Subgroups — #14540110

Order 2176.

398 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763359	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763106	C₂	1	2	yes	maximal, complemented
3	#11763104	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763099	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763043	C₂	1	2	yes	maximal, complemented
6	#11763040	C₂	1	2	yes	maximal, complemented
7	#11763021	C₂	1	2	yes	maximal, complemented
8	#11763009	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763004	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊#3622	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₇×#529	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#63252	—	2	1	no	—
13	C₁₇×#126	—	2	1	no	—
14	C₁₇×#125	C₂²	1	1	yes	characteristic
15	C₁₇×#124	C₂²	1	1	yes	characteristic
16	C₁₇×#123	—	2	1	no	—
17	C₁₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
18	C₁₇×#121	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₁₇	—	2	1	no	—
21	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
22	C₁₇×#117	—	2	1	no	—
23	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
24	C₂×#63230	C₂²	1	2	yes	—
25	C₂×#63228	C₂²	1	2	yes	—
26	#10587616	—	2	1	no	—
27	#10587616	C₂²	1	2	yes	—
28	C₂²×#3622	C₂²	1	2	yes	—
29	#10587561	—	2	1	no	—
30	#10587560	C₂²	1	1	yes	characteristic
31	C₄×D₆₈	—	2	1	no	—
32	C₄×D₆₈	C₂²	1	2	yes	—
33	#10587555	C₂²	1	1	yes	characteristic
34	#10587506	C₂²	1	2	yes	—
35	#10587503	C₂²	1	2	yes	—
36	#10587480	C₂²	1	2	yes	—
37	#10587480	C₂²	1	2	yes	—
38	#10587479	C₂²	1	2	yes	—
39	#10587478	C₂²	1	2	yes	—
40	C₄⋊#3622	C₂²	1	1	yes	characteristic
41	C₄⋊#3622	C₂²	1	1	yes	characteristic
42	C₄⋊#3622	C₂²	1	2	yes	—
43	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₁₇	—	2	1	no	—
45	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	—	2	1	no	—
47	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
50	C₂×C₂²⋊D₁₇	—	2	2	no	—
51	#63252	—	4	4	no	—
52	C₂×D₆₈	—	2	1	no	—
53	C₂×D₆₈	—	2	1	no	—
54	C₂×D₆₈	C₂³	1	2	yes	—
55	C₂×C₄×D₁₇	—	2	2	no	—
56	C₂×Dic₃₄	—	2	1	no	—
57	C₄⋊C₄×C₁₇	—	2	1	no	—
58	C₄⋊C₄×C₁₇	—	2	1	no	—
59	C₄⋊C₄×C₁₇	—	2	1	no	—
60	C₄⋊C₄×C₁₇	—	2	1	no	—
61	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
63	C₂²⋊C₄×C₁₇	—	2	1	no	—
64	C₂²⋊C₄×C₁₇	—	2	1	no	—
65	C₂²⋊C₄×C₁₇	—	2	1	no	—
66	C₂²⋊C₄×C₁₇	—	2	1	no	—
67	C₂²⋊C₄×C₁₇	—	2	1	no	—
68	C₄²×C₁₇	—	2	1	no	—
69	C₄²×C₁₇	D₄	1	1	yes	characteristic
70	C₄²×C₁₇	D₄	1	1	yes	characteristic
71	C₄²×C₁₇	C₂³	1	1	yes	characteristic
72	C₄²×C₁₇	C₂³	1	1	yes	characteristic
73	#63230	—	4	2	no	—
74	#63228	—	4	2	no	—
75	#63227	—	2	2	no	—
76	#63227	—	2	2	no	—
77	C₄⋊Dic₁₇	—	2	1	no	—
78	C₄⋊Dic₁₇	C₂³	1	2	yes	—
79	C₂×#3622	—	4	2	no	—
80	C₂×#3622	C₂³	1	2	yes	—
81	C₂×#3622	C₂³	1	2	yes	—
82	C₂³×C₁₇	—	2	1	no	—
83	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×D₁₇	—	2	2	no	—
85	Q₈×C₁₇	—	2	1	no	—
86	Q₈×C₁₇	—	2	1	no	—
87	D₄×C₁₇	—	2	1	no	—
88	D₄×C₁₇	—	2	1	no	—
89	D₄×C₁₇	—	4	2	no	—
90	C₂×C₄×C₁₇	—	2	2	no	—
91	C₂×C₄×C₁₇	—	2	2	no	—
92	C₂×C₄×C₁₇	—	2	1	no	—
93	C₂×C₄×C₁₇	—	2	1	no	—
94	C₂×C₄×C₁₇	—	2	1	no	—
95	C₂×C₄×C₁₇	—	2	1	no	—
96	C₂×C₄×C₁₇	—	2	1	no	—
97	C₂×C₄×C₁₇	—	4	2	no	—
98	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
104	C₂²⋊D₁₇	—	4	4	no	—
105	C₂×Dic₁₇	—	2	2	no	—
106	D₆₈	—	2	2	no	—
107	D₆₈	—	2	2	no	—
108	D₆₈	—	4	2	no	—
109	D₆₈	—	4	2	no	—
110	C₄×D₁₇	—	4	4	no	—
111	Dic₃₄	—	4	2	no	—
112	#3622	—	2	2	no	—
113	#3622	—	2	2	no	—
114	#3247	—	17	1	no	maximal, Sylow
115	C₂²×C₁₇	—	2	1	no	—
116	C₂²×C₁₇	—	2	1	no	—
117	C₂²×C₁₇	—	4	1	no	—
118	C₂²×C₁₇	—	4	1	no	—
119	C₂²×C₁₇	—	4	1	no	—
120	C₂²×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
121	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×D₁₇	—	4	2	no	—
123	C₂×D₁₇	—	4	2	no	—
124	C₂×D₁₇	—	4	2	no	—
125	C₄×C₁₇	—	2	1	no	—
126	C₄×C₁₇	—	2	1	no	—
127	C₄×C₁₇	—	2	1	no	—
128	C₄×C₁₇	—	2	1	no	—
129	C₄×C₁₇	—	4	1	no	—
130	C₄×C₁₇	—	4	1	no	—
131	C₄×C₁₇	—	4	1	no	—
132	C₄×C₁₇	#135	1	2	yes	—
133	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
134	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
135	Dic₁₇	—	4	2	no	—
136	#532	—	17	1	no	—
137	#529	D₁₇	1	1	yes	characteristic, complemented
138	#478	—	17	2	no	—
139	#477	—	17	1	no	—
140	#476	—	17	1	no	—
141	#466	—	17	2	no	—
142	#465	—	17	2	no	—
143	#463	—	17	2	no	—
144	#460	—	17	1	no	—
145	#459	—	17	1	no	—
146	C₂×C₄⋊C₈	—	17	1	no	—
147	C₂×C₁₇	—	2	2	no	—
148	C₂×C₁₇	—	8	1	no	—
149	C₂×C₁₇	#537	1	1	yes	characteristic
150	C₂×C₁₇	C₂×#135	1	1	yes	characteristic
151	C₂×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
152	D₁₇	—	8	2	no	—
153	C₂×#41	—	34	1	no	—
154	C₂×SD₁₆	—	17	2	no	—
155	C₂×D₈	—	17	2	no	—
156	C₂²×C₈	—	17	2	no	—
157	C₄⋊Q₈	—	17	1	no	—
158	#127	—	17	1	no	—
159	#126	—	2	1	no	—
160	#125	C₂×D₁₇	1	1	yes	characteristic
161	#124	—	34	1	no	—
162	#124	C₂×D₁₇	1	1	yes	characteristic
163	#123	—	2	1	no	—
164	C₂²⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
165	#121	—	17	2	no	—
166	#121	—	34	1	no	—
167	#121	C₂×D₁₇	1	1	yes	characteristic
168	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	—	17	2	no	—
171	C₄×D₄	—	34	1	no	—
172	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
173	#117	—	2	1	no	—
174	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
175	#107	—	17	2	no	—
176	#106	—	17	2	no	—
177	C₄⋊C₈	—	17	1	no	—
178	C₄⋊C₈	—	17	1	no	—
179	C₄⋊C₈	—	17	2	no	—
180	#103	—	17	2	no	—
181	#102	—	17	2	no	—
182	#102	—	17	2	no	—
183	#102	—	17	2	no	—
184	C₁₇	#3247	1	1	yes	characteristic, complemented, Sylow
185	#41	—	68	4	no	—
186	C₂×Q₈	—	34	1	no	—
187	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	17	2	no	—
190	C₂×D₄	—	34	2	no	—
191	C₂×D₄	—	34	1	no	—
192	C₂×D₄	—	34	1	no	—
193	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	34	2	no	—
196	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
197	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
199	SD₁₆	—	68	2	no	—
200	D₈	—	68	2	no	—
201	C₂×C₈	—	17	2	no	—
202	C₂×C₈	—	17	2	no	—
203	C₂×C₈	—	68	2	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	17	2	no	—
209	C₄⋊C₄	—	34	1	no	—
210	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
211	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
212	C₂²⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₂²⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	34	2	no	—
218	C₂²⋊C₄	—	34	2	no	—
219	C₄²	—	2	1	no	—
220	C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
221	C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
222	C₄²	C₂²×D₁₇	1	1	yes	characteristic
223	C₄²	C₂²×D₁₇	1	1	yes	characteristic
224	C₂³	—	2	1	no	—
225	C₂³	—	34	2	no	—
226	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
227	Q₈	—	2	1	no	—
228	Q₈	—	2	1	no	—
229	Q₈	—	68	2	no	—
230	D₄	—	2	1	no	—
231	D₄	—	2	1	no	—
232	D₄	—	4	2	no	—
233	D₄	—	34	2	no	—
234	D₄	—	34	2	no	—
235	D₄	—	68	2	no	—
236	D₄	—	68	4	no	—
237	D₄	—	68	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	34	2	no	—
247	C₂×C₄	—	68	4	no	—
248	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
249	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
250	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
252	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
253	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
254	C₈	—	34	2	no	—
255	C₈	—	34	2	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	68	2	no	—
262	C₂²	—	68	2	no	—
263	C₂²	—	68	2	no	—
264	C₂²	#10587635	1	2	yes	—
265	C₂²	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	68	2	no	—
274	C₄	#10587636	1	2	yes	—
275	C₄	#10587696	1	1	yes	characteristic
276	C₄	#10587697	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	8	1	no	—
279	C₂	—	136	2	no	—
280	C₂	#11763387	1	1	yes	characteristic
281	C₂	C₂×#10587636	1	1	yes	characteristic
282	C₂	C₂×#10587635	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#529	C₂
socle	C₂²×C₁₇	C₂²×D₄
hypercenter	#529	D₁₇
hypocenter	C₁₇	#3247
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #529

Complements

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

N	H ≅ G/N	type
1. #11763359	C₂	semidirect
2. #11763106	C₂	semidirect
3. #11763104	C₂	semidirect
4. #11763099	C₂	semidirect
5. #11763043	C₂	semidirect
6. #11763040	C₂	semidirect
7. #11763021	C₂	semidirect
8. #11763009	C₂	semidirect
9. #11763004	C₂	semidirect
10. C₂×C₄⋊#3622	C₂	semidirect
11. C₁₇×#529	C₂	semidirect
137. #529	D₁₇	semidirect
184. C₁₇	#3247	semidirect

Maximal subgroups

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

subgroup	index
#11763359	2
#11763106	2
#11763104	2
#11763099	2
#11763043	2
#11763040	2
#11763021	2
#11763009	2
#11763004	2
C₂×C₄⋊#3622	2
C₁₇×#529	2
#3247	17