Subgroups — C₁₇×#3512 #14526518

Order 2176.

1032 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#553	C₂	1	4	yes	maximal, complemented
2	C₁₇×#553	C₂	1	8	yes	maximal, complemented
3	C₁₇×#552	C₂	1	4	yes	maximal, complemented
4	C₁₇×#541	C₂	1	2	yes	maximal, complemented
5	C₁₇×#540	C₂	1	2	yes	maximal, complemented
6	C₁₇×#537	C₂	1	2	yes	maximal, complemented
7	C₁₇×#532	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₇×#529	C₂	1	4	yes	maximal, complemented
9	C₂×C₁₇×#125	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₇×#520	C₂	1	2	yes	maximal, complemented
11	C₄×#41×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₁₇	C₂²	1	1	yes	characteristic
13	C₂×#41×C₁₇	C₂²	1	2	yes	—
14	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₁₇×#127	C₂²	1	1	yes	characteristic
16	C₁₇×#126	C₂²	1	8	yes	—
17	C₁₇×#126	C₂²	1	8	yes	—
18	C₁₇×#125	C₂²	1	2	yes	—
19	C₁₇×#125	C₂²	1	2	yes	—
20	C₁₇×#125	C₂²	1	4	yes	—
21	C₁₇×#125	C₂²	1	8	yes	—
22	C₁₇×#124	C₂²	1	2	yes	—
23	C₁₇×#123	C₂²	1	8	yes	—
24	C₁₇×#123	C₂²	1	8	yes	—
25	C₁₇×C₂²⋊Q₈	C₂²	1	4	yes	—
26	C₁₇×C₂²⋊Q₈	C₂²	1	4	yes	—
27	C₁₇×#121	C₂²	1	4	yes	—
28	C₁₇×#121	C₂²	1	4	yes	—
29	C₁₇×#121	C₂²	1	8	yes	—
30	C₁₇×#121	C₂²	1	4	yes	—
31	C₁₇×#121	C₂²	1	8	yes	—
32	C₄×Q₈×C₁₇	C₂²	1	2	yes	—
33	C₄×Q₈×C₁₇	C₂²	1	4	yes	—
34	C₄×D₄×C₁₇	C₂²	1	2	yes	—
35	C₄×D₄×C₁₇	C₂²	1	4	yes	—
36	C₄×D₄×C₁₇	C₂²	1	4	yes	—
37	C₄×D₄×C₁₇	C₂²	1	8	yes	—
38	C₄×D₄×C₁₇	C₂²	1	8	yes	—
39	C₄×D₄×C₁₇	C₂²	1	4	yes	—
40	C₁₇×#117	C₂²	1	1	yes	characteristic
41	C₁₇×#117	C₂²	1	2	yes	—
42	C₁₇×#117	C₂²	1	4	yes	—
43	C₁₇×#117	C₂²	1	8	yes	—
44	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
45	C₂×C₄⋊C₄×C₁₇	C₂²	1	4	yes	—
46	C₂×C₄⋊C₄×C₁₇	C₂²	1	4	yes	—
47	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
48	C₂×C₄²×C₁₇	C₂²	1	2	yes	—
49	#41×C₁₇	—	2	4	no	—
50	#41×C₁₇	—	2	8	no	—
51	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
52	C₂×Q₈×C₁₇	C₂³	1	2	yes	—
53	C₂×D₄×C₁₇	C₂³	1	2	yes	—
54	C₂×D₄×C₁₇	C₂³	1	2	yes	—
55	C₂×D₄×C₁₇	C₂³	1	4	yes	—
56	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
57	C₂×D₄×C₁₇	C₂³	1	4	yes	—
58	C₂×D₄×C₁₇	C₂³	1	8	yes	—
59	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
60	C₂²×C₄×C₁₇	C₂³	1	4	yes	—
61	C₂²×C₄×C₁₇	C₂³	1	4	yes	—
62	C₂²×C₄×C₁₇	C₂³	1	8	yes	—
63	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
64	C₂²×C₄×C₁₇	C₂³	1	4	yes	—
65	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₁₇	C₂³	1	8	yes	—
67	C₄⋊C₄×C₁₇	C₂³	1	4	yes	—
68	C₄⋊C₄×C₁₇	C₂³	1	4	yes	—
69	C₄⋊C₄×C₁₇	C₂³	1	4	yes	—
70	C₄⋊C₄×C₁₇	C₂³	1	8	yes	—
71	C₄⋊C₄×C₁₇	C₂³	1	8	yes	—
72	C₄⋊C₄×C₁₇	C₂³	1	8	yes	—
73	C₄⋊C₄×C₁₇	C₂³	1	4	yes	—
74	C₄⋊C₄×C₁₇	C₂³	1	2	yes	—
75	C₂²⋊C₄×C₁₇	C₂³	1	8	yes	—
76	C₂²⋊C₄×C₁₇	C₂³	1	8	yes	—
77	C₂²⋊C₄×C₁₇	C₂³	1	4	yes	—
78	C₂²⋊C₄×C₁₇	C₂³	1	8	yes	—
79	C₂²⋊C₄×C₁₇	C₂³	1	4	yes	—
80	C₂²⋊C₄×C₁₇	C₂³	1	4	yes	—
81	C₂²⋊C₄×C₁₇	C₂³	1	2	yes	—
82	C₄²×C₁₇	C₂³	1	1	yes	characteristic
83	C₄²×C₁₇	C₂³	1	1	yes	characteristic
84	C₄²×C₁₇	C₂³	1	2	yes	—
85	C₄²×C₁₇	C₂³	1	8	yes	—
86	C₄²×C₁₇	C₂³	1	4	yes	—
87	C₄²×C₁₇	C₂³	1	2	yes	—
88	C₂³×C₁₇	C₂⁴	1	4	yes	—
89	C₂³×C₁₇	C₂⁴	1	2	yes	—
90	C₂³×C₁₇	C₂⁴	1	1	yes	characteristic
91	Q₈×C₁₇	—	2	2	no	—
92	Q₈×C₁₇	—	2	4	no	—
93	D₄×C₁₇	—	2	4	no	—
94	D₄×C₁₇	—	2	8	no	—
95	D₄×C₁₇	—	2	2	no	—
96	D₄×C₁₇	—	2	4	no	—
97	D₄×C₁₇	—	4	8	no	—
98	D₄×C₁₇	—	4	4	no	—
99	C₂×C₄×C₁₇	—	2	8	no	—
100	C₂×C₄×C₁₇	—	2	4	no	—
101	C₂×C₄×C₁₇	—	4	4	no	—
102	C₂×C₄×C₁₇	—	4	8	no	—
103	C₂×C₄×C₁₇	—	4	2	no	—
104	C₂×C₄×C₁₇	—	4	4	no	—
105	C₂×C₄×C₁₇	#41	1	4	yes	—
106	C₂×C₄×C₁₇	C₂⁴	1	2	yes	—
107	C₂×C₄×C₁₇	C₂⁴	1	4	yes	—
108	C₂×C₄×C₁₇	C₂⁴	1	4	yes	—
109	C₂×C₄×C₁₇	C₂⁴	1	2	yes	—
110	C₂×C₄×C₁₇	C₂⁴	1	2	yes	—
111	C₂×C₄×C₁₇	C₂⁴	1	8	yes	—
112	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
113	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
114	#3512	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂²×C₁₇	—	2	2	no	—
116	C₂²×C₁₇	—	2	4	no	—
117	C₂²×C₁₇	—	2	2	no	—
118	C₂²×C₁₇	—	2	4	no	—
119	C₂²×C₁₇	—	2	4	no	—
120	C₂²×C₁₇	—	2	4	no	—
121	C₂²×C₁₇	C₂×#41	1	2	yes	—
122	C₂²×C₁₇	C₂⁵	1	1	yes	characteristic
123	C₄×C₁₇	—	2	4	no	—
124	C₄×C₁₇	—	2	2	no	—
125	C₄×C₁₇	—	2	2	no	—
126	C₄×C₁₇	—	2	4	no	—
127	C₄×C₁₇	—	2	8	no	—
128	C₄×C₁₇	—	2	2	no	—
129	C₄×C₁₇	C₂×#41	1	2	yes	—
130	C₄×C₁₇	C₂×#41	1	2	yes	—
131	#553	C₂×C₁₇	1	4	yes	complemented
132	#553	C₂×C₁₇	1	8	yes	complemented
133	#552	C₂×C₁₇	1	4	yes	complemented
134	#541	C₂×C₁₇	1	2	yes	complemented
135	#540	C₂×C₁₇	1	2	yes	complemented
136	#537	C₂×C₁₇	1	2	yes	complemented
137	#532	C₂×C₁₇	1	1	yes	characteristic, complemented
138	#529	C₂×C₁₇	1	4	yes	complemented
139	C₂×#125	C₂×C₁₇	1	1	yes	characteristic, complemented
140	#520	C₂×C₁₇	1	2	yes	complemented
141	C₄×#41	C₂×C₁₇	1	1	yes	characteristic, complemented
142	C₂×C₁₇	—	2	2	no	—
143	C₂×C₁₇	—	4	4	no	—
144	C₂×C₁₇	—	4	2	no	—
145	C₂×C₁₇	C₂²×#41	1	1	yes	characteristic
146	C₂×C₁₇	#585	1	2	yes	—
147	C₂×#41	C₂²×C₁₇	1	1	yes	characteristic
148	C₂×#41	C₂²×C₁₇	1	2	yes	—
149	C₄⋊Q₈	C₂²×C₁₇	1	1	yes	characteristic
150	#127	C₂²×C₁₇	1	1	yes	characteristic
151	#126	C₂²×C₁₇	1	8	yes	—
152	#126	C₂²×C₁₇	1	8	yes	—
153	#125	C₂²×C₁₇	1	2	yes	—
154	#125	C₂²×C₁₇	1	2	yes	—
155	#125	C₂²×C₁₇	1	4	yes	—
156	#125	C₂²×C₁₇	1	8	yes	—
157	#124	C₂²×C₁₇	1	2	yes	—
158	#123	C₂²×C₁₇	1	8	yes	—
159	#123	C₂²×C₁₇	1	8	yes	—
160	C₂²⋊Q₈	C₂²×C₁₇	1	4	yes	—
161	C₂²⋊Q₈	C₂²×C₁₇	1	4	yes	—
162	#121	C₂²×C₁₇	1	4	yes	—
163	#121	C₂²×C₁₇	1	4	yes	—
164	#121	C₂²×C₁₇	1	8	yes	—
165	#121	C₂²×C₁₇	1	4	yes	—
166	#121	C₂²×C₁₇	1	8	yes	—
167	C₄×Q₈	C₂²×C₁₇	1	2	yes	—
168	C₄×Q₈	C₂²×C₁₇	1	4	yes	—
169	C₄×D₄	C₂²×C₁₇	1	4	yes	—
170	C₄×D₄	C₂²×C₁₇	1	2	yes	—
171	C₄×D₄	C₂²×C₁₇	1	4	yes	—
172	C₄×D₄	C₂²×C₁₇	1	8	yes	—
173	C₄×D₄	C₂²×C₁₇	1	8	yes	—
174	C₄×D₄	C₂²×C₁₇	1	4	yes	—
175	#117	C₂²×C₁₇	1	1	yes	characteristic
176	#117	C₂²×C₁₇	1	2	yes	—
177	#117	C₂²×C₁₇	1	4	yes	—
178	#117	C₂²×C₁₇	1	8	yes	—
179	C₂×C₄⋊C₄	C₂²×C₁₇	1	2	yes	—
180	C₂×C₄⋊C₄	C₂²×C₁₇	1	4	yes	—
181	C₂×C₄⋊C₄	C₂²×C₁₇	1	4	yes	—
182	C₂×C₄²	C₂²×C₁₇	1	1	yes	characteristic
183	C₂×C₄²	C₂²×C₁₇	1	2	yes	—
184	C₁₇	#3512	1	1	yes	characteristic, direct product factor, Sylow
185	#41	—	2	4	no	—
186	#41	—	2	8	no	—
187	C₂×Q₈	C₂³×C₁₇	1	1	yes	characteristic
188	C₂×Q₈	C₂³×C₁₇	1	2	yes	—
189	C₂×D₄	C₂³×C₁₇	1	2	yes	—
190	C₂×D₄	C₂³×C₁₇	1	4	yes	—
191	C₂×D₄	C₂³×C₁₇	1	1	yes	characteristic
192	C₂×D₄	C₂³×C₁₇	1	2	yes	—
193	C₂×D₄	C₂³×C₁₇	1	8	yes	—
194	C₂×D₄	C₂³×C₁₇	1	4	yes	—
195	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
196	C₂²×C₄	C₂³×C₁₇	1	4	yes	—
197	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
198	C₂²×C₄	C₂³×C₁₇	1	4	yes	—
199	C₂²×C₄	C₂³×C₁₇	1	4	yes	—
200	C₂²×C₄	C₂³×C₁₇	1	8	yes	—
201	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
202	C₄⋊C₄	C₂³×C₁₇	1	2	yes	—
203	C₄⋊C₄	C₂³×C₁₇	1	4	yes	—
204	C₄⋊C₄	C₂³×C₁₇	1	8	yes	—
205	C₄⋊C₄	C₂³×C₁₇	1	4	yes	—
206	C₄⋊C₄	C₂³×C₁₇	1	8	yes	—
207	C₄⋊C₄	C₂³×C₁₇	1	8	yes	—
208	C₄⋊C₄	C₂³×C₁₇	1	4	yes	—
209	C₄⋊C₄	C₂³×C₁₇	1	8	yes	—
210	C₄⋊C₄	C₂³×C₁₇	1	4	yes	—
211	C₂²⋊C₄	C₂³×C₁₇	1	4	yes	—
212	C₂²⋊C₄	C₂³×C₁₇	1	2	yes	—
213	C₂²⋊C₄	C₂³×C₁₇	1	4	yes	—
214	C₂²⋊C₄	C₂³×C₁₇	1	8	yes	—
215	C₂²⋊C₄	C₂³×C₁₇	1	8	yes	—
216	C₂²⋊C₄	C₂³×C₁₇	1	4	yes	—
217	C₂²⋊C₄	C₂³×C₁₇	1	8	yes	—
218	C₄²	C₂³×C₁₇	1	1	yes	characteristic
219	C₄²	C₂³×C₁₇	1	2	yes	—
220	C₄²	C₂³×C₁₇	1	4	yes	—
221	C₄²	C₂³×C₁₇	1	1	yes	characteristic
222	C₄²	C₂³×C₁₇	1	2	yes	—
223	C₄²	C₂³×C₁₇	1	8	yes	—
224	C₂³	C₂⁴×C₁₇	1	4	yes	—
225	C₂³	C₂⁴×C₁₇	1	2	yes	—
226	C₂³	C₂⁴×C₁₇	1	1	yes	characteristic
227	Q₈	—	2	2	no	—
228	Q₈	—	2	4	no	—
229	D₄	—	2	4	no	—
230	D₄	—	2	8	no	—
231	D₄	—	2	2	no	—
232	D₄	—	2	4	no	—
233	D₄	—	4	8	no	—
234	D₄	—	4	4	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	8	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	#41×C₁₇	1	4	yes	—
242	C₂×C₄	C₂⁴×C₁₇	1	2	yes	—
243	C₂×C₄	C₂⁴×C₁₇	1	4	yes	—
244	C₂×C₄	C₂⁴×C₁₇	1	2	yes	—
245	C₂×C₄	C₂⁴×C₁₇	1	4	yes	—
246	C₂×C₄	C₂⁴×C₁₇	1	2	yes	—
247	C₂×C₄	C₂⁴×C₁₇	1	8	yes	—
248	C₂×C₄	C₂⁴×C₁₇	1	1	yes	characteristic
249	C₂×C₄	C₂⁴×C₁₇	1	1	yes	characteristic
250	C₂²	—	2	2	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	2	4	no	—
256	C₂²	C₂×#41×C₁₇	1	2	yes	—
257	C₂²	C₂⁵×C₁₇	1	1	yes	characteristic
258	C₄	—	2	4	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	2	8	no	—
263	C₄	—	2	2	no	—
264	C₄	C₂×#41×C₁₇	1	2	yes	—
265	C₄	C₂×#41×C₁₇	1	2	yes	—
266	C₂	—	2	2	no	—
267	C₂	—	4	4	no	—
268	C₂	—	4	2	no	—
269	C₂	C₂²×#41×C₁₇	1	1	yes	characteristic
270	C₂	C₁₇×#585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₇	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₇×#553	C₂	semidirect
2. C₁₇×#553	C₂	semidirect
3. C₁₇×#552	C₂	semidirect
4. C₁₇×#541	C₂	semidirect
5. C₁₇×#540	C₂	semidirect
6. C₁₇×#537	C₂	semidirect
7. C₁₇×#532	C₂	semidirect
8. C₁₇×#529	C₂	semidirect
9. C₂×C₁₇×#125	C₂	semidirect
10. C₁₇×#520	C₂	semidirect
11. C₄×#41×C₁₇	C₂	semidirect
114. #3512	C₁₇	direct (N × H)
131. #553	C₂×C₁₇	semidirect
132. #553	C₂×C₁₇	semidirect
133. #552	C₂×C₁₇	semidirect
134. #541	C₂×C₁₇	semidirect
135. #540	C₂×C₁₇	semidirect
136. #537	C₂×C₁₇	semidirect
137. #532	C₂×C₁₇	semidirect
138. #529	C₂×C₁₇	semidirect
139. C₂×#125	C₂×C₁₇	semidirect
140. #520	C₂×C₁₇	semidirect
141. C₄×#41	C₂×C₁₇	semidirect
184. C₁₇	#3512	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₇×#553	2
C₁₇×#552	2
C₁₇×#541	2
C₁₇×#540	2
C₁₇×#537	2
C₁₇×#532	2
C₁₇×#529	2
C₂×C₁₇×#125	2
C₁₇×#520	2
C₄×#41×C₁₇	2
#3512	17

Subgroups — C17×#3512 #14526518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#3512 #14526518