Subgroups — #14537441

Order 2176.

402 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#117×D₁₇	C₂	1	1	yes	maximal, characteristic
2	#11762869	C₂	1	2	yes	maximal, complemented
3	#11762869	C₂	1	2	yes	maximal, complemented
4	#11762865	C₂	1	2	yes	maximal, complemented
5	#11762861	C₂	1	2	yes	maximal, complemented
6	#11762836	C₂	1	1	yes	maximal, characteristic, complemented
7	#11762772	C₂	1	1	yes	maximal, characteristic
8	C₂×#10587491	C₂	1	1	yes	maximal, characteristic
9	#11762767	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762764	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₇×#424	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₇	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₁₇×#117	C₂²	1	1	yes	characteristic
16	#10587612	C₂²	1	1	yes	characteristic
17	C₂×#63223	C₂²	1	1	yes	characteristic
18	C₂²×#3622	C₂²	1	1	yes	characteristic
19	#10587578	C₄	1	1	yes	characteristic
20	#10587578	C₄	1	1	yes	characteristic
21	C₄⋊C₄×D₁₇	C₄	1	1	yes	characteristic
22	C₄⋊C₄×D₁₇	C₄	1	1	yes	characteristic
23	C₂²⋊C₄×D₁₇	C₄	1	2	yes	—
24	#10587563	C₄	1	2	yes	—
25	#10587558	C₂²	1	2	yes	—
26	C₄²×D₁₇	C₂²	1	2	yes	—
27	C₁₇×C₄⋊C₈	C₂²	1	2	yes	complemented
28	C₁₇×C₄⋊C₈	C₂²	1	2	yes	complemented
29	#10587491	C₂²	1	1	yes	characteristic
30	#10587491	C₂²	1	1	yes	characteristic
31	#10587491	C₂²	1	1	yes	characteristic
32	#10587491	C₂²	1	1	yes	characteristic
33	#10587491	C₂²	1	2	yes	—
34	#10587491	C₂²	1	2	yes	—
35	#10587486	C₂²	1	2	yes	complemented
36	#10587486	C₂²	1	2	yes	complemented
37	C₈×Dic₁₇	C₂²	1	2	yes	complemented
38	C₈×Dic₁₇	C₂²	1	2	yes	complemented
39	C₄⋊#3622	C₂²	1	2	yes	complemented
40	C₄⋊#3622	C₂²	1	2	yes	complemented
41	C₂³×D₁₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
43	C₂²×Dic₁₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₇	—	2	2	no	—
45	C₂×C₄×D₁₇	—	2	2	no	—
46	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₇	C₂×C₄	1	2	yes	—
49	C₂×C₄×D₁₇	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
51	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
52	M₁₆×C₁₇	—	2	2	no	—
53	C₂×C₈×C₁₇	—	2	2	no	—
54	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₁₇	C₂³	1	2	yes	—
57	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
58	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
59	C₂²⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
60	C₄²×C₁₇	C₂³	1	2	yes	—
61	C₂²⋊Dic₁₇	C₂×C₄	1	2	yes	—
62	#63227	C₂×C₄	1	2	yes	—
63	#63227	C₂×C₄	1	2	yes	—
64	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
65	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
66	#63225	C₂×C₄	1	2	yes	—
67	#63225	C₂×C₄	1	2	yes	—
68	C₄×Dic₁₇	C₂×C₄	1	2	yes	—
69	C₄×Dic₁₇	C₂×C₄	1	2	yes	—
70	C₄×Dic₁₇	C₂³	1	2	yes	—
71	#63223	—	2	2	no	—
72	C₂×#3622	—	2	2	no	—
73	C₂×#3622	C₂³	1	1	yes	characteristic
74	C₂×#3622	C₂³	1	2	yes	—
75	C₂×#3622	C₂³	1	1	yes	characteristic
76	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×D₁₇	—	2	2	no	—
78	C₂²×D₁₇	—	2	2	no	—
79	C₂²×D₁₇	—	4	1	no	—
80	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₇	#41	1	2	yes	—
86	C₂×C₄×C₁₇	#41	1	2	yes	—
87	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
88	C₂×Dic₁₇	—	2	2	no	—
89	C₂×Dic₁₇	C₂²×C₄	1	2	yes	—
90	C₂×Dic₁₇	C₂²×C₄	1	2	yes	—
91	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
93	C₄×D₁₇	—	2	4	no	—
94	C₄×D₁₇	—	2	2	no	—
95	C₄×D₁₇	—	2	2	no	—
96	C₄×D₁₇	—	2	1	no	—
97	C₄×D₁₇	—	2	1	no	—
98	C₄×D₁₇	—	2	1	no	—
99	C₄×D₁₇	—	2	1	no	—
100	C₄×D₁₇	—	4	2	no	—
101	C₈×C₁₇	—	2	1	no	—
102	C₈×C₁₇	—	2	1	no	—
103	C₈×C₁₇	—	2	2	no	—
104	#3622	—	2	1	no	—
105	#3622	—	2	1	no	—
106	#3622	—	2	2	no	—
107	#2924	—	17	1	no	maximal, Sylow
108	C₂²×C₁₇	—	2	1	no	—
109	C₂²×C₁₇	—	2	1	no	—
110	C₂²×C₁₇	#117	1	2	yes	—
111	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×D₁₇	—	2	1	no	—
113	C₂×D₁₇	—	2	1	no	—
114	C₂×D₁₇	—	2	1	no	—
115	C₂×D₁₇	—	2	1	no	—
116	C₂×D₁₇	—	2	1	no	—
117	C₂×D₁₇	—	4	4	no	—
118	C₂×D₁₇	#131	1	2	yes	—
119	C₄×C₁₇	—	2	2	no	—
120	C₄×C₁₇	—	2	2	no	—
121	C₄×C₁₇	#117	1	2	yes	—
122	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
123	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
124	Dic₁₇	—	2	1	no	—
125	Dic₁₇	—	2	1	no	—
126	Dic₁₇	—	2	2	no	—
127	Dic₁₇	—	2	2	no	—
128	C₂×#117	—	17	1	no	—
129	#433	—	17	2	no	—
130	#433	—	17	2	no	—
131	#432	—	17	2	no	—
132	#431	—	17	2	no	—
133	#424	—	17	1	no	—
134	#424	D₁₇	1	1	yes	characteristic, complemented
135	#407	—	17	1	no	—
136	C₂×C₂²⋊C₈	—	17	1	no	—
137	#405	—	17	1	no	—
138	#405	—	17	1	no	—
139	C₂×C₁₇	—	2	2	no	—
140	C₂×C₁₇	C₂×#117	1	1	yes	characteristic
141	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
142	C₂×C₁₇	#568	1	1	yes	characteristic
143	D₁₇	—	2	2	no	—
144	D₁₇	—	4	1	no	—
145	C₂³×C₄	—	17	1	no	—
146	C₂×M₁₆	—	17	1	no	—
147	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
148	C₂²×C₈	—	17	1	no	—
149	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
150	#117	—	17	1	no	—
151	#117	—	17	1	no	—
152	#117	—	17	1	no	—
153	#117	—	17	2	no	—
154	#117	—	17	2	no	—
155	#117	C₂×D₁₇	1	1	yes	characteristic
156	C₂×C₄⋊C₄	—	17	1	no	—
157	C₂×C₄⋊C₄	—	17	1	no	—
158	C₂×C₂²⋊C₄	—	17	2	no	—
159	C₂×C₄²	—	17	2	no	—
160	C₄⋊C₈	—	17	2	no	—
161	C₄⋊C₈	—	17	2	no	—
162	C₄⋊C₈	C₂×D₁₇	1	2	yes	complemented
163	C₄⋊C₈	C₂×D₁₇	1	2	yes	complemented
164	C₂²⋊C₈	—	17	1	no	—
165	C₂²⋊C₈	—	17	1	no	—
166	C₂²⋊C₈	—	17	1	no	—
167	C₂²⋊C₈	—	17	1	no	—
168	C₂²⋊C₈	—	17	2	no	—
169	C₂²⋊C₈	—	17	2	no	—
170	#97	—	17	2	no	—
171	#97	—	17	2	no	—
172	C₄×C₈	—	17	2	no	—
173	C₄×C₈	—	17	2	no	—
174	C₁₇	#2924	1	1	yes	characteristic, complemented, Sylow
175	C₂⁴	—	17	1	no	—
176	C₂²×C₄	—	17	1	no	—
177	C₂²×C₄	—	17	2	no	—
178	C₂²×C₄	—	17	1	no	—
179	C₂²×C₄	—	17	1	no	—
180	C₂²×C₄	—	17	1	no	—
181	C₂²×C₄	—	17	1	no	—
182	C₂²×C₄	—	17	2	no	—
183	C₂²×C₄	—	34	2	no	—
184	C₂²×C₄	—	34	2	no	—
185	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
186	M₁₆	—	2	2	no	—
187	M₁₆	—	34	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	17	1	no	—
190	C₂×C₈	—	17	2	no	—
191	C₂×C₈	—	17	1	no	—
192	C₂×C₈	—	34	2	no	—
193	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
194	C₂×C₈	C₂²×D₁₇	1	2	yes	—
195	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
196	C₄⋊C₄	—	17	2	no	—
197	C₄⋊C₄	—	17	2	no	—
198	C₄⋊C₄	—	17	1	no	—
199	C₄⋊C₄	—	17	1	no	—
200	C₄⋊C₄	C₄×D₁₇	1	1	yes	characteristic
201	C₄⋊C₄	C₄×D₁₇	1	1	yes	characteristic
202	C₂²⋊C₄	—	17	2	no	—
203	C₂²⋊C₄	—	17	2	no	—
204	C₂²⋊C₄	—	17	2	no	—
205	C₂²⋊C₄	C₄×D₁₇	1	2	yes	—
206	C₄²	—	17	2	no	—
207	C₄²	—	17	2	no	—
208	C₄²	—	17	2	no	—
209	C₄²	C₂²×D₁₇	1	2	yes	—
210	C₂³	—	17	1	no	—
211	C₂³	—	17	1	no	—
212	C₂³	—	34	2	no	—
213	C₂³	—	34	2	no	—
214	C₂³	—	68	1	no	—
215	C₂³	C₂×C₄×D₁₇	1	1	yes	characteristic
216	C₂×C₄	—	17	1	no	—
217	C₂×C₄	—	17	2	no	—
218	C₂×C₄	—	17	2	no	—
219	C₂×C₄	—	17	1	no	—
220	C₂×C₄	—	34	4	no	—
221	C₂×C₄	—	34	2	no	—
222	C₂×C₄	—	34	1	no	—
223	C₂×C₄	—	34	1	no	—
224	C₂×C₄	—	34	1	no	—
225	C₂×C₄	—	34	1	no	—
226	C₂×C₄	—	34	2	no	—
227	C₂×C₄	—	34	2	no	—
228	C₂×C₄	—	68	2	no	—
229	C₂×C₄	C₂×C₄×D₁₇	1	2	yes	—
230	C₂×C₄	C₂×C₄×D₁₇	1	2	yes	—
231	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
232	C₂×C₄	#63254	1	2	yes	—
233	C₂×C₄	#63256	1	2	yes	—
234	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	2	no	—
238	C₈	—	34	1	no	—
239	C₈	—	34	1	no	—
240	C₈	—	34	2	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	17	2	no	—
244	C₂²	—	34	1	no	—
245	C₂²	—	34	1	no	—
246	C₂²	—	34	1	no	—
247	C₂²	—	34	1	no	—
248	C₂²	—	34	1	no	—
249	C₂²	—	68	4	no	—
250	C₂²	#10587578	1	2	yes	—
251	C₂²	C₂²×C₄×D₁₇	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	34	1	no	—
255	C₄	—	34	1	no	—
256	C₄	—	34	2	no	—
257	C₄	—	34	2	no	—
258	C₄	#10587578	1	2	yes	—
259	C₄	C₂×#63254	1	1	yes	characteristic
260	C₄	C₂×#63256	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	34	2	no	—
263	C₂	—	68	1	no	—
264	C₂	C₂×#10587578	1	1	yes	characteristic
265	C₂	#131×D₁₇	1	1	yes	characteristic
266	C₂	#11763574	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₇
commutator, socle	C₂²×C₁₇	C₂³×C₄
Fitting	C₁₇×#424	C₂
prime radical	C₂³×D₁₇	C₂×C₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	#424	D₁₇
hypocenter	C₁₇	#2924
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
2. #11762869	C₂	semidirect
3. #11762869	C₂	semidirect
4. #11762865	C₂	semidirect
5. #11762861	C₂	semidirect
6. #11762836	C₂	semidirect
9. #11762767	C₂	semidirect
10. #11762764	C₂	semidirect
11. C₁₇×#424	C₂	semidirect
27. C₁₇×C₄⋊C₈	C₂²	semidirect
28. C₁₇×C₄⋊C₈	C₂²	semidirect
35. #10587486	C₂²	semidirect
36. #10587486	C₂²	semidirect
37. C₈×Dic₁₇	C₂²	semidirect
38. C₈×Dic₁₇	C₂²	semidirect
39. C₄⋊#3622	C₂²	semidirect
40. C₄⋊#3622	C₂²	semidirect
134. #424	D₁₇	semidirect
162. C₄⋊C₈	C₂×D₁₇	semidirect
163. C₄⋊C₈	C₂×D₁₇	semidirect
174. C₁₇	#2924	semidirect

Maximal subgroups

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

subgroup	index
#117×D₁₇	2
#11762869	2
#11762865	2
#11762861	2
#11762836	2
#11762772	2
C₂×#10587491	2
#11762767	2
#11762764	2
C₁₇×#424	2
#2924	17