Subgroups — #14538992

Order 2176.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763289	C₂	1	1	yes	maximal, characteristic
2	#11763142	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763137	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763113	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763106	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763016	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763011	C₂	1	1	yes	maximal, characteristic, complemented
8	#11762969	C₂	1	1	yes	maximal, characteristic
9	#11762969	C₂	1	1	yes	maximal, characteristic
10	#11762914	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762907	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#63231	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#63230	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762870	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#541	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#63252	C₂²	1	1	yes	characteristic
17	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₁₇×#126	—	2	1	no	—
19	C₁₇×#126	—	2	1	no	—
20	C₁₇×#125	—	2	1	no	—
21	C₁₇×#125	—	2	1	no	—
22	C₁₇×#125	C₂²	1	1	yes	characteristic
23	C₁₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
24	C₁₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
25	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
26	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
27	C₁₇×#117	C₂²	1	1	yes	characteristic
28	C₂×#63231	C₂²	1	1	yes	characteristic
29	C₂×#63231	C₂²	1	1	yes	characteristic
30	C₂×#63230	C₂²	1	1	yes	characteristic
31	C₂×#63230	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
33	#10587578	—	2	1	no	—
34	C₄⋊C₄×D₁₇	—	2	1	no	—
35	#10587567	—	2	1	no	—
36	#10587567	—	2	1	no	—
37	C₄×D₆₈	C₂²	1	1	yes	characteristic
38	C₄×D₆₈	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
40	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
41	#10587506	C₂²	1	1	yes	characteristic
42	#10587506	C₂²	1	1	yes	characteristic
43	#10587506	C₂²	1	1	yes	characteristic
44	#10587506	C₂²	1	1	yes	characteristic
45	C₂²⋊#3622	C₂²	1	1	yes	characteristic
46	C₂²⋊#3622	C₂²	1	1	yes	characteristic
47	#10587481	C₂²	1	1	yes	characteristic
48	#10587481	C₂²	1	1	yes	characteristic
49	#10587480	C₂²	1	1	yes	characteristic
50	#10587480	C₂²	1	1	yes	characteristic
51	#10587479	C₂²	1	1	yes	characteristic
52	#10587479	C₂²	1	1	yes	characteristic
53	#10587478	C₂²	1	1	yes	characteristic
54	#10587478	C₂²	1	1	yes	characteristic
55	C₄⋊#3622	C₂²	1	1	yes	characteristic
56	C₄⋊#3622	C₂²	1	1	yes	characteristic
57	#10587474	C₂²	1	1	yes	characteristic
58	C₄×#3622	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₇	—	2	1	no	—
63	C₂²×Dic₁₇	—	2	1	no	—
64	#63252	—	2	2	no	—
65	#63252	—	2	2	no	—
66	C₂×D₆₈	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₁₇	—	2	1	no	—
68	C₂×C₄×D₁₇	—	2	1	no	—
69	C₂×C₄×D₁₇	—	2	1	no	—
70	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₇	—	2	1	no	—
72	C₄⋊C₄×C₁₇	—	2	1	no	—
73	C₄⋊C₄×C₁₇	—	2	1	no	—
74	C₄⋊C₄×C₁₇	—	2	1	no	—
75	C₄⋊C₄×C₁₇	—	2	1	no	—
76	C₄⋊C₄×C₁₇	—	2	1	no	—
77	C₄⋊C₄×C₁₇	—	2	1	no	—
78	C₄⋊C₄×C₁₇	—	2	1	no	—
79	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
80	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₁₇	—	2	1	no	—
84	C₂²⋊C₄×C₁₇	—	2	1	no	—
85	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
86	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
87	C₄²×C₁₇	—	2	1	no	—
88	C₄²×C₁₇	—	2	1	no	—
89	C₄²×C₁₇	C₂³	1	1	yes	characteristic
90	C₄²×C₁₇	C₂³	1	1	yes	characteristic
91	#63231	—	2	2	no	—
92	#63231	—	2	2	no	—
93	#63230	—	2	2	no	—
94	#63230	—	2	2	no	—
95	#63227	—	2	1	no	—
96	#63227	—	2	1	no	—
97	C₄⋊Dic₁₇	—	2	1	no	—
98	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
99	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
100	#63225	—	2	1	no	—
101	#63225	—	2	1	no	—
102	C₄×Dic₁₇	—	2	1	no	—
103	C₄×Dic₁₇	—	2	1	no	—
104	C₂×#3622	C₂³	1	1	yes	characteristic
105	C₂×#3622	C₂³	1	1	yes	characteristic
106	C₂×#3622	C₂³	1	1	yes	characteristic
107	C₂×#3622	C₂³	1	1	yes	characteristic
108	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₁₇	—	2	1	no	—
110	Q₈×C₁₇	—	2	1	no	—
111	Q₈×C₁₇	—	2	1	no	—
112	Q₈×C₁₇	—	2	1	no	—
113	Q₈×C₁₇	—	2	1	no	—
114	C₂×C₄×C₁₇	—	2	1	no	—
115	C₂×C₄×C₁₇	—	2	1	no	—
116	C₂×C₄×C₁₇	—	2	1	no	—
117	C₂×C₄×C₁₇	—	2	1	no	—
118	C₂×C₄×C₁₇	—	2	1	no	—
119	C₂×C₄×C₁₇	—	2	1	no	—
120	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
126	C₂²⋊D₁₇	—	2	4	no	—
127	C₂×Dic₁₇	—	2	1	no	—
128	C₂×Dic₁₇	—	2	1	no	—
129	C₂×Dic₁₇	—	2	1	no	—
130	C₂×Dic₁₇	—	8	1	no	—
131	D₆₈	—	2	1	no	—
132	D₆₈	#41	1	2	yes	—
133	C₄×D₁₇	—	2	4	no	—
134	C₄×D₁₇	—	8	1	no	—
135	C₄×D₁₇	—	8	1	no	—
136	Dic₃₄	—	2	1	no	—
137	Dic₃₄	#41	1	2	yes	—
138	#3622	—	2	1	no	—
139	#3622	—	2	1	no	—
140	#3622	—	2	1	no	—
141	#3622	—	2	1	no	—
142	#3120	—	17	1	no	maximal, Sylow
143	C₂²×C₁₇	—	2	1	no	—
144	C₂²×C₁₇	—	2	1	no	—
145	C₂²×C₁₇	—	2	1	no	—
146	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×D₁₇	—	2	2	no	—
148	C₂×D₁₇	—	4	1	no	—
149	C₂×D₁₇	—	4	1	no	—
150	C₄×C₁₇	—	2	1	no	—
151	C₄×C₁₇	—	2	1	no	—
152	C₄×C₁₇	—	2	1	no	—
153	C₄×C₁₇	—	2	1	no	—
154	C₄×C₁₇	—	2	1	no	—
155	C₄×C₁₇	—	4	1	no	—
156	C₄×C₁₇	—	4	1	no	—
157	C₄×C₁₇	—	4	1	no	—
158	C₄×C₁₇	—	4	1	no	—
159	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
160	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
161	Dic₁₇	—	2	2	no	—
162	Dic₁₇	—	4	1	no	—
163	Dic₁₇	—	4	1	no	—
164	#541	D₁₇	1	1	yes	characteristic, complemented
165	#520	—	17	1	no	—
166	#484	—	17	1	no	—
167	#483	—	17	1	no	—
168	#479	—	17	1	no	—
169	#478	—	17	1	no	—
170	#462	—	17	1	no	—
171	#461	—	17	1	no	—
172	#452	—	17	1	no	—
173	#452	—	17	1	no	—
174	#441	—	17	1	no	—
175	#440	—	17	1	no	—
176	C₄×Q₁₆	—	17	1	no	—
177	C₄×SD₁₆	—	17	1	no	—
178	#433	—	17	1	no	—
179	C₂×C₁₇	—	4	1	no	—
180	C₂×C₁₇	#525	1	1	yes	characteristic
181	C₂×C₁₇	#577	1	1	yes	characteristic
182	C₂×C₁₇	#578	1	1	yes	characteristic
183	D₁₇	—	4	2	no	—
184	C₂×#41	—	17	1	no	—
185	C₂×Q₁₆	—	17	1	no	—
186	C₂×Q₁₆	—	17	1	no	—
187	C₂×SD₁₆	—	17	1	no	—
188	C₂×SD₁₆	—	17	1	no	—
189	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
190	#126	—	2	1	no	—
191	#126	—	2	1	no	—
192	#125	—	2	1	no	—
193	#125	—	2	1	no	—
194	#125	C₂×D₁₇	1	1	yes	characteristic
195	C₂²⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
196	C₂²⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
197	C₄×Q₈	—	17	1	no	—
198	C₄×Q₈	—	17	1	no	—
199	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
200	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
201	C₄×D₄	—	17	1	no	—
202	C₄×D₄	—	17	1	no	—
203	C₄×D₄	—	34	1	no	—
204	C₄×D₄	—	34	1	no	—
205	#117	—	34	1	no	—
206	#117	C₂×D₁₇	1	1	yes	characteristic
207	C₂×C₄⋊C₄	—	17	1	no	—
208	C₂×C₄⋊C₄	—	34	1	no	—
209	#107	—	17	1	no	—
210	#107	—	17	1	no	—
211	#106	—	17	1	no	—
212	#106	—	17	1	no	—
213	C₄⋊C₈	—	17	1	no	—
214	C₄⋊C₈	—	17	1	no	—
215	#103	—	17	1	no	—
216	#103	—	17	1	no	—
217	#103	—	17	1	no	—
218	#103	—	17	1	no	—
219	#103	—	17	1	no	—
220	#103	—	17	1	no	—
221	#102	—	17	1	no	—
222	#102	—	17	1	no	—
223	C₂²⋊C₈	—	17	1	no	—
224	C₂²⋊C₈	—	17	1	no	—
225	#97	—	17	1	no	—
226	C₄×C₈	—	17	1	no	—
227	C₁₇	#3120	1	1	yes	characteristic, complemented, Sylow
228	#41	—	34	2	no	—
229	#41	—	34	2	no	—
230	C₂×Q₈	—	17	1	no	—
231	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
232	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
233	C₂×D₄	—	17	1	no	—
234	C₂×D₄	—	34	1	no	—
235	C₂²×C₄	—	34	1	no	—
236	C₂²×C₄	—	34	1	no	—
237	C₂²×C₄	—	34	1	no	—
238	C₂²×C₄	—	34	1	no	—
239	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
240	Q₁₆	—	34	2	no	—
241	Q₁₆	—	34	2	no	—
242	SD₁₆	—	34	2	no	—
243	SD₁₆	—	34	2	no	—
244	C₂×C₈	—	17	1	no	—
245	C₂×C₈	—	17	1	no	—
246	C₂×C₈	—	17	1	no	—
247	C₂×C₈	—	17	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	17	1	no	—
257	C₄⋊C₄	—	17	1	no	—
258	C₄⋊C₄	—	34	1	no	—
259	C₄⋊C₄	—	34	1	no	—
260	C₄⋊C₄	—	34	1	no	—
261	C₄⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
262	C₄⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	34	1	no	—
268	C₂²⋊C₄	—	34	1	no	—
269	C₂²⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
270	C₂²⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
271	C₄²	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	34	1	no	—
274	C₄²	—	34	1	no	—
275	C₄²	C₂²×D₁₇	1	1	yes	characteristic
276	C₄²	C₂²×D₁₇	1	1	yes	characteristic
277	C₂³	—	34	1	no	—
278	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	17	2	no	—
284	Q₈	—	34	1	no	—
285	D₄	—	17	2	no	—
286	D₄	—	34	4	no	—
287	D₄	—	34	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	34	4	no	—
295	C₂×C₄	—	34	1	no	—
296	C₂×C₄	—	34	1	no	—
297	C₂×C₄	—	34	1	no	—
298	C₂×C₄	—	136	1	no	—
299	C₂×C₄	—	136	1	no	—
300	C₂×C₄	—	136	1	no	—
301	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
302	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
303	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
304	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
305	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
307	C₈	—	34	1	no	—
308	C₈	—	34	1	no	—
309	C₈	—	34	1	no	—
310	C₈	—	34	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	34	2	no	—
315	C₂²	—	68	1	no	—
316	C₂²	—	68	1	no	—
317	C₂²	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	34	2	no	—
328	C₄	—	68	1	no	—
329	C₄	—	68	1	no	—
330	C₄	#41×D₁₇	1	1	yes	characteristic
331	C₄	#41×D₁₇	1	1	yes	characteristic
332	C₂	—	4	1	no	—
333	C₂	—	68	2	no	—
334	C₂	#11763316	1	1	yes	characteristic
335	C₂	#11763617	1	1	yes	characteristic
336	C₂	#11763624	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×#541	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×#63252	C₂²
hypercenter	#541	D₁₇
hypocenter	C₁₇	#3120
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #541

Complements

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

N	H ≅ G/N	type
2. #11763142	C₂	semidirect
3. #11763137	C₂	semidirect
4. #11763113	C₂	semidirect
5. #11763106	C₂	semidirect
6. #11763016	C₂	semidirect
7. #11763011	C₂	semidirect
10. #11762914	C₂	semidirect
11. #11762907	C₂	semidirect
12. C₄×#63231	C₂	semidirect
13. C₄×#63230	C₂	semidirect
14. #11762870	C₂	semidirect
15. C₁₇×#541	C₂	semidirect
164. #541	D₁₇	semidirect
227. C₁₇	#3120	semidirect

Maximal subgroups

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

subgroup	index
#11763289	2
#11763142	2
#11763137	2
#11763113	2
#11763106	2
#11763016	2
#11763011	2
#11762969	2
#11762914	2
#11762907	2
C₄×#63231	2
C₄×#63230	2
#11762870	2
C₁₇×#541	2
#3120	17