Subgroups — #14541729

Order 2176.

334 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763537	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763219	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763215	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763208	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763113	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763113	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763109	C₂	1	1	yes	maximal, characteristic
8	#11763109	C₂	1	1	yes	maximal, characteristic
9	#11763099	C₂	1	1	yes	maximal, characteristic
10	#11763091	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762940	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762909	C₂	1	1	yes	maximal, characteristic
13	#11762909	C₂	1	1	yes	maximal, characteristic
14	C₄×#63229	C₂	1	1	yes	maximal, characteristic
15	C₁₇×#554	C₂	1	1	yes	maximal, characteristic
16	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₇×#125	C₂²	1	1	yes	characteristic
18	C₁₇×#125	C₂²	1	1	yes	characteristic
19	C₁₇×C₂²⋊Q₈	—	2	1	no	—
20	C₁₇×C₂²⋊Q₈	—	2	1	no	—
21	C₁₇×C₂²⋊Q₈	—	2	1	no	—
22	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
27	#10587630	—	2	1	no	—
28	C₂×#63229	C₂²	1	1	yes	characteristic
29	#10587575	—	2	1	no	—
30	#10587575	—	2	1	no	—
31	#10587573	—	2	1	no	—
32	#10587556	C₂²	1	1	yes	characteristic
33	#10587556	C₂²	1	1	yes	characteristic
34	#10587555	C₂²	1	1	yes	characteristic
35	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
36	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
37	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
38	#10587503	C₂²	1	1	yes	characteristic
39	#10587503	C₂²	1	1	yes	characteristic
40	#10587503	C₂²	1	1	yes	characteristic
41	#10587481	C₂²	1	1	yes	characteristic
42	#10587481	C₂²	1	1	yes	characteristic
43	#10587481	C₂²	1	1	yes	characteristic
44	#10587479	C₂²	1	1	yes	characteristic
45	#10587479	C₂²	1	1	yes	characteristic
46	#10587479	C₂²	1	1	yes	characteristic
47	#10587479	C₂²	1	1	yes	characteristic
48	#10587479	C₂²	1	1	yes	characteristic
49	#10587478	C₂²	1	1	yes	characteristic
50	#10587478	C₂²	1	1	yes	characteristic
51	#10587478	C₂²	1	1	yes	characteristic
52	#10587478	C₂²	1	1	yes	characteristic
53	C₄⋊#3622	C₂²	1	1	yes	characteristic
54	C₄⋊#3622	C₂²	1	1	yes	characteristic
55	C₄⋊#3622	C₂²	1	1	yes	characteristic
56	#10587474	C₂²	1	1	yes	characteristic
57	#10587474	C₂²	1	1	yes	characteristic
58	C₄×#3622	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	—	2	1	no	—
60	C₂×Q₈×C₁₇	D₄	1	1	yes	characteristic
61	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	—	2	1	no	—
63	C₂²×C₄×C₁₇	—	2	1	no	—
64	C₂²×C₄×C₁₇	—	2	1	no	—
65	C₂×Dic₃₄	—	2	1	no	—
66	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₁₇	—	2	1	no	—
68	C₄⋊C₄×C₁₇	—	2	1	no	—
69	C₄⋊C₄×C₁₇	—	2	1	no	—
70	C₄⋊C₄×C₁₇	—	2	1	no	—
71	C₄⋊C₄×C₁₇	—	2	1	no	—
72	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
73	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₇	—	2	1	no	—
79	C₂²⋊C₄×C₁₇	—	2	1	no	—
80	C₂²⋊C₄×C₁₇	—	2	1	no	—
81	C₄²×C₁₇	C₂³	1	1	yes	characteristic
82	C₄²×C₁₇	C₂³	1	1	yes	characteristic
83	C₄²×C₁₇	C₂³	1	1	yes	characteristic
84	#63229	Q₈	1	4	yes	complemented
85	C₄⋊Dic₁₇	—	2	1	no	—
86	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
87	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
89	#63225	—	2	1	no	—
90	#63225	—	2	1	no	—
91	#63225	—	2	1	no	—
92	#63225	—	2	1	no	—
93	#63225	—	2	1	no	—
94	#63225	—	2	1	no	—
95	#63225	—	2	1	no	—
96	C₄×Dic₁₇	—	2	1	no	—
97	C₄×Dic₁₇	—	2	1	no	—
98	C₄×Dic₁₇	—	2	1	no	—
99	C₂×#3622	C₂³	1	1	yes	characteristic
100	C₂×#3622	C₂³	1	1	yes	characteristic
101	C₂×#3622	C₂³	1	1	yes	characteristic
102	C₂×#3622	C₂³	1	1	yes	characteristic
103	C₂³×C₁₇	—	2	1	no	—
104	Q₈×C₁₇	—	4	1	no	—
105	Q₈×C₁₇	—	8	1	no	—
106	D₄×C₁₇	—	2	1	no	—
107	D₄×C₁₇	C₂×Q₈	1	2	yes	—
108	C₂×C₄×C₁₇	—	2	1	no	—
109	C₂×C₄×C₁₇	—	2	1	no	—
110	C₂×C₄×C₁₇	—	2	1	no	—
111	C₂×C₄×C₁₇	—	4	2	no	—
112	C₂×C₄×C₁₇	—	8	1	no	—
113	C₂×C₄×C₁₇	—	8	1	no	—
114	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
121	C₂×Dic₁₇	—	2	1	no	—
122	C₂×Dic₁₇	—	2	1	no	—
123	C₂×Dic₁₇	—	2	1	no	—
124	C₂×Dic₁₇	—	2	1	no	—
125	Dic₃₄	—	2	1	no	—
126	Dic₃₄	—	4	2	no	—
127	Dic₃₄	C₂×Q₈	1	2	yes	—
128	#3622	—	2	1	no	—
129	#3622	—	2	1	no	—
130	#3622	—	2	1	no	—
131	#3622	C₂×Q₈	1	2	yes	—
132	#3388	—	17	1	no	maximal, Sylow
133	C₂²×C₁₇	—	2	2	no	—
134	C₂²×C₁₇	—	4	1	no	—
135	C₂²×C₁₇	—	4	1	no	—
136	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
137	C₄×C₁₇	—	2	1	no	—
138	C₄×C₁₇	—	2	1	no	—
139	C₄×C₁₇	—	2	1	no	—
140	C₄×C₁₇	—	2	1	no	—
141	C₄×C₁₇	—	2	1	no	—
142	C₄×C₁₇	—	2	1	no	—
143	C₄×C₁₇	—	4	1	no	—
144	C₄×C₁₇	—	4	1	no	—
145	C₄×C₁₇	—	4	1	no	—
146	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
147	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
148	Dic₁₇	—	2	2	no	—
149	Dic₁₇	—	4	1	no	—
150	Dic₁₇	—	4	1	no	—
151	Dic₁₇	—	4	1	no	—
152	#557	—	17	1	no	—
153	#554	D₁₇	1	1	yes	characteristic
154	#501	—	17	1	no	—
155	#501	—	17	1	no	—
156	#499	—	17	1	no	—
157	#479	—	17	1	no	—
158	#479	—	17	1	no	—
159	#478	—	17	1	no	—
160	#478	—	17	1	no	—
161	#476	—	17	1	no	—
162	#475	—	17	1	no	—
163	#446	—	17	1	no	—
164	#440	—	17	1	no	—
165	#440	—	17	1	no	—
166	C₄×SD₁₆	—	17	1	no	—
167	C₂×C₁₇	—	4	2	no	—
168	C₂×C₁₇	D₄×Q₈	1	1	yes	characteristic
169	C₂×C₁₇	#575	1	1	yes	characteristic
170	C₂×C₁₇	#577	1	1	yes	characteristic
171	C₂×SD₁₆	—	17	1	no	—
172	C₄⋊Q₈	—	17	1	no	—
173	C₄⋊Q₈	—	34	1	no	—
174	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
175	#125	—	17	1	no	—
176	#125	—	17	1	no	—
177	#125	—	34	1	no	—
178	#125	—	34	1	no	—
179	#125	C₂×D₁₇	1	1	yes	characteristic
180	#125	C₂×D₁₇	1	1	yes	characteristic
181	C₂²⋊Q₈	—	2	1	no	—
182	C₂²⋊Q₈	—	2	1	no	—
183	C₂²⋊Q₈	—	2	1	no	—
184	C₄×Q₈	—	17	1	no	—
185	C₄×Q₈	—	17	1	no	—
186	C₄×Q₈	—	17	1	no	—
187	C₄×Q₈	—	34	1	no	—
188	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
189	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
190	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
191	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
192	C₂×C₄⋊C₄	—	2	1	no	—
193	#107	—	17	1	no	—
194	#107	—	17	1	no	—
195	#107	—	17	1	no	—
196	#107	—	17	1	no	—
197	#106	—	17	1	no	—
198	#106	—	17	1	no	—
199	#106	—	17	1	no	—
200	#106	—	17	1	no	—
201	#106	—	17	1	no	—
202	C₄⋊C₈	—	17	1	no	—
203	C₄⋊C₈	—	17	1	no	—
204	C₄⋊C₈	—	17	1	no	—
205	#103	—	17	1	no	—
206	#103	—	17	1	no	—
207	#103	—	17	1	no	—
208	#102	—	17	1	no	—
209	#102	—	17	1	no	—
210	#102	—	17	1	no	—
211	#97	—	17	1	no	—
212	#97	—	17	1	no	—
213	C₄×C₈	—	17	1	no	—
214	C₁₇	#3388	1	1	yes	characteristic, complemented, Sylow
215	C₂×Q₈	—	2	1	no	—
216	C₂×Q₈	—	17	1	no	—
217	C₂×Q₈	—	34	1	no	—
218	C₂×Q₈	C₂²⋊D₁₇	1	1	yes	characteristic
219	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	SD₁₆	—	17	4	no	—
224	C₂×C₈	—	17	1	no	—
225	C₂×C₈	—	17	1	no	—
226	C₂×C₈	—	17	1	no	—
227	C₂×C₈	—	17	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	17	1	no	—
234	C₄⋊C₄	—	17	1	no	—
235	C₄⋊C₄	—	17	1	no	—
236	C₄⋊C₄	—	34	1	no	—
237	C₄⋊C₄	—	34	1	no	—
238	C₄⋊C₄	—	34	1	no	—
239	C₄⋊C₄	—	34	1	no	—
240	C₄⋊C₄	—	34	1	no	—
241	C₄⋊C₄	—	34	1	no	—
242	C₄⋊C₄	—	34	1	no	—
243	C₄⋊C₄	—	34	1	no	—
244	C₄⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
245	C₄⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
246	C₄⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
247	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
248	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₄²	—	34	1	no	—
254	C₄²	—	34	1	no	—
255	C₄²	—	34	1	no	—
256	C₄²	C₂²×D₁₇	1	1	yes	characteristic
257	C₄²	C₂²×D₁₇	1	1	yes	characteristic
258	C₄²	C₂²×D₁₇	1	1	yes	characteristic
259	C₂³	—	2	1	no	—
260	Q₈	—	4	1	no	—
261	Q₈	—	8	1	no	—
262	Q₈	—	17	2	no	—
263	Q₈	—	34	1	no	—
264	Q₈	—	68	2	no	—
265	D₄	—	2	1	no	—
266	D₄	Q₈×D₁₇	1	2	yes	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	34	1	no	—
274	C₂×C₄	—	34	1	no	—
275	C₂×C₄	—	34	1	no	—
276	C₂×C₄	—	34	1	no	—
277	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
278	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
279	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
280	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
281	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
282	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
283	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
284	C₈	—	17	2	no	—
285	C₈	—	34	1	no	—
286	C₈	—	34	1	no	—
287	C₈	—	34	1	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	4	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	34	2	no	—
302	C₄	—	68	1	no	—
303	C₄	—	68	1	no	—
304	C₄	—	68	1	no	—
305	C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
306	C₄	#10587694	1	1	yes	characteristic
307	C₂	—	4	2	no	—
308	C₂	Q₈×C₂²⋊D₁₇	1	1	yes	characteristic
309	C₂	#11763607	1	1	yes	characteristic
310	C₂	#11763620	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×#554	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×D₄×C₁₇	C₂³
hypercenter	#554	D₁₇
hypocenter	C₁₇	#3388
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #554

Complements

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

N	H ≅ G/N	type
1. #11763537	C₂	semidirect
2. #11763219	C₂	semidirect
3. #11763215	C₂	semidirect
4. #11763208	C₂	semidirect
5. #11763113	C₂	semidirect
6. #11763113	C₂	semidirect
10. #11763091	C₂	semidirect
11. #11762940	C₂	semidirect
84. #63229	Q₈	semidirect
214. C₁₇	#3388	semidirect

Maximal subgroups

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

subgroup	index
#11763537	2
#11763219	2
#11763215	2
#11763208	2
#11763113	2
#11763109	2
#11763099	2
#11763091	2
#11762940	2
#11762909	2
C₄×#63229	2
C₁₇×#554	2
#3388	17