Subgroups — #32155489

Order 18176.

498 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22601329	C₂	1	1	yes	maximal, characteristic
2	#22590609	C₂	1	2	yes	maximal
3	#22590592	C₂	1	1	yes	maximal, characteristic, complemented
4	#22590402	C₂	1	2	yes	maximal
5	#22590390	C₂	1	1	yes	maximal, characteristic, complemented
6	#22590387	C₂	1	2	yes	maximal, complemented
7	#22590382	C₂	1	1	yes	maximal, characteristic
8	#22590249	C₂	1	1	yes	maximal, characteristic, complemented
9	#22590209	C₂	1	2	yes	maximal, complemented
10	C₄×#14556071	C₂	1	1	yes	maximal, characteristic
11	C₇₁×#3395	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986722	—	2	1	no	—
13	C₂×#14556074	C₂²	1	2	yes	—
14	C₂×#14556074	C₂²	1	2	yes	—
15	C₂×#14556074	C₂²	1	2	yes	—
16	C₂×#14556071	C₂²	1	1	yes	characteristic
17	C₂×#14556071	C₂²	1	2	yes	—
18	#17986383	C₂²	1	2	yes	—
19	#17986379	C₂²	1	1	yes	characteristic
20	#17986222	—	2	2	no	—
21	#17986205	—	2	1	no	—
22	C₄×Dic₂₈₄	C₂²	1	2	yes	—
23	C₄×D₅₆₈	C₂²	1	1	yes	characteristic
24	#17985826	C₂²	1	1	yes	characteristic
25	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	1	yes	characteristic
26	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
27	#17985803	C₂²	1	2	yes	—
28	#17985803	C₂²	1	1	yes	characteristic
29	#17985802	C₂²	1	2	yes	—
30	#17985799	C₂²	1	1	yes	characteristic
31	#17985775	C₂²	1	2	yes	—
32	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
33	#557×C₇₁	—	2	1	no	—
34	#494×C₇₁	C₂²	1	1	yes	characteristic
35	#494×C₇₁	C₂²	1	2	yes	—
36	#462×C₇₁	—	2	1	no	—
37	#462×C₇₁	—	2	2	no	—
38	C₈×Q₈×C₇₁	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₇₁	C₂²	1	2	yes	—
40	C₄×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₈₄	—	2	1	no	—
42	C₂×Dic₂₈₄	C₂³	1	2	yes	—
43	C₂×#11765854	—	2	2	no	—
44	C₂×D₅₆₈	C₂³	1	1	yes	characteristic
45	#14556128	—	2	1	no	—
46	#14556123	—	4	2	no	—
47	#14556116	—	2	2	no	—
48	#14556104	—	4	1	no	—
49	C₄×Dic₁₄₂	—	2	2	no	—
50	C₄×D₂₈₄	—	2	1	no	—
51	#14556086	—	4	1	no	—
52	#14556074	—	2	4	no	—
53	#14556074	—	4	2	no	—
54	#14556074	—	4	2	no	—
55	#14556071	—	2	2	no	—
56	#14556071	—	4	2	no	—
57	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
59	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
60	#14556062	C₂³	1	1	yes	characteristic
61	#14556053	—	2	2	no	—
62	#14556050	—	2	1	no	—
63	C₂×Q₁₆×C₇₁	—	2	1	no	—
64	C₂×Q₁₆×C₇₁	—	2	2	no	—
65	C₂×Q₁₆×C₇₁	C₂³	1	1	yes	characteristic
66	C₂×Q₁₆×C₇₁	C₂³	1	2	yes	—
67	C₄⋊Q₈×C₇₁	—	2	1	no	—
68	C₄⋊Q₈×C₇₁	—	2	2	no	—
69	#125×C₇₁	—	4	1	no	—
70	#125×C₇₁	—	4	2	no	—
71	C₄×Q₈×C₇₁	—	2	1	no	—
72	C₄×Q₈×C₇₁	—	2	2	no	—
73	C₄×Q₈×C₇₁	—	4	1	no	—
74	C₄×Q₈×C₇₁	D₄	1	1	yes	characteristic
75	#107×C₇₁	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₇₁	D₄	1	1	yes	characteristic
77	C₄⋊C₈×C₇₁	D₄	1	2	yes	—
78	#103×C₇₁	—	2	1	no	—
79	#103×C₇₁	—	2	2	no	—
80	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₇₁	C₂³	1	2	yes	—
82	C₂×Dic₁₄₂	—	2	2	no	—
83	C₂×D₂₈₄	—	2	1	no	—
84	C₂×C₄×D₇₁	—	4	1	no	—
85	Dic₂₈₄	—	2	2	no	—
86	Dic₂₈₄	—	2	2	no	—
87	Dic₂₈₄	—	8	1	no	—
88	#11765854	—	8	2	no	—
89	D₅₆₈	—	2	1	no	—
90	D₅₆₈	—	2	1	no	—
91	C₄⋊Dic₇₁	—	2	1	no	—
92	C₄⋊Dic₇₁	—	4	1	no	—
93	C₄⋊Dic₇₁	—	4	1	no	—
94	Dic₇₁⋊C₄	—	4	2	no	—
95	(C₂×D₇₁)⋊C₄	—	4	2	no	—
96	(C₂×D₇₁)⋊C₄	—	4	1	no	—
97	(C₂×D₇₁)⋊C₄	—	4	2	no	—
98	C₄×Dic₇₁	—	4	2	no	—
99	C₇₁⋊C₁₆	—	2	2	no	—
100	C₇₁⋊C₁₆	—	2	1	no	—
101	C₇₁⋊C₁₆	#41	1	2	yes	—
102	C₂×Q₈×C₇₁	—	2	1	no	—
103	C₂×Q₈×C₇₁	—	2	2	no	—
104	C₂×Q₈×C₇₁	D₈	1	1	yes	characteristic
105	Q₁₆×C₇₁	—	2	1	no	—
106	Q₁₆×C₇₁	—	2	2	no	—
107	Q₁₆×C₇₁	—	2	2	no	—
108	Q₁₆×C₇₁	—	2	1	no	—
109	Q₁₆×C₇₁	—	8	2	no	—
110	Q₁₆×C₇₁	—	8	1	no	—
111	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₇₁	—	2	1	no	—
115	C₄⋊C₄×C₇₁	—	4	2	no	—
116	C₄⋊C₄×C₇₁	—	4	1	no	—
117	C₄⋊C₄×C₇₁	—	4	2	no	—
118	C₄⋊C₄×C₇₁	—	4	1	no	—
119	C₄⋊C₄×C₇₁	—	4	2	no	—
120	C₄⋊C₄×C₇₁	—	4	1	no	—
121	C₄⋊C₄×C₇₁	D₈	1	1	yes	characteristic
122	C₄⋊C₄×C₇₁	D₈	1	2	yes	—
123	C₄²×C₇₁	—	4	2	no	—
124	C₄²×C₇₁	—	4	1	no	—
125	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₇₁	C₂×D₄	1	2	yes	—
127	C₂²×D₇₁	—	4	1	no	—
128	Q₈×C₇₁	—	2	2	no	—
129	Q₈×C₇₁	—	4	1	no	—
130	Q₈×C₇₁	—	4	2	no	—
131	Q₈×C₇₁	—	4	1	no	—
132	Q₈×C₇₁	—	4	2	no	—
133	C₂×C₄×C₇₁	—	4	1	no	—
134	C₂×C₄×C₇₁	—	4	2	no	—
135	C₂×C₄×C₇₁	—	4	1	no	—
136	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
141	C₂×Dic₇₁	—	4	1	no	—
142	C₂×Dic₇₁	—	4	2	no	—
143	D₂₈₄	—	4	1	no	—
144	D₂₈₄	—	4	1	no	—
145	C₄×D₇₁	—	8	2	no	—
146	Dic₁₄₂	—	4	2	no	—
147	Dic₁₄₂	—	4	2	no	—
148	C₈×C₇₁	—	2	2	no	—
149	C₈×C₇₁	—	2	1	no	—
150	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
151	C₈×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
153	C₂×D₇₁	—	8	1	no	—
154	C₂×D₇₁	—	8	1	no	—
155	C₂×D₇₁	—	8	1	no	—
156	C₄×C₇₁	—	2	2	no	—
157	C₄×C₇₁	—	2	2	no	—
158	C₄×C₇₁	—	2	1	no	—
159	C₄×C₇₁	—	8	1	no	—
160	C₄×C₇₁	—	8	2	no	—
161	C₄×C₇₁	—	8	1	no	—
162	C₄×C₇₁	#510	1	2	yes	—
163	C₄×C₇₁	#550	1	1	yes	characteristic
164	C₄×C₇₁	#576	1	1	yes	characteristic
165	Dic₇₁	—	8	2	no	—
166	Dic₇₁	—	8	1	no	—
167	#33965	—	71	1	no	maximal, Sylow
168	C₂×C₇₁	#3391	1	1	yes	characteristic
169	C₂×C₇₁	C₂×#510	1	1	yes	characteristic
170	C₂×C₇₁	#3417	1	1	yes	characteristic
171	D₇₁	—	16	1	no	—
172	#3395	D₇₁	1	1	yes	characteristic, complemented
173	#3393	—	71	1	no	—
174	#2251	—	71	2	no	—
175	#2247	—	71	1	no	—
176	#2211	—	71	2	no	—
177	#2209	—	71	2	no	—
178	#2209	—	71	1	no	—
179	#2208	—	71	1	no	—
180	#2183	—	71	1	no	—
181	#2176	—	71	2	no	—
182	C₄×SD₃₂	—	71	1	no	—
183	C₇₁	#33965	1	1	yes	characteristic, complemented, Sylow
184	#557	—	2	1	no	—
185	#556	—	142	1	no	—
186	C₂×Q₃₂	—	71	2	no	—
187	C₂×Q₃₂	—	71	2	no	—
188	C₂×Q₃₂	—	71	2	no	—
189	C₂×SD₃₂	—	71	2	no	—
190	C₂×SD₃₂	—	71	1	no	—
191	#495	—	71	2	no	—
192	#494	—	71	1	no	—
193	#494	C₂×D₇₁	1	1	yes	characteristic
194	#494	C₂×D₇₁	1	2	yes	—
195	#464	—	142	2	no	—
196	#462	—	2	1	no	—
197	#462	—	2	2	no	—
198	#461	—	142	1	no	—
199	C₈×Q₈	C₂×D₇₁	1	1	yes	characteristic
200	C₄×Q₁₆	—	71	2	no	—
201	C₄×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
202	C₄×Q₁₆	C₂×D₇₁	1	2	yes	—
203	C₄×D₈	—	71	1	no	—
204	#367	—	71	1	no	—
205	C₄⋊C₁₆	—	71	1	no	—
206	C₄⋊C₁₆	—	71	2	no	—
207	#358	—	71	1	no	—
208	#358	—	71	2	no	—
209	#358	—	71	2	no	—
210	#357	—	71	1	no	—
211	#346	—	71	2	no	—
212	C₄×C₁₆	—	71	1	no	—
213	C₂×Q₁₆	—	2	1	no	—
214	C₂×Q₁₆	—	2	2	no	—
215	C₂×Q₁₆	—	71	2	no	—
216	C₂×Q₁₆	—	142	1	no	—
217	C₂×Q₁₆	C₂²×D₇₁	1	1	yes	characteristic
218	C₂×Q₁₆	C₂²×D₇₁	1	2	yes	—
219	C₂×SD₁₆	—	142	2	no	—
220	C₂×D₈	—	71	1	no	—
221	C₄⋊Q₈	—	2	1	no	—
222	C₄⋊Q₈	—	2	2	no	—
223	C₄⋊Q₈	—	142	1	no	—
224	#126	—	284	2	no	—
225	#125	—	4	1	no	—
226	#125	—	4	2	no	—
227	#124	—	142	2	no	—
228	C₂²⋊Q₈	—	284	1	no	—
229	C₄×Q₈	—	2	1	no	—
230	C₄×Q₈	—	2	2	no	—
231	C₄×Q₈	—	4	1	no	—
232	C₄×Q₈	—	142	2	no	—
233	C₄×Q₈	C₂²⋊D₇₁	1	1	yes	characteristic
234	C₄×D₄	—	142	1	no	—
235	#117	—	284	1	no	—
236	Q₃₂	—	284	2	no	—
237	Q₃₂	—	284	2	no	—
238	Q₃₂	—	142	4	no	—
239	SD₃₂	—	284	2	no	—
240	SD₃₂	—	142	2	no	—
241	C₂×C₁₆	—	71	1	no	—
242	C₂×C₁₆	—	71	1	no	—
243	C₂×C₁₆	—	71	2	no	—
244	#107	—	71	1	no	—
245	#107	C₂²×D₇₁	1	1	yes	characteristic
246	C₄⋊C₈	C₂²⋊D₇₁	1	1	yes	characteristic
247	C₄⋊C₈	C₂²⋊D₇₁	1	2	yes	—
248	#103	—	2	1	no	—
249	#103	—	2	2	no	—
250	#103	—	142	2	no	—
251	#102	—	142	1	no	—
252	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
253	C₄×C₈	C₂²×D₇₁	1	2	yes	—
254	C₂×Q₈	—	2	1	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	142	2	no	—
257	C₂×Q₈	#11765850	1	1	yes	characteristic
258	C₂×D₄	—	142	1	no	—
259	C₂²×C₄	—	284	1	no	—
260	Q₁₆	—	2	1	no	—
261	Q₁₆	—	2	2	no	—
262	Q₁₆	—	2	2	no	—
263	Q₁₆	—	2	1	no	—
264	Q₁₆	—	8	2	no	—
265	Q₁₆	—	8	1	no	—
266	Q₁₆	—	568	1	no	—
267	Q₁₆	—	142	2	no	—
268	Q₁₆	—	142	2	no	—
269	SD₁₆	—	568	2	no	—
270	D₈	—	142	1	no	—
271	D₈	—	142	1	no	—
272	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
273	C₂×C₈	C₂×C₂²⋊D₇₁	1	2	yes	—
274	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	284	2	no	—
283	C₄⋊C₄	—	284	1	no	—
284	C₄⋊C₄	—	284	1	no	—
285	C₄⋊C₄	—	142	1	no	—
286	C₄⋊C₄	#11765850	1	1	yes	characteristic
287	C₄⋊C₄	#11765850	1	2	yes	—
288	C₂²⋊C₄	—	284	2	no	—
289	C₂²⋊C₄	—	284	1	no	—
290	C₂²⋊C₄	—	284	2	no	—
291	C₄²	—	4	2	no	—
292	C₄²	—	4	1	no	—
293	C₄²	—	284	2	no	—
294	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
295	C₄²	C₂×C₂²⋊D₇₁	1	2	yes	—
296	C₁₆	—	71	2	no	—
297	C₁₆	—	142	2	no	—
298	C₁₆	—	142	1	no	—
299	C₂³	—	284	1	no	—
300	Q₈	—	2	2	no	—
301	Q₈	—	4	1	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	4	1	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	284	2	no	—
306	Q₈	—	284	2	no	—
307	D₄	—	284	1	no	—
308	D₄	—	284	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	284	1	no	—
313	C₂×C₄	—	284	2	no	—
314	C₂×C₄	—	568	2	no	—
315	C₂×C₄	C₂×#11765850	1	2	yes	—
316	C₂×C₄	C₂×#11765850	1	2	yes	—
317	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
318	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
319	C₂×C₄	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
320	C₈	—	2	2	no	—
321	C₈	—	2	1	no	—
322	C₈	#41×D₇₁	1	1	yes	characteristic
323	C₈	#14556174	1	1	yes	characteristic
324	C₂²	—	568	1	no	—
325	C₂²	—	568	1	no	—
326	C₂²	—	568	1	no	—
327	C₂²	C₂²×#11765850	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	2	no	—
331	C₄	—	8	1	no	—
332	C₄	—	8	1	no	—
333	C₄	—	8	2	no	—
334	C₄	—	568	2	no	—
335	C₄	—	568	1	no	—
336	C₄	#17986440	1	2	yes	—
337	C₄	#17986668	1	1	yes	characteristic
338	C₄	#17986802	1	1	yes	characteristic
339	C₂	—	1136	1	no	—
340	C₂	#22601312	1	1	yes	characteristic
341	C₂	C₂×#17986440	1	1	yes	characteristic
342	C₂	#22601491	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11765850
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₇₁×#3395	C₂
socle	C₂²×C₇₁	C₂²×D₈
prime radical	C₂×D₅₆₈	C₂³
hypercenter	#3395	D₇₁
hypocenter	C₇₁	#33965
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3395

Complements

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

N	H ≅ G/N	type
3. #22590592	C₂	semidirect
5. #22590390	C₂	semidirect
6. #22590387	C₂	semidirect
8. #22590249	C₂	semidirect
9. #22590209	C₂	semidirect
11. C₇₁×#3395	C₂	semidirect
172. #3395	D₇₁	semidirect
183. C₇₁	#33965	semidirect

Maximal subgroups

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

subgroup	index
#22601329	2
#22590609	2
#22590592	2
#22590402	2
#22590390	2
#22590387	2
#22590382	2
#22590249	2
#22590209	2
C₄×#14556071	2
C₇₁×#3395	2
#33965	71