Subgroups — C₁₇×#1876 #14524882

Order 2176.

324 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₇×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₇×#104	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₇×#104	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#408	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₇×#341	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#336	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×#41×C₁₇	C₂²	1	1	yes	characteristic
10	C₁₇×#135	—	2	2	no	—
11	C₁₇×#135	—	2	2	no	—
12	C₂×Q₁₆×C₁₇	C₄	1	1	yes	characteristic, complemented
13	C₂×SD₁₆×C₁₇	C₄	1	1	yes	characteristic, complemented
14	C₂×SD₁₆×C₁₇	C₄	1	1	yes	characteristic, complemented
15	C₂×D₈×C₁₇	C₄	1	1	yes	characteristic, complemented
16	C₁₇×#131	—	2	2	no	—
17	C₁₇×#131	—	2	2	no	—
18	C₂×M₁₆×C₁₇	—	2	1	no	—
19	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	—	2	1	no	—
22	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
23	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
25	C₁₇×#104	—	2	2	no	—
26	C₁₇×#104	—	2	2	no	—
27	C₁₇×C₂²⋊C₈	—	2	1	no	—
28	C₁₇×C₂²⋊C₈	—	2	1	no	—
29	#41×C₁₇	—	2	1	no	—
30	#41×C₁₇	—	2	1	no	—
31	#41×C₁₇	—	2	1	no	—
32	#41×C₁₇	—	2	1	no	—
33	#41×C₁₇	—	2	1	no	—
34	#41×C₁₇	—	2	1	no	—
35	#41×C₁₇	D₄	1	2	yes	—
36	#41×C₁₇	D₄	1	2	yes	—
37	C₂×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
38	C₂×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
39	C₂×D₄×C₁₇	—	2	1	no	—
40	C₂×D₄×C₁₇	—	2	1	no	—
41	C₂×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₇	—	2	1	no	—
44	C₂²×C₄×C₁₇	—	2	1	no	—
45	C₂²×C₄×C₁₇	—	2	1	no	—
46	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
47	Q₁₆×C₁₇	—	2	2	no	—
48	SD₁₆×C₁₇	—	2	2	no	—
49	SD₁₆×C₁₇	—	2	2	no	—
50	D₈×C₁₇	—	2	2	no	—
51	M₁₆×C₁₇	—	2	4	no	—
52	M₁₆×C₁₇	—	2	1	no	—
53	M₁₆×C₁₇	—	2	1	no	—
54	M₁₆×C₁₇	—	2	1	no	—
55	M₁₆×C₁₇	D₄	1	2	yes	—
56	C₂×C₈×C₁₇	—	2	4	no	—
57	C₂×C₈×C₁₇	—	2	1	no	—
58	C₂×C₈×C₁₇	—	2	1	no	—
59	C₂×C₈×C₁₇	—	2	1	no	—
60	C₂×C₈×C₁₇	—	2	1	no	—
61	C₂×C₈×C₁₇	—	2	1	no	—
62	C₂×C₈×C₁₇	—	2	1	no	—
63	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
66	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
67	C₄²×C₁₇	—	2	1	no	—
68	C₄²×C₁₇	—	2	1	no	—
69	C₂³×C₁₇	—	2	1	no	—
70	C₂³×C₁₇	—	2	1	no	—
71	C₂³×C₁₇	#41	1	1	yes	characteristic
72	Q₈×C₁₇	—	2	1	no	—
73	Q₈×C₁₇	—	2	1	no	—
74	Q₈×C₁₇	—	2	1	no	—
75	Q₈×C₁₇	C₂²⋊C₄	1	2	yes	—
76	D₄×C₁₇	—	2	2	no	—
77	D₄×C₁₇	—	2	2	no	—
78	D₄×C₁₇	—	2	1	no	—
79	D₄×C₁₇	—	2	1	no	—
80	D₄×C₁₇	—	2	1	no	—
81	D₄×C₁₇	—	4	1	no	—
82	D₄×C₁₇	—	4	1	no	—
83	D₄×C₁₇	C₂²⋊C₄	1	2	yes	—
84	C₂×C₄×C₁₇	—	2	1	no	—
85	C₂×C₄×C₁₇	—	2	2	no	—
86	C₂×C₄×C₁₇	—	2	2	no	—
87	C₂×C₄×C₁₇	—	2	2	no	—
88	C₂×C₄×C₁₇	—	2	1	no	—
89	C₂×C₄×C₁₇	—	4	1	no	—
90	C₂×C₄×C₁₇	—	4	1	no	—
91	C₂×C₄×C₁₇	—	4	1	no	—
92	C₂×C₄×C₁₇	—	4	1	no	—
93	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
99	C₈×C₁₇	—	2	1	no	—
100	C₈×C₁₇	—	2	1	no	—
101	C₈×C₁₇	—	2	1	no	—
102	C₈×C₁₇	—	2	2	no	—
103	C₈×C₁₇	—	2	1	no	—
104	C₈×C₁₇	—	4	1	no	—
105	#1876	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂²×C₁₇	—	2	1	no	—
107	C₂²×C₁₇	—	2	1	no	—
108	C₂²×C₁₇	—	2	2	no	—
109	C₂²×C₁₇	—	4	1	no	—
110	C₂²×C₁₇	—	4	1	no	—
111	C₂²×C₁₇	—	4	1	no	—
112	C₂²×C₁₇	—	4	1	no	—
113	C₂²×C₁₇	—	4	1	no	—
114	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
115	C₂²×C₁₇	#121	1	1	yes	characteristic
116	C₂²×C₁₇	#123	1	1	yes	characteristic
117	C₄×C₁₇	—	2	2	no	—
118	C₄×C₁₇	—	4	2	no	—
119	C₄×C₁₇	—	4	1	no	—
120	C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
122	C₄×C₁₇	#120	1	1	yes	characteristic
123	C₄×C₁₇	#121	1	1	yes	characteristic
124	C₂×#135	C₂×C₁₇	1	1	yes	characteristic
125	C₂×#131	C₂×C₁₇	1	1	yes	characteristic, complemented
126	C₂×#104	C₂×C₁₇	1	1	yes	characteristic, complemented
127	C₂×#104	C₂×C₁₇	1	1	yes	characteristic, complemented
128	#408	C₂×C₁₇	1	1	yes	characteristic, complemented
129	#341	C₂×C₁₇	1	1	yes	characteristic, complemented
130	#336	C₂×C₁₇	1	1	yes	characteristic, complemented
131	C₂×C₁₇	—	2	1	no	—
132	C₂×C₁₇	—	2	1	no	—
133	C₂×C₁₇	—	4	2	no	—
134	C₂×C₁₇	—	8	1	no	—
135	C₂×C₁₇	#386	1	1	yes	characteristic
136	C₂×C₁₇	#443	1	1	yes	characteristic
137	C₂×C₁₇	#444	1	1	yes	characteristic
138	C₂×#41	C₂²×C₁₇	1	1	yes	characteristic
139	C₂×#41	C₂²×C₁₇	1	1	yes	characteristic
140	#135	—	2	2	no	—
141	#135	—	2	2	no	—
142	C₂×Q₁₆	C₄×C₁₇	1	1	yes	characteristic, complemented
143	C₂×SD₁₆	C₄×C₁₇	1	1	yes	characteristic, complemented
144	C₂×SD₁₆	C₄×C₁₇	1	1	yes	characteristic, complemented
145	C₂×D₈	C₄×C₁₇	1	1	yes	characteristic, complemented
146	#131	—	2	2	no	—
147	#131	—	2	2	no	—
148	C₂×M₁₆	—	2	1	no	—
149	C₂×M₁₆	C₂²×C₁₇	1	1	yes	characteristic
150	C₂×M₁₆	C₂²×C₁₇	1	1	yes	characteristic
151	C₂²×C₈	—	2	1	no	—
152	C₂²×C₈	C₂²×C₁₇	1	1	yes	characteristic
153	C₂²×C₈	C₂²×C₁₇	1	1	yes	characteristic
154	C₂×C₄²	C₂²×C₁₇	1	1	yes	characteristic
155	#104	—	2	2	no	—
156	#104	—	2	2	no	—
157	C₂²⋊C₈	—	2	1	no	—
158	C₂²⋊C₈	—	2	1	no	—
159	C₁₇	#1876	1	1	yes	characteristic, direct product factor, Sylow
160	#41	—	2	1	no	—
161	#41	—	2	1	no	—
162	#41	—	2	1	no	—
163	#41	—	2	1	no	—
164	#41	—	2	1	no	—
165	#41	—	2	1	no	—
166	#41	D₄×C₁₇	1	2	yes	—
167	#41	D₄×C₁₇	1	2	yes	—
168	C₂×Q₈	C₂×C₄×C₁₇	1	1	yes	characteristic
169	C₂×Q₈	C₂×C₄×C₁₇	1	1	yes	characteristic
170	C₂×D₄	—	2	1	no	—
171	C₂×D₄	—	2	1	no	—
172	C₂×D₄	C₂×C₄×C₁₇	1	1	yes	characteristic
173	C₂×D₄	C₂×C₄×C₁₇	1	1	yes	characteristic
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
178	Q₁₆	—	2	2	no	—
179	SD₁₆	—	2	2	no	—
180	SD₁₆	—	2	2	no	—
181	D₈	—	2	2	no	—
182	M₁₆	—	2	4	no	—
183	M₁₆	—	2	1	no	—
184	M₁₆	—	2	1	no	—
185	M₁₆	—	2	1	no	—
186	M₁₆	D₄×C₁₇	1	2	yes	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
195	C₂×C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
196	C₂×C₈	D₄×C₁₇	1	1	yes	characteristic
197	C₂×C₈	D₄×C₁₇	1	1	yes	characteristic
198	C₄²	—	2	1	no	—
199	C₄²	—	2	1	no	—
200	C₂³	—	2	1	no	—
201	C₂³	—	2	1	no	—
202	C₂³	#41×C₁₇	1	1	yes	characteristic
203	Q₈	—	2	1	no	—
204	Q₈	—	2	1	no	—
205	Q₈	—	2	1	no	—
206	Q₈	C₂²⋊C₄×C₁₇	1	2	yes	—
207	D₄	—	2	2	no	—
208	D₄	—	2	2	no	—
209	D₄	—	2	1	no	—
210	D₄	—	2	1	no	—
211	D₄	—	2	1	no	—
212	D₄	—	4	1	no	—
213	D₄	—	4	1	no	—
214	D₄	C₂²⋊C₄×C₁₇	1	2	yes	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
225	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
226	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
227	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
228	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
229	C₂×C₄	#41×C₁₇	1	1	yes	characteristic
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	2	1	no	—
233	C₈	—	2	2	no	—
234	C₈	—	2	1	no	—
235	C₈	—	4	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	4	1	no	—
240	C₂²	—	4	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	C₄×D₄×C₁₇	1	1	yes	characteristic
245	C₂²	C₁₇×#121	1	1	yes	characteristic
246	C₂²	C₁₇×#123	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	1	no	—
250	C₄	C₂×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
251	C₄	C₄×D₄×C₁₇	1	1	yes	characteristic
252	C₄	C₁₇×#120	1	1	yes	characteristic
253	C₄	C₁₇×#121	1	1	yes	characteristic
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	4	2	no	—
257	C₂	—	8	1	no	—
258	C₂	C₁₇×#386	1	1	yes	characteristic
259	C₂	C₁₇×#443	1	1	yes	characteristic
260	C₂	C₁₇×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₁₇
Frattini, cosocle	C₂²×C₄	C₂³×C₁₇
socle	C₂²×C₁₇	C₄×D₄
prime radical	C₂×C₁₇×#135	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₇ ◁ C₂²×C₈×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₇×#131	C₂	semidirect
3. C₂×C₁₇×#104	C₂	semidirect
4. C₂×C₁₇×#104	C₂	semidirect
5. C₁₇×#408	C₂	semidirect
6. C₁₇×#341	C₂	semidirect
7. C₁₇×#336	C₂	semidirect
12. C₂×Q₁₆×C₁₇	C₄	semidirect
13. C₂×SD₁₆×C₁₇	C₄	semidirect
14. C₂×SD₁₆×C₁₇	C₄	semidirect
15. C₂×D₈×C₁₇	C₄	semidirect
105. #1876	C₁₇	direct (N × H)
125. C₂×#131	C₂×C₁₇	semidirect
126. C₂×#104	C₂×C₁₇	semidirect
127. C₂×#104	C₂×C₁₇	semidirect
128. #408	C₂×C₁₇	semidirect
129. #341	C₂×C₁₇	semidirect
130. #336	C₂×C₁₇	semidirect
142. C₂×Q₁₆	C₄×C₁₇	semidirect
143. C₂×SD₁₆	C₄×C₁₇	semidirect
144. C₂×SD₁₆	C₄×C₁₇	semidirect
145. C₂×D₈	C₄×C₁₇	semidirect
159. C₁₇	#1876	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×#135	2
C₂×C₁₇×#131	2
C₂×C₁₇×#104	2
C₁₇×#408	2
C₁₇×#341	2
C₁₇×#336	2
#1876	17

Subgroups — C17×#1876 #14524882

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#1876 #14524882