Subgroups — #17192048

Order 4352.

508 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₁₃₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#14531020	C₂	1	2	yes	maximal, complemented
3	#14531011	C₂	1	1	yes	maximal, characteristic, complemented
4	#14530936	C₂	1	2	yes	maximal, complemented
5	#14530921	C₂	1	1	yes	maximal, characteristic, complemented
6	#14530836	C₂	1	2	yes	maximal
7	#14530826	C₂	1	1	yes	maximal, characteristic, complemented
8	#14530693	C₂	1	1	yes	maximal, characteristic, complemented
9	#14530646	C₂	1	2	yes	maximal, complemented
10	C₄×#10587497	C₂	1	1	yes	maximal, characteristic
11	C₁₇×#3393	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈×D₆₈	—	2	1	no	—
13	C₂×#10587499	C₂²	1	2	yes	—
14	C₂×#10587497	C₂²	1	1	yes	characteristic
15	#11763214	C₂²	1	1	yes	characteristic
16	#11763214	C₂²	1	2	yes	—
17	#11763090	—	2	1	no	—
18	#11763090	—	2	2	no	—
19	C₄×D₁₃₆	C₂²	1	1	yes	characteristic
20	C₄×D₁₃₆	C₂²	1	2	yes	—
21	#11762634	C₂²	1	1	yes	characteristic
22	#11762634	C₂²	1	2	yes	—
23	C₄⋊#63214	C₂²	1	1	yes	characteristic
24	C₄⋊#63214	C₂²	1	2	yes	—
25	#11762613	C₂²	1	2	yes	—
26	#11762613	C₂²	1	2	yes	—
27	#11762613	C₂²	1	2	yes	—
28	#11762610	C₂²	1	2	yes	—
29	#11762610	C₂²	1	1	yes	characteristic
30	#11762609	C₂²	1	1	yes	characteristic
31	#11762585	C₂²	1	2	yes	—
32	C₄×#63214	C₂²	1	1	yes	characteristic
33	C₁₇×#556	—	2	1	no	—
34	C₁₇×#495	C₂²	1	2	yes	—
35	C₁₇×#494	C₂²	1	1	yes	characteristic
36	C₁₇×#464	—	2	2	no	—
37	C₁₇×#461	—	2	1	no	—
38	C₈×Q₈×C₁₇	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₁₇	C₂²	1	2	yes	—
40	C₄×D₈×C₁₇	C₂²	1	1	yes	characteristic
41	C₂×Q₈×D₁₇	—	4	1	no	—
42	C₂×Q₁₆×C₁₇	—	2	1	no	—
43	C₂×Q₁₆×C₁₇	C₂³	1	2	yes	—
44	C₂×SD₁₆×C₁₇	—	2	2	no	—
45	C₂×D₈×C₁₇	C₂³	1	1	yes	characteristic
46	C₁₇×C₄⋊Q₈	—	2	1	no	—
47	C₁₇×#126	—	4	2	no	—
48	C₁₇×#124	—	2	2	no	—
49	C₁₇×C₂²⋊Q₈	—	4	1	no	—
50	C₄×Q₈×C₁₇	—	2	2	no	—
51	C₄×Q₈×C₁₇	D₄	1	1	yes	characteristic
52	C₄×D₄×C₁₇	—	2	1	no	—
53	C₁₇×#117	—	4	1	no	—
54	C₂×D₁₃₆	C₂³	1	1	yes	characteristic
55	#10587583	—	4	2	no	—
56	#10587583	—	4	1	no	—
57	C₄×D₆₈	—	2	2	no	—
58	C₄×D₆₈	—	2	1	no	—
59	#10587555	—	2	2	no	—
60	#10587555	—	2	1	no	—
61	C₁₇×#107	C₂³	1	1	yes	characteristic
62	C₁₇×C₄⋊C₈	D₄	1	1	yes	characteristic
63	C₁₇×C₄⋊C₈	D₄	1	2	yes	—
64	C₁₇×#103	—	2	2	no	—
65	C₁₇×#102	—	2	1	no	—
66	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₁₇	C₂³	1	2	yes	—
68	#10587499	—	2	4	no	—
69	#10587497	—	2	2	no	—
70	#10587492	—	2	2	no	—
71	#10587492	—	2	1	no	—
72	#10587489	—	2	2	no	—
73	#10587489	—	2	1	no	—
74	#10587489	C₂³	1	2	yes	—
75	#10587489	C₂³	1	1	yes	characteristic
76	C₂×#63214	C₂³	1	1	yes	characteristic
77	C₂×#63214	C₂³	1	2	yes	—
78	C₂×#63214	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₁₇	—	2	2	no	—
80	C₂×Q₈×C₁₇	D₈	1	1	yes	characteristic
81	C₂×D₄×C₁₇	—	2	1	no	—
82	C₂²×C₄×C₁₇	—	4	1	no	—
83	Q₈×D₁₇	—	8	4	no	—
84	C₂×D₆₈	—	2	1	no	—
85	C₂×C₄×D₁₇	—	4	2	no	—
86	C₂×C₄×D₁₇	—	4	1	no	—
87	C₂×Dic₃₄	—	4	2	no	—
88	C₂×Dic₃₄	—	4	1	no	—
89	Q₁₆×C₁₇	—	2	2	no	—
90	Q₁₆×C₁₇	—	2	2	no	—
91	Q₁₆×C₁₇	—	8	1	no	—
92	SD₁₆×C₁₇	—	8	2	no	—
93	D₈×C₁₇	—	2	1	no	—
94	D₈×C₁₇	—	2	1	no	—
95	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₁₇	—	2	1	no	—
99	C₄⋊C₄×C₁₇	—	4	2	no	—
100	C₄⋊C₄×C₁₇	—	4	1	no	—
101	C₄⋊C₄×C₁₇	—	4	1	no	—
102	C₄⋊C₄×C₁₇	D₈	1	2	yes	—
103	C₄⋊C₄×C₁₇	D₈	1	1	yes	characteristic
104	C₂²⋊C₄×C₁₇	—	4	2	no	—
105	C₂²⋊C₄×C₁₇	—	4	2	no	—
106	C₂²⋊C₄×C₁₇	—	4	1	no	—
107	C₄²×C₁₇	—	4	2	no	—
108	C₄²×C₁₇	C₂×D₄	1	2	yes	—
109	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
110	#63227	—	4	2	no	—
111	#63227	—	4	1	no	—
112	C₄⋊Dic₁₇	—	2	1	no	—
113	C₄⋊Dic₁₇	—	2	2	no	—
114	C₄⋊Dic₁₇	—	4	2	no	—
115	C₄⋊Dic₁₇	—	4	1	no	—
116	D₁₃₆	—	2	1	no	—
117	D₁₃₆	#41	1	2	yes	—
118	#63214	—	2	1	no	—
119	#63214	—	2	2	no	—
120	#63214	—	2	1	no	—
121	#33934	—	17	1	no	maximal, Sylow
122	C₂³×C₁₇	—	4	1	no	—
123	C₂²×D₁₇	—	4	1	no	—
124	Q₈×C₁₇	—	2	2	no	—
125	Q₈×C₁₇	—	4	2	no	—
126	Q₈×C₁₇	—	4	2	no	—
127	D₄×C₁₇	—	4	1	no	—
128	D₄×C₁₇	—	4	1	no	—
129	C₂×C₄×C₁₇	—	4	1	no	—
130	C₂×C₄×C₁₇	—	4	2	no	—
131	C₂×C₄×C₁₇	—	8	2	no	—
132	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×Dic₁₇	—	4	1	no	—
138	C₂×Dic₁₇	—	4	2	no	—
139	D₆₈	—	2	2	no	—
140	D₆₈	—	4	1	no	—
141	C₄×D₁₇	—	8	4	no	—
142	C₄×D₁₇	—	8	2	no	—
143	Dic₃₄	—	8	4	no	—
144	Dic₃₄	—	8	2	no	—
145	C₈×C₁₇	—	2	2	no	—
146	C₈×C₁₇	—	2	1	no	—
147	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
148	C₈×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
149	#3393	D₁₇	1	1	yes	characteristic, complemented
150	Q₈×D₈	—	17	1	no	—
151	#2243	—	17	2	no	—
152	#2241	—	17	1	no	—
153	#2227	—	17	2	no	—
154	#2224	—	17	1	no	—
155	#2210	—	17	2	no	—
156	#2208	—	17	1	no	—
157	#2183	—	17	1	no	—
158	#2175	—	17	2	no	—
159	C₄×D₁₆	—	17	1	no	—
160	C₂²×C₁₇	—	8	1	no	—
161	C₂²×C₁₇	—	8	1	no	—
162	C₂²×C₁₇	—	8	1	no	—
163	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
164	C₂×D₁₇	—	4	2	no	—
165	C₂×D₁₇	—	8	1	no	—
166	C₂×D₁₇	—	8	1	no	—
167	C₄×C₁₇	—	2	2	no	—
168	C₄×C₁₇	—	2	2	no	—
169	C₄×C₁₇	—	2	1	no	—
170	C₄×C₁₇	—	8	2	no	—
171	C₄×C₁₇	—	8	1	no	—
172	C₄×C₁₇	#509	1	2	yes	—
173	C₄×C₁₇	#548	1	1	yes	characteristic
174	C₄×C₁₇	#577	1	1	yes	characteristic
175	Dic₁₇	—	8	1	no	—
176	Dic₁₇	—	8	2	no	—
177	#556	—	2	1	no	—
178	D₄×Q₈	—	34	1	no	—
179	C₂×SD₃₂	—	17	2	no	—
180	C₂×D₁₆	—	17	1	no	—
181	#500	—	17	1	no	—
182	#500	—	17	2	no	—
183	#495	C₂×D₁₇	1	2	yes	—
184	#494	C₂×D₁₇	1	1	yes	characteristic
185	#474	—	34	1	no	—
186	#474	—	34	2	no	—
187	#464	—	2	2	no	—
188	#461	—	2	1	no	—
189	C₈×Q₈	C₂×D₁₇	1	1	yes	characteristic
190	C₄×Q₁₆	C₂×D₁₇	1	2	yes	—
191	C₄×D₈	—	17	1	no	—
192	C₄×D₈	—	17	2	no	—
193	C₄×D₈	C₂×D₁₇	1	1	yes	characteristic
194	#366	—	17	1	no	—
195	#366	—	17	2	no	—
196	C₄⋊C₁₆	—	17	1	no	—
197	C₄⋊C₁₆	—	17	2	no	—
198	#358	—	17	2	no	—
199	#358	—	17	2	no	—
200	#358	—	17	2	no	—
201	#357	—	17	1	no	—
202	#357	—	17	2	no	—
203	#357	—	17	1	no	—
204	#346	—	17	2	no	—
205	C₄×C₁₆	—	17	1	no	—
206	C₂×C₁₇	—	16	1	no	—
207	C₂×C₁₇	#3358	1	1	yes	characteristic
208	C₂×C₁₇	C₂×#509	1	1	yes	characteristic
209	C₂×C₁₇	#3417	1	1	yes	characteristic
210	D₁₇	—	8	2	no	—
211	C₂²×Q₈	—	68	1	no	—
212	C₂×Q₁₆	—	2	1	no	—
213	C₂×Q₁₆	C₂²×D₁₇	1	2	yes	—
214	C₂×SD₁₆	—	2	2	no	—
215	C₂×D₈	—	17	1	no	—
216	C₂×D₈	C₂²×D₁₇	1	1	yes	characteristic
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	—	34	1	no	—
219	C₄⋊Q₈	—	34	2	no	—
220	#126	—	4	2	no	—
221	#124	—	2	2	no	—
222	C₂²⋊Q₈	—	4	1	no	—
223	C₂²⋊Q₈	—	68	1	no	—
224	C₂²⋊Q₈	—	68	2	no	—
225	C₄×Q₈	—	2	2	no	—
226	C₄×Q₈	C₂²⋊D₁₇	1	1	yes	characteristic
227	C₄×D₄	—	2	1	no	—
228	C₄×D₄	—	34	1	no	—
229	C₄×D₄	—	34	2	no	—
230	#117	—	4	1	no	—
231	SD₃₂	—	34	4	no	—
232	D₁₆	—	34	2	no	—
233	C₂×C₁₆	—	17	1	no	—
234	C₂×C₁₆	—	17	1	no	—
235	C₂×C₁₆	—	17	2	no	—
236	#107	—	17	1	no	—
237	#107	—	17	2	no	—
238	#107	—	34	1	no	—
239	#107	—	34	2	no	—
240	#107	C₂²×D₁₇	1	1	yes	characteristic
241	C₄⋊C₈	C₂²⋊D₁₇	1	1	yes	characteristic
242	C₄⋊C₈	C₂²⋊D₁₇	1	2	yes	—
243	#103	—	2	2	no	—
244	#102	—	2	1	no	—
245	#102	—	34	1	no	—
246	#102	—	34	2	no	—
247	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₁₇	1	2	yes	—
249	C₁₇	#33934	1	1	yes	characteristic, complemented, Sylow
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	68	2	no	—
252	C₂×Q₈	—	68	1	no	—
253	C₂×Q₈	—	136	4	no	—
254	C₂×Q₈	#63228	1	1	yes	characteristic
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	34	1	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	68	2	no	—
259	C₂²×C₄	—	68	1	no	—
260	Q₁₆	—	2	2	no	—
261	Q₁₆	—	2	2	no	—
262	Q₁₆	—	8	1	no	—
263	SD₁₆	—	8	2	no	—
264	D₈	—	2	1	no	—
265	D₈	—	2	1	no	—
266	D₈	—	17	2	no	—
267	D₈	—	34	1	no	—
268	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
269	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
270	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	34	1	no	—
276	C₄⋊C₄	—	34	2	no	—
277	C₄⋊C₄	—	68	2	no	—
278	C₄⋊C₄	—	68	1	no	—
279	C₄⋊C₄	#63228	1	1	yes	characteristic
280	C₄⋊C₄	#63228	1	2	yes	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	1	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	68	2	no	—
285	C₂²⋊C₄	—	68	1	no	—
286	C₄²	—	4	2	no	—
287	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
288	C₄²	C₂×C₂²⋊D₁₇	1	2	yes	—
289	C₁₆	—	34	1	no	—
290	C₁₆	—	34	2	no	—
291	C₁₆	—	34	1	no	—
292	C₂³	—	4	1	no	—
293	C₂³	—	68	1	no	—
294	Q₈	—	2	2	no	—
295	Q₈	—	4	2	no	—
296	Q₈	—	4	2	no	—
297	Q₈	—	136	4	no	—
298	Q₈	—	136	2	no	—
299	D₄	—	4	1	no	—
300	D₄	—	4	1	no	—
301	D₄	—	34	2	no	—
302	D₄	—	68	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	—	68	2	no	—
307	C₂×C₄	—	68	1	no	—
308	C₂×C₄	—	136	4	no	—
309	C₂×C₄	—	136	2	no	—
310	C₂×C₄	C₂×#63228	1	2	yes	—
311	C₂×C₄	C₂×#63228	1	1	yes	characteristic
312	C₂×C₄	C₂×#63228	1	2	yes	—
313	C₂×C₄	C₂×#63228	1	1	yes	characteristic
314	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	—	2	1	no	—
317	C₈	#41×D₁₇	1	1	yes	characteristic
318	C₈	#10587697	1	1	yes	characteristic
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	1	no	—
322	C₂²	—	68	2	no	—
323	C₂²	—	136	1	no	—
324	C₂²	—	136	1	no	—
325	C₂²	C₂²×#63228	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	8	1	no	—
330	C₄	—	8	2	no	—
331	C₄	—	136	2	no	—
332	C₄	—	136	1	no	—
333	C₄	#11763244	1	2	yes	—
334	C₄	#11763466	1	1	yes	characteristic
335	C₄	#11763620	1	1	yes	characteristic
336	C₂	—	16	1	no	—
337	C₂	—	136	2	no	—
338	C₂	#14541415	1	1	yes	characteristic
339	C₂	C₂×#11763244	1	1	yes	characteristic
340	C₂	#14541935	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#63228
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₁₇×#3393	C₂
socle	C₂²×C₁₇	C₂²×D₈
prime radical	C₂×#10587497	C₂²
hypercenter	#3393	D₁₇
hypocenter	C₁₇	#33934
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3393

Complements

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

N	H ≅ G/N	type
1. Q₈×D₁₃₆	C₂	semidirect
2. #14531020	C₂	semidirect
3. #14531011	C₂	semidirect
4. #14530936	C₂	semidirect
5. #14530921	C₂	semidirect
7. #14530826	C₂	semidirect
8. #14530693	C₂	semidirect
9. #14530646	C₂	semidirect
11. C₁₇×#3393	C₂	semidirect
149. #3393	D₁₇	semidirect
249. C₁₇	#33934	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₁₃₆	2
#14531020	2
#14531011	2
#14530936	2
#14530921	2
#14530836	2
#14530826	2
#14530693	2
#14530646	2
C₄×#10587497	2
C₁₇×#3393	2
#33934	17