	—	2	2	no	—
83	C₂×C₄×C₁₇	—	2	2	no	—
84	C₂×C₄×C₁₇	—	2	2	no	—
85	C₂×C₄×C₁₇	—	2	2	no	—
86	C₂×C₄×C₁₇	—	2	1	no	—
87	C₂×C₄×C₁₇	—	2	1	no	—
88	C₂×C₄×C₁₇	—	2	2	no	—
89	C₂×C₄×C₁₇	—	2	2	no	—
90	C₂×C₄×C₁₇	—	2	2	no	—
91	C₂×C₄×C₁₇	—	4	8	no	—
92	C₂×C₄×C₁₇	C₂×D₄	1	4	yes	—
93	C₂×C₄×C₁₇	C₂×D₄	1	4	yes	—
94	C₂×C₄×C₁₇	#41	1	2	yes	—
95	#2680	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂²×C₁₇	—	2	4	no	—
97	C₂²×C₁₇	—	2	4	no	—
98	C₂²×C₁₇	—	2	4	no	—
99	C₂²×C₁₇	—	4	2	no	—
100	C₂²×C₁₇	—	4	2	no	—
101	C₂²×C₁₇	—	4	2	no	—
102	C₂²×C₁₇	—	4	2	no	—
103	C₂²×C₁₇	C₂²×D₄	1	2	yes	—
104	C₂²×C₁₇	C₂×#41	1	1	yes	characteristic
105	C₂²×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
106	C₂²×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
107	C₂²×C₁₇	2₋¹⁺⁴	1	2	yes	—
108	C₄×C₁₇	—	2	2	no	—
109	C₄×C₁₇	—	2	4	no	—
110	C₄×C₁₇	—	2	4	no	—
111	C₄×C₁₇	—	4	2	no	—
112	C₄×C₁₇	—	4	2	no	—
113	C₄×C₁₇	—	4	1	no	—
114	C₄×C₁₇	—	4	1	no	—
115	C₄×C₁₇	—	4	2	no	—
116	C₂×#123	C₂×C₁₇	1	2	yes	—
117	C₂×C₂²⋊Q₈	C₂×C₁₇	1	2	yes	complemented
118	#398	C₂×C₁₇	1	2	yes	complemented
119	#397	C₂×C₁₇	1	2	yes	complemented
120	#395	C₂×C₁₇	1	1	yes	characteristic, complemented
121	#394	C₂×C₁₇	1	1	yes	characteristic
122	#388	C₂×C₁₇	1	2	yes	complemented
123	#385	C₂×C₁₇	1	2	yes	complemented
124	#384	C₂×C₁₇	1	1	yes	characteristic, complemented
125	C₂×C₁₇	—	4	4	no	—
126	C₂×C₁₇	#536	1	2	yes	—
127	C₂×C₁₇	#539	1	1	yes	characteristic
128	C₂×C₁₇	D₄²	1	1	yes	characteristic
129	C₂×C₁₇	#547	1	2	yes	—
130	C₂×C₁₇	#563	1	1	yes	characteristic
131	C₂²×Q₈	C₂²×C₁₇	1	1	yes	characteristic
132	C₂²×D₄	C₂²×C₁₇	1	1	yes	characteristic
133	C₂³×C₄	C₂²×C₁₇	1	2	yes	—
134	#123	—	2	8	no	—
135	C₂²⋊Q₈	—	2	8	no	—
136	C₂×C₄⋊C₄	C₂²×C₁₇	1	2	yes	—
137	C₂×C₄⋊C₄	C₂²×C₁₇	1	2	yes	—
138	C₂×C₄⋊C₄	C₂²×C₁₇	1	2	yes	—
139	C₂×C₄⋊C₄	C₂²×C₁₇	1	2	yes	—
140	C₂×C₄⋊C₄	C₂²×C₁₇	1	2	yes	—
141	C₂×C₄⋊C₄	C₂²×C₁₇	1	1	yes	characteristic
142	C₂×C₄⋊C₄	C₂²×C₁₇	1	1	yes	characteristic
143	C₂×C₂²⋊C₄	C₂²×C₁₇	1	2	yes	—
144	C₂×C₂²⋊C₄	C₂²×C₁₇	1	2	yes	—
145	C₂×C₂²⋊C₄	C₂²×C₁₇	1	2	yes	—
146	C₂×C₂²⋊C₄	C₂²×C₁₇	1	2	yes	—
147	C₂×C₂²⋊C₄	C₂²×C₁₇	1	2	yes	—
148	C₂×C₄²	C₂²×C₁₇	1	1	yes	characteristic
149	#95	C₂²×C₁₇	1	2	yes	—
150	#95	C₂²×C₁₇	1	2	yes	—
151	#95	C₂²×C₁₇	1	2	yes	—
152	#95	C₂²×C₁₇	1	1	yes	characteristic
153	#95	C₂²×C₁₇	1	1	yes	characteristic
154	C₁₇	#2680	1	1	yes	characteristic, direct product factor, Sylow
155	C₂⁴	C₂³×C₁₇	1	2	yes	—
156	C₂×Q₈	—	2	4	no	—
157	C₂×Q₈	—	4	1	no	—
158	C₂×D₄	—	2	4	no	—
159	C₂×D₄	—	4	1	no	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
164	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
165	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
166	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
167	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
169	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
170	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	4	no	—
174	C₄⋊C₄	—	2	4	no	—
175	C₄⋊C₄	—	2	4	no	—
176	C₄⋊C₄	—	4	1	no	—
177	C₄⋊C₄	—	4	1	no	—
178	C₂²⋊C₄	—	2	4	no	—
179	C₂²⋊C₄	—	2	4	no	—
180	C₂²⋊C₄	—	2	4	no	—
181	C₂²⋊C₄	—	2	4	no	—
182	C₂²⋊C₄	D₄×C₁₇	1	8	yes	—
183	C₄²	—	2	2	no	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	2	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	2	no	—
189	C₂³	—	2	2	no	—
190	C₂³	C₂×D₄×C₁₇	1	4	yes	—
191	C₂³	C₂⁴×C₁₇	1	1	yes	characteristic
192	Q₈	—	4	4	no	—
193	D₄	—	4	4	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	4	8	no	—
214	C₂×C₄	C₂×D₄×C₁₇	1	4	yes	—
215	C₂×C₄	C₂×D₄×C₁₇	1	4	yes	—
216	C₂×C₄	#41×C₁₇	1	2	yes	—
217	C₂²	—	2	4	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	2	4	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	C₂²×D₄×C₁₇	1	2	yes	—
225	C₂²	C₂×#41×C₁₇	1	1	yes	characteristic
226	C₂²	C₁₇×H(5,𝔽₂)	1	1	yes	characteristic
227	C₂²	C₁₇×2₋¹⁺⁴	1	1	yes	characteristic
228	C₂²	C₁₇×2₋¹⁺⁴	1	2	yes	—
229	C₄	—	2	2	no	—
230	C₄	—	2	4	no	—
231	C₄	—	2	4	no	—
232	C₄	—	4	2	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	2	no	—
237	C₂	—	4	4	no	—
238	C₂	C₁₇×#536	1	2	yes	—
239	C₂	C₁₇×#539	1	1	yes	characteristic
240	C₂	D₄²×C₁₇	1	1	yes	characteristic
241	C₂	C₁₇×#547	1	2	yes	—
242	C₂	C₁₇×#563	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₇
prime radical	C₂²×D₄×C₁₇	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₇×C₂²⋊Q₈	C₂	semidirect
3. C₁₇×#398	C₂	semidirect
4. C₁₇×#397	C₂	semidirect
5. C₁₇×#395	C₂	semidirect
7. C₁₇×#388	C₂	semidirect
8. C₁₇×#385	C₂	semidirect
9. C₁₇×#384	C₂	semidirect
95. #2680	C₁₇	direct (N × H)
117. C₂×C₂²⋊Q₈	C₂×C₁₇	semidirect
118. #398	C₂×C₁₇	semidirect
119. #397	C₂×C₁₇	semidirect
120. #395	C₂×C₁₇	semidirect
122. #388	C₂×C₁₇	semidirect
123. #385	C₂×C₁₇	semidirect
124. #384	C₂×C₁₇	semidirect
154. C₁₇	#2680	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×#123	2
C₂×C₁₇×C₂²⋊Q₈	2
C₁₇×#398	2
C₁₇×#397	2
C₁₇×#395	2
C₁₇×#394	2
C₁₇×#388	2
C₁₇×#385	2
C₁₇×#384	2
#2680	17

Subgroups — C17×#2680 #14525686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#2680 #14525686