Subgroups — #14538239

Order 2176.

496 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10587636	C₂	1	2	yes	maximal, complemented
2	#11763359	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763031	C₂	1	2	yes	maximal, complemented
4	#11763024	C₂	1	2	yes	maximal, complemented
5	#11763019	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763014	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763009	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763004	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄⋊#3622	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762803	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×#63252	C₂²	1	2	yes	—
14	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₄×D₄×C₁₇	—	2	1	no	—
17	C₄×D₄×C₁₇	—	2	1	no	—
18	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
20	C₁₇×#117	—	2	1	no	—
21	C₁₇×#117	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₁₇	—	2	1	no	—
23	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
24	#10587636	—	2	8	no	—
25	C₂×#63231	C₂²	1	2	yes	—
26	C₂×#63230	C₂²	1	2	yes	—
27	C₂×#63229	C₂²	1	2	yes	—
28	C₂×#63228	C₂²	1	2	yes	—
29	#10587616	—	2	1	no	—
30	#10587616	—	2	1	no	—
31	C₂²×#3622	C₂²	1	2	yes	—
32	#10587561	C₂²	1	2	yes	—
33	#10587560	C₂²	1	1	yes	characteristic
34	C₄×D₆₈	—	2	1	no	—
35	C₄×D₆₈	—	2	1	no	—
36	#10587555	C₂²	1	1	yes	characteristic
37	#10587481	C₂²	1	2	yes	—
38	#10587481	C₂²	1	2	yes	—
39	#10587480	C₂²	1	2	yes	—
40	#10587480	C₂²	1	2	yes	—
41	C₄⋊#3622	C₂²	1	1	yes	characteristic
42	C₄⋊#3622	C₂²	1	1	yes	characteristic
43	C₄⋊#3622	C₂²	1	2	yes	—
44	#41×C₁₇	—	2	2	no	—
45	#41×C₁₇	D₄	1	4	yes	complemented
46	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₁₇	—	2	1	no	—
48	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₇	—	2	1	no	—
50	C₂²×C₄×C₁₇	—	2	1	no	—
51	C₂²×C₄×C₁₇	—	2	1	no	—
52	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
55	C₂×C₂²⋊D₁₇	—	2	2	no	—
56	#63252	—	2	4	no	—
57	#63252	—	2	4	no	—
58	C₂×D₆₈	—	2	1	no	—
59	C₂×D₆₈	—	2	1	no	—
60	C₂×D₆₈	C₂³	1	2	yes	—
61	C₂×C₄×D₁₇	—	2	2	no	—
62	C₂×Dic₃₄	C₂³	1	2	yes	—
63	C₄⋊C₄×C₁₇	—	2	1	no	—
64	C₄⋊C₄×C₁₇	—	2	1	no	—
65	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
67	C₂²⋊C₄×C₁₇	—	2	1	no	—
68	C₂²⋊C₄×C₁₇	—	2	1	no	—
69	C₂²⋊C₄×C₁₇	—	2	1	no	—
70	C₄²×C₁₇	—	2	1	no	—
71	C₄²×C₁₇	—	2	1	no	—
72	C₄²×C₁₇	—	2	1	no	—
73	C₄²×C₁₇	D₄	1	1	yes	characteristic
74	C₄²×C₁₇	D₄	1	1	yes	characteristic
75	C₄²×C₁₇	C₂³	1	1	yes	characteristic
76	C₄²×C₁₇	C₂³	1	1	yes	characteristic
77	#63231	—	2	4	no	—
78	#63230	—	2	4	no	—
79	#63229	—	2	4	no	—
80	#63228	—	2	4	no	—
81	#63227	—	2	2	no	—
82	#63227	—	2	2	no	—
83	C₄⋊Dic₁₇	—	2	1	no	—
84	C₄⋊Dic₁₇	—	2	1	no	—
85	C₂×#3622	—	2	4	no	—
86	C₂×#3622	C₂³	1	2	yes	—
87	C₂×#3622	C₂³	1	2	yes	—
88	C₂³×C₁₇	—	2	1	no	—
89	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×D₁₇	—	2	2	no	—
91	Q₈×C₁₇	—	2	1	no	—
92	Q₈×C₁₇	C₂×D₄	1	2	yes	—
93	D₄×C₁₇	—	2	4	no	—
94	D₄×C₁₇	—	2	1	no	—
95	D₄×C₁₇	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₁₇	—	2	1	no	—
97	C₂×C₄×C₁₇	—	2	4	no	—
98	C₂×C₄×C₁₇	—	2	2	no	—
99	C₂×C₄×C₁₇	—	2	1	no	—
100	C₂×C₄×C₁₇	—	2	1	no	—
101	C₂×C₄×C₁₇	—	4	1	no	—
102	C₂×C₄×C₁₇	—	4	2	no	—
103	C₂×C₄×C₁₇	—	4	2	no	—
104	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₇	#41	1	2	yes	—
111	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
112	C₂²⋊D₁₇	—	4	4	no	—
113	C₂×Dic₁₇	—	2	2	no	—
114	D₆₈	—	2	2	no	—
115	D₆₈	—	2	2	no	—
116	D₆₈	—	4	2	no	—
117	D₆₈	—	4	2	no	—
118	C₄×D₁₇	—	4	4	no	—
119	Dic₃₄	—	2	2	no	—
120	Dic₃₄	—	2	2	no	—
121	#3622	—	2	2	no	—
122	#3622	—	2	2	no	—
123	#3035	—	17	1	no	maximal, Sylow
124	C₂²×C₁₇	—	2	1	no	—
125	C₂²×C₁₇	—	2	1	no	—
126	C₂²×C₁₇	—	2	2	no	—
127	C₂²×C₁₇	—	4	1	no	—
128	C₂²×C₁₇	—	4	1	no	—
129	C₂²×C₁₇	#121	1	2	yes	—
130	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×D₁₇	—	4	2	no	—
132	C₂×D₁₇	—	4	2	no	—
133	C₂×D₁₇	—	4	2	no	—
134	C₄×C₁₇	—	2	2	no	—
135	C₄×C₁₇	—	2	1	no	—
136	C₄×C₁₇	—	2	1	no	—
137	C₄×C₁₇	—	2	1	no	—
138	C₄×C₁₇	—	4	1	no	—
139	C₄×C₁₇	—	4	1	no	—
140	C₄×C₁₇	#121	1	2	yes	—
141	C₄×C₁₇	#135	1	2	yes	complemented
142	C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
143	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
144	Dic₁₇	—	4	2	no	—
145	C₂×#135	—	17	2	no	—
146	#532	—	17	1	no	—
147	C₄×#41	D₁₇	1	1	yes	characteristic, complemented
148	#464	—	17	2	no	—
149	#463	—	17	2	no	—
150	#462	—	17	1	no	—
151	#461	—	17	1	no	—
152	#460	—	17	1	no	—
153	#459	—	17	1	no	—
154	C₂×C₄⋊C₈	—	17	1	no	—
155	#416	—	17	2	no	—
156	C₂×C₁₇	—	2	2	no	—
157	C₂×C₁₇	—	4	2	no	—
158	C₂×C₁₇	C₂×#121	1	1	yes	characteristic
159	C₂×C₁₇	C₂×#135	1	1	yes	characteristic
160	C₂×C₁₇	#575	1	1	yes	characteristic
161	D₁₇	—	8	2	no	—
162	C₂×#41	—	17	2	no	—
163	C₂×#41	C₂×D₁₇	1	1	yes	characteristic
164	#135	—	34	8	no	—
165	C₂×Q₁₆	—	17	2	no	—
166	C₂×SD₁₆	—	17	2	no	—
167	C₂×SD₁₆	—	17	2	no	—
168	C₂×D₈	—	17	2	no	—
169	C₂²×C₈	—	17	2	no	—
170	C₄⋊Q₈	—	17	1	no	—
171	#127	—	17	1	no	—
172	#124	—	17	2	no	—
173	#121	—	34	1	no	—
174	#121	—	34	1	no	—
175	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
176	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
177	C₄×D₄	—	2	1	no	—
178	C₄×D₄	—	2	1	no	—
179	C₄×D₄	—	34	1	no	—
180	C₄×D₄	—	34	1	no	—
181	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
182	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
183	#117	—	2	1	no	—
184	#117	C₂×D₁₇	1	1	yes	characteristic
185	C₂×C₄²	—	2	1	no	—
186	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
187	C₄⋊C₈	—	17	1	no	—
188	C₄⋊C₈	—	17	1	no	—
189	C₄⋊C₈	—	17	2	no	—
190	#103	—	17	2	no	—
191	#103	—	17	2	no	—
192	#102	—	17	2	no	—
193	#102	—	17	2	no	—
194	C₁₇	#3035	1	1	yes	characteristic, complemented, Sylow
195	#41	—	2	2	no	—
196	#41	—	34	4	no	—
197	#41	—	34	4	no	—
198	#41	D₆₈	1	4	yes	complemented
199	C₂×Q₈	—	17	2	no	—
200	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	17	2	no	—
203	C₂×D₄	—	34	2	no	—
204	C₂×D₄	—	34	1	no	—
205	C₂×D₄	—	34	1	no	—
206	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	34	2	no	—
211	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
212	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
213	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
214	Q₁₆	—	34	4	no	—
215	SD₁₆	—	34	4	no	—
216	SD₁₆	—	34	4	no	—
217	D₈	—	34	4	no	—
218	C₂×C₈	—	17	2	no	—
219	C₂×C₈	—	17	2	no	—
220	C₂×C₈	—	34	4	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	34	1	no	—
224	C₄⋊C₄	—	34	1	no	—
225	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
226	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
227	C₂²⋊C₄	—	2	1	no	—
228	C₂²⋊C₄	—	2	1	no	—
229	C₂²⋊C₄	—	2	1	no	—
230	C₂²⋊C₄	—	34	2	no	—
231	C₂²⋊C₄	—	34	2	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	1	no	—
235	C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
236	C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
237	C₄²	C₂²×D₁₇	1	1	yes	characteristic
238	C₄²	C₂²×D₁₇	1	1	yes	characteristic
239	C₂³	—	2	1	no	—
240	C₂³	—	34	2	no	—
241	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
242	Q₈	—	2	1	no	—
243	Q₈	—	34	2	no	—
244	Q₈	—	34	2	no	—
245	Q₈	C₂×D₆₈	1	2	yes	—
246	D₄	—	2	4	no	—
247	D₄	—	2	1	no	—
248	D₄	—	34	2	no	—
249	D₄	—	34	2	no	—
250	D₄	—	68	2	no	—
251	D₄	—	68	4	no	—
252	D₄	—	68	2	no	—
253	D₄	C₂×D₆₈	1	2	yes	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	34	2	no	—
263	C₂×C₄	—	68	4	no	—
264	C₂×C₄	C₂×D₆₈	1	2	yes	—
265	C₂×C₄	#63252	1	2	yes	—
266	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
267	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
268	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
270	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
271	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
272	C₈	—	34	2	no	—
273	C₈	—	34	2	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	68	2	no	—
280	C₂²	—	68	2	no	—
281	C₂²	—	68	2	no	—
282	C₂²	#10587616	1	2	yes	—
283	C₂²	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	4	1	no	—
290	C₄	—	68	2	no	—
291	C₄	#10587616	1	2	yes	—
292	C₄	#10587636	1	2	yes	complemented
293	C₄	C₂²×D₆₈	1	1	yes	characteristic
294	C₄	C₂×#63252	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	4	2	no	—
297	C₂	—	136	2	no	—
298	C₂	C₂×#10587616	1	1	yes	characteristic
299	C₂	C₂×#10587636	1	1	yes	characteristic
300	C₂	#11763607	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₄×#41×C₁₇	C₂
socle	C₂²×C₁₇	C₂²×D₄
hypercenter	C₄×#41	D₁₇
hypocenter	C₁₇	#3035
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#10587636	C₂	semidirect
2. #11763359	C₂	semidirect
3. #11763031	C₂	semidirect
4. #11763024	C₂	semidirect
5. #11763019	C₂	semidirect
6. #11763014	C₂	semidirect
7. #11763009	C₂	semidirect
8. #11763004	C₂	semidirect
9. C₂×C₄⋊#3622	C₂	semidirect
10. #11762803	C₂	semidirect
11. C₄×#41×C₁₇	C₂	semidirect
45. #41×C₁₇	D₄	semidirect
141. C₄×C₁₇	#135	semidirect
147. C₄×#41	D₁₇	semidirect
194. C₁₇	#3035	semidirect
198. #41	D₆₈	semidirect
292. C₄	#10587636	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10587636	2
#11763359	2
#11763031	2
#11763024	2
#11763019	2
#11763014	2
#11763009	2
#11763004	2
C₂×C₄⋊#3622	2
#11762803	2
C₄×#41×C₁₇	2
#3035	17