Subgroups — #32097415

Order 18176.

608 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22598864	C₂	1	1	yes	maximal, characteristic, complemented
2	#22589512	C₂	1	4	yes	maximal, complemented
3	#22589239	C₂	1	2	yes	maximal
4	#22588465	C₂	1	2	yes	maximal, complemented
5	#22588394	C₂	1	2	yes	maximal, complemented
6	#22588386	C₂	1	2	yes	maximal, complemented
7	#22588224	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#444×C₇₁	C₂	1	1	yes	maximal, characteristic
9	#17986556	C₂²	1	2	yes	—
10	#17986408	—	2	4	no	—
11	C₂×#14556064	C₂²	1	1	yes	characteristic
12	#17986037	C₂²	1	2	yes	—
13	#17986037	C₂²	1	1	yes	characteristic
14	#17986037	C₂²	1	2	yes	—
15	#17986016	C₂²	1	4	yes	—
16	#17985973	C₂²	1	4	yes	—
17	#17985954	C₂²	1	2	yes	—
18	#17985771	C₂²	1	4	yes	—
19	#17985759	C₂²	1	2	yes	—
20	#17985756	C₂²	1	4	yes	—
21	C₂×#135×C₇₁	C₂²	1	1	yes	characteristic
22	C₂×#131×C₇₁	C₂²	1	2	yes	—
23	#444×C₇₁	—	2	4	no	—
24	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
25	C₂×#104×C₇₁	C₂²	1	2	yes	—
26	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
27	C₂×#11765849	C₂³	1	2	yes	—
28	#14556128	—	2	2	no	—
29	#14556120	—	2	2	no	—
30	#14556107	—	4	2	no	—
31	C₄×Dic₁₄₂	—	4	2	no	—
32	#14556087	C₂³	1	2	yes	—
33	#14556087	C₂³	1	4	yes	—
34	#14556064	—	2	2	no	—
35	#14556062	—	2	2	no	—
36	#14556062	—	2	2	no	—
37	#14556062	—	2	1	no	—
38	#14556060	—	2	2	no	—
39	#14556060	—	2	2	no	—
40	#14556060	—	2	1	no	—
41	#14556055	—	4	4	no	—
42	#14556043	—	4	4	no	—
43	#14556038	—	2	2	no	—
44	C₈×Dic₇₁	—	2	2	no	—
45	C₂×#41×C₇₁	C₂³	1	2	yes	—
46	#135×C₇₁	—	2	2	no	—
47	#135×C₇₁	—	2	2	no	—
48	C₂×Q₁₆×C₇₁	Q₈	1	1	yes	characteristic, complemented
49	C₂×SD₁₆×C₇₁	Q₈	1	2	yes	complemented
50	C₂×D₈×C₇₁	Q₈	1	1	yes	characteristic, complemented
51	#131×C₇₁	—	2	2	no	—
52	#131×C₇₁	—	2	2	no	—
53	#131×C₇₁	—	2	2	no	—
54	#131×C₇₁	—	2	2	no	—
55	C₂×M₁₆×C₇₁	—	2	2	no	—
56	C₂×M₁₆×C₇₁	C₂³	1	2	yes	—
57	C₂²×C₈×C₇₁	—	2	2	no	—
58	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
61	#108×C₇₁	—	2	2	no	—
62	#104×C₇₁	—	2	4	no	—
63	#97×C₇₁	D₄	1	4	yes	—
64	C₂×Dic₁₄₂	—	4	2	no	—
65	#11765849	—	2	2	no	—
66	#11765849	—	2	2	no	—
67	C₂×C₇₁⋊C₈	—	2	2	no	—
68	C₄⋊Dic₇₁	—	2	4	no	—
69	C₄⋊Dic₇₁	—	2	2	no	—
70	C₄⋊Dic₇₁	—	4	2	no	—
71	Dic₇₁⋊C₄	—	4	4	no	—
72	C₂²⋊Dic₇₁	—	2	2	no	—
73	C₂²⋊Dic₇₁	—	2	4	no	—
74	C₄×Dic₇₁	—	2	2	no	—
75	C₄×Dic₇₁	—	2	4	no	—
76	#41×C₇₁	—	2	2	no	—
77	#41×C₇₁	—	2	2	no	—
78	#41×C₇₁	—	2	2	no	—
79	#41×C₇₁	—	2	2	no	—
80	C₂×Q₈×C₇₁	C₂×Q₈	1	2	yes	—
81	C₂×D₄×C₇₁	—	2	2	no	—
82	C₂×D₄×C₇₁	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₇₁	—	2	1	no	—
84	C₂²×C₄×C₇₁	—	2	2	no	—
85	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
86	Q₁₆×C₇₁	—	2	2	no	—
87	SD₁₆×C₇₁	—	2	4	no	—
88	D₈×C₇₁	—	2	2	no	—
89	M₁₆×C₇₁	—	2	2	no	—
90	M₁₆×C₇₁	—	2	2	no	—
91	M₁₆×C₇₁	—	4	2	no	—
92	M₁₆×C₇₁	—	4	2	no	—
93	C₂×C₈×C₇₁	—	2	2	no	—
94	C₂×C₈×C₇₁	—	4	2	no	—
95	C₂×C₈×C₇₁	—	4	2	no	—
96	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
99	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₈×C₇₁	#41	1	2	yes	—
101	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
102	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
103	C₂×C₈×C₇₁	#41	1	2	yes	—
104	C₄²×C₇₁	—	2	1	no	—
105	C₄²×C₇₁	C₂×D₄	1	2	yes	—
106	C₂³×C₇₁	—	2	2	no	—
107	C₂³×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
108	Q₈×C₇₁	—	2	2	no	—
109	Q₈×C₇₁	—	2	2	no	—
110	D₄×C₇₁	—	2	2	no	—
111	D₄×C₇₁	—	2	2	no	—
112	D₄×C₇₁	—	4	2	no	—
113	D₄×C₇₁	—	4	2	no	—
114	C₂×C₄×C₇₁	—	2	2	no	—
115	C₂×C₄×C₇₁	—	2	2	no	—
116	C₂×C₄×C₇₁	—	4	2	no	—
117	C₂×C₄×C₇₁	—	4	2	no	—
118	C₂×C₄×C₇₁	—	8	1	no	—
119	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂×Dic₇₁	—	4	4	no	—
126	C₂×Dic₇₁	—	4	2	no	—
127	Dic₁₄₂	—	8	4	no	—
128	C₈×C₇₁	—	4	2	no	—
129	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
130	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
131	C₈×C₇₁	#124	1	2	yes	—
132	C₈×C₇₁	#124	1	2	yes	—
133	C₇₁⋊C₈	—	4	2	no	—
134	C₂²×C₇₁	—	4	1	no	—
135	C₂²×C₇₁	—	4	2	no	—
136	C₂²×C₇₁	—	8	2	no	—
137	C₂²×C₇₁	#535	1	1	yes	characteristic
138	C₂²×C₇₁	#554	1	1	yes	characteristic
139	C₂²×C₇₁	#555	1	1	yes	characteristic
140	C₄×C₇₁	—	4	2	no	—
141	C₄×C₇₁	—	4	2	no	—
142	C₄×C₇₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
143	C₄×C₇₁	C₂×#124	1	1	yes	characteristic
144	C₄×C₇₁	#554	1	1	yes	characteristic
145	C₄×C₇₁	#555	1	1	yes	characteristic
146	Dic₇₁	—	8	4	no	—
147	Dic₇₁	—	8	2	no	—
148	#29155	—	71	1	no	maximal, Sylow
149	C₂×C₇₁	—	2	1	no	—
150	C₂×C₇₁	—	2	1	no	—
151	C₂×C₇₁	—	2	1	no	—
152	C₂×C₇₁	—	8	2	no	—
153	C₂×C₇₁	#2563	1	1	yes	characteristic
154	#3162	—	71	1	no	—
155	C₂×#444	D₇₁	1	1	yes	characteristic
156	#2031	—	71	4	no	—
157	#1976	—	71	2	no	—
158	#1833	—	71	2	no	—
159	#1814	—	71	2	no	—
160	#1812	—	71	2	no	—
161	#1773	—	71	1	no	—
162	C₇₁	#29155	1	1	yes	characteristic, complemented, Sylow
163	C₂×#135	C₂×D₇₁	1	1	yes	characteristic
164	C₂×#131	C₂×D₇₁	1	2	yes	—
165	#533	—	71	2	no	—
166	#501	—	142	4	no	—
167	#444	—	2	4	no	—
168	C₂×#108	—	71	1	no	—
169	C₂×#108	C₂×D₇₁	1	1	yes	characteristic
170	#427	—	71	1	no	—
171	#427	—	71	2	no	—
172	#427	—	71	2	no	—
173	#421	—	71	4	no	—
174	C₂×#104	C₂×D₇₁	1	2	yes	—
175	#410	—	71	4	no	—
176	#405	—	71	2	no	—
177	C₂×#97	C₂×D₇₁	1	1	yes	characteristic
178	#344	—	71	4	no	—
179	#339	—	71	2	no	—
180	H(3,ℤ/4)	—	71	4	no	—
181	C₂×#41	C₂²×D₇₁	1	2	yes	—
182	#135	—	2	2	no	—
183	#135	—	2	2	no	—
184	C₂×Q₁₆	Dic₁₄₂	1	1	yes	characteristic, complemented
185	C₂×SD₁₆	Dic₁₄₂	1	2	yes	complemented
186	C₂×D₈	Dic₁₄₂	1	1	yes	characteristic, complemented
187	#131	—	2	2	no	—
188	#131	—	2	2	no	—
189	#131	—	2	2	no	—
190	#131	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	71	2	no	—
193	C₂×M₁₆	C₂²×D₇₁	1	2	yes	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
197	C₄⋊Q₈	—	142	2	no	—
198	#125	—	142	2	no	—
199	C₂²⋊Q₈	—	284	2	no	—
200	C₄×Q₈	—	284	2	no	—
201	#117	—	71	4	no	—
202	#117	—	71	2	no	—
203	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
204	#108	—	2	2	no	—
205	#108	—	142	2	no	—
206	#107	—	142	2	no	—
207	#107	—	142	2	no	—
208	#107	—	142	1	no	—
209	#106	—	142	2	no	—
210	#106	—	142	2	no	—
211	#106	—	142	1	no	—
212	#104	—	2	4	no	—
213	#104	—	284	4	no	—
214	#99	—	284	4	no	—
215	#97	—	142	2	no	—
216	#97	C₂²⋊D₇₁	1	4	yes	—
217	C₄×C₈	—	142	2	no	—
218	#41	—	2	2	no	—
219	#41	—	2	2	no	—
220	#41	—	2	2	no	—
221	#41	—	2	2	no	—
222	C₂×Q₈	—	284	2	no	—
223	C₂×Q₈	C₂×Dic₁₄₂	1	2	yes	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	C₂×Dic₁₄₂	1	2	yes	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
229	Q₁₆	—	2	2	no	—
230	SD₁₆	—	2	4	no	—
231	D₈	—	2	2	no	—
232	M₁₆	—	2	2	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	4	2	no	—
235	M₁₆	—	4	2	no	—
236	M₁₆	—	142	2	no	—
237	M₁₆	—	142	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	142	2	no	—
242	C₂×C₈	C₂×C₂²⋊D₇₁	1	2	yes	—
243	C₂×C₈	C₂×C₂²⋊D₇₁	1	2	yes	—
244	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
245	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
246	C₂×C₈	#11765864	1	1	yes	characteristic
247	C₂×C₈	#11765864	1	1	yes	characteristic
248	C₂×C₈	#11765864	1	2	yes	—
249	C₂×C₈	#11765865	1	2	yes	—
250	C₄⋊C₄	—	284	4	no	—
251	C₄⋊C₄	—	284	2	no	—
252	C₄⋊C₄	—	142	4	no	—
253	C₄⋊C₄	—	142	2	no	—
254	C₂²⋊C₄	—	142	2	no	—
255	C₂²⋊C₄	—	142	4	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	142	2	no	—
258	C₄²	—	142	4	no	—
259	C₄²	C₂×C₂²⋊D₇₁	1	2	yes	—
260	C₂³	—	2	2	no	—
261	C₂³	#14556174	1	1	yes	characteristic
262	Q₈	—	2	2	no	—
263	Q₈	—	2	2	no	—
264	Q₈	—	568	4	no	—
265	D₄	—	2	2	no	—
266	D₄	—	2	2	no	—
267	D₄	—	4	2	no	—
268	D₄	—	4	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	284	4	no	—
275	C₂×C₄	—	284	2	no	—
276	C₂×C₄	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
277	C₂×C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
278	C₂×C₄	C₂×#11765864	1	1	yes	characteristic
279	C₂×C₄	C₂×#11765864	1	1	yes	characteristic
280	C₂×C₄	C₂×#11765865	1	1	yes	characteristic
281	C₂×C₄	#14556174	1	1	yes	characteristic
282	C₈	—	4	2	no	—
283	C₈	—	284	2	no	—
284	C₈	#14556107	1	2	yes	—
285	C₈	#14556107	1	2	yes	—
286	C₈	#14556115	1	2	yes	—
287	C₈	#14556115	1	2	yes	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	8	2	no	—
291	C₂²	#17986563	1	1	yes	characteristic
292	C₂²	#17986706	1	1	yes	characteristic
293	C₂²	#17986714	1	1	yes	characteristic
294	C₄	—	4	2	no	—
295	C₄	—	4	2	no	—
296	C₄	—	568	4	no	—
297	C₄	—	568	2	no	—
298	C₄	C₂×#14556107	1	1	yes	characteristic
299	C₄	C₂×#14556115	1	1	yes	characteristic
300	C₄	#17986706	1	1	yes	characteristic
301	C₄	#17986714	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	8	2	no	—
306	C₂	#22593513	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17986714
commutator, cosocle	C₂²×C₄×C₇₁	C₂⁴
Frattini	C₂²×C₄	C₂³×D₇₁
Fitting	C₂×#444×C₇₁	C₂
socle	C₂×C₇₁	#2563
prime radical	C₂×#135×C₇₁	C₂²
hypercenter	C₂×#444	D₇₁
hypocenter	C₇₁	#29155
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#444

Complements

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

N	H ≅ G/N	type
1. #22598864	C₂	semidirect
2. #22589512	C₂	semidirect
4. #22588465	C₂	semidirect
5. #22588394	C₂	semidirect
6. #22588386	C₂	semidirect
7. #22588224	C₂	semidirect
48. C₂×Q₁₆×C₇₁	Q₈	semidirect
49. C₂×SD₁₆×C₇₁	Q₈	semidirect
50. C₂×D₈×C₇₁	Q₈	semidirect
162. C₇₁	#29155	semidirect
184. C₂×Q₁₆	Dic₁₄₂	semidirect
185. C₂×SD₁₆	Dic₁₄₂	semidirect
186. C₂×D₈	Dic₁₄₂	semidirect

Maximal subgroups

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

subgroup	index
#22598864	2
#22589512	2
#22589239	2
#22588465	2
#22588394	2
#22588386	2
#22588224	2
C₂×#444×C₇₁	2
#29155	71