Subgroups — #32006255

Order 18176.

750 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986401	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556120	C₂	1	2	yes	maximal
3	#22589305	C₂	1	2	yes	maximal
4	#22589083	C₂	1	2	yes	maximal
5	#22589078	C₂	1	2	yes	maximal
6	#22588504	C₂	1	1	yes	maximal, characteristic
7	#22588237	C₂	1	2	yes	maximal
8	#22588237	C₂	1	2	yes	maximal
9	C₄×#97×C₇₁	C₂	1	1	yes	maximal, characteristic
10	C₂×#14556120	C₂²	1	2	yes	—
11	#17986401	C₄	1	8	yes	complemented
12	C₂×#14556062	C₂²	1	2	yes	—
13	C₂×#14556062	C₂²	1	2	yes	—
14	C₂×#14556060	C₂²	1	2	yes	—
15	C₂×#14556060	C₂²	1	2	yes	—
16	#17985897	C₂²	1	4	yes	—
17	#17985883	—	2	4	no	—
18	#17985865	C₂²	1	2	yes	—
19	C₄×Dic₇₁⋊C₄	—	2	4	no	—
20	C₄×C₄⋊Dic₇₁	C₂²	1	4	yes	—
21	#17985762	C₂²	1	4	yes	—
22	#17985762	C₂²	1	4	yes	—
23	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
24	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
25	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
26	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
29	C₄³×C₇₁	C₂²	1	1	yes	characteristic
30	#14556120	—	2	4	no	—
31	#14556120	C₂×C₄	1	8	yes	—
32	C₂×C₄⋊Dic₇₁	C₂³	1	4	yes	—
33	C₂×C₄⋊Dic₇₁	C₂³	1	4	yes	—
34	C₂×Dic₇₁⋊C₄	—	2	4	no	—
35	C₂×Dic₇₁⋊C₄	—	2	4	no	—
36	C₂×C₄×Dic₇₁	—	2	4	no	—
37	#14556062	—	2	4	no	—
38	#14556062	C₂×C₄	1	8	yes	—
39	#14556060	—	2	4	no	—
40	#14556060	C₂×C₄	1	8	yes	—
41	C₇₁⋊#95	—	2	4	no	—
42	C₇₁⋊#95	—	2	4	no	—
43	C₇₁⋊#95	—	2	4	no	—
44	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₇₁	D₄	1	4	yes	—
49	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
52	#97×C₇₁	—	2	2	no	—
53	#97×C₇₁	—	2	2	no	—
54	#97×C₇₁	Q₈	1	4	yes	—
55	#97×C₇₁	Q₈	1	4	yes	—
56	C₄×C₈×C₇₁	—	2	2	no	—
57	C₄×C₈×C₇₁	C₂×C₄	1	4	yes	—
58	C₂²×Dic₇₁	—	2	4	no	—
59	C₂²×Dic₇₁	—	2	4	no	—
60	C₄⋊Dic₇₁	—	2	4	no	—
61	C₄⋊Dic₇₁	—	2	4	no	—
62	C₄⋊Dic₇₁	—	2	4	no	—
63	C₄⋊Dic₇₁	C₂²×C₄	1	8	yes	—
64	Dic₇₁⋊C₄	—	2	16	no	—
65	Dic₇₁⋊C₄	—	4	8	no	—
66	C₄×Dic₇₁	—	4	8	no	—
67	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
71	C₂×C₈×C₇₁	—	2	1	no	—
72	C₂×C₈×C₇₁	—	2	1	no	—
73	C₂×C₈×C₇₁	—	2	1	no	—
74	C₂×C₈×C₇₁	—	2	1	no	—
75	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
78	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
79	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
80	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
81	C₂×C₈×C₇₁	#41	1	2	yes	—
82	C₂×C₈×C₇₁	#41	1	2	yes	—
83	C₄²×C₇₁	—	2	1	no	—
84	C₄²×C₇₁	—	2	1	no	—
85	C₄²×C₇₁	—	4	4	no	—
86	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
87	C₄²×C₇₁	C₂×Q₈	1	2	yes	—
88	C₄²×C₇₁	C₂×Q₈	1	2	yes	—
89	C₄²×C₇₁	#41	1	2	yes	—
90	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₇₁	—	2	4	no	—
92	C₂×C₄×C₇₁	—	2	4	no	—
93	C₂×C₄×C₇₁	—	2	2	no	—
94	C₂×C₄×C₇₁	—	2	4	no	—
95	C₂×C₄×C₇₁	C₄×D₄	1	4	yes	—
96	C₂×C₄×C₇₁	C₄⋊Q₈	1	4	yes	—
97	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
101	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
102	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
103	C₂×Dic₇₁	—	2	8	no	—
104	C₂×Dic₇₁	—	4	4	no	—
105	C₂×Dic₇₁	—	4	4	no	—
106	C₂×Dic₇₁	—	4	4	no	—
107	C₂×Dic₇₁	—	4	4	no	—
108	C₂×Dic₇₁	—	4	4	no	—
109	C₈×C₇₁	—	2	2	no	—
110	C₈×C₇₁	—	2	2	no	—
111	C₈×C₇₁	C₄×Q₈	1	4	yes	—
112	C₈×C₇₁	C₄×Q₈	1	4	yes	—
113	C₂²×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
114	C₂²×C₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
115	C₂²×C₇₁	#532	1	1	yes	characteristic
116	C₂²×C₇₁	#575	1	4	yes	—
117	C₄×C₇₁	—	2	4	no	—
118	C₄×C₇₁	—	2	4	no	—
119	C₄×C₇₁	—	4	2	no	—
120	C₄×C₇₁	C₂×C₄×Q₈	1	1	yes	characteristic
121	C₄×C₇₁	C₂×C₄×Q₈	1	1	yes	characteristic
122	C₄×C₇₁	#533	1	1	yes	characteristic
123	C₄×C₇₁	#533	1	1	yes	characteristic
124	Dic₇₁	—	4	8	no	—
125	Dic₇₁	—	8	4	no	—
126	#21728	—	71	1	no	maximal, Sylow
127	C₂×C₇₁	C₄×C₄⋊Q₈	1	1	yes	characteristic
128	C₂×C₇₁	#2943	1	2	yes	—
129	C₂×C₇₁	#3166	1	2	yes	—
130	C₂×C₇₁	#3183	1	2	yes	—
131	C₂×#499	—	71	1	no	—
132	C₄×#125	—	71	2	no	—
133	#1991	—	71	2	no	—
134	#1944	—	71	2	no	—
135	#1943	—	71	2	no	—
136	#1846	—	71	1	no	—
137	#1776	—	71	2	no	—
138	#1776	—	71	2	no	—
139	C₄×#97	D₇₁	1	1	yes	characteristic
140	C₇₁	#21728	1	1	yes	characteristic, complemented, Sylow
141	C₂×#125	—	71	2	no	—
142	#499	—	71	8	no	—
143	C₂×#107	—	71	2	no	—
144	C₂×#107	—	71	2	no	—
145	C₂×#106	—	71	2	no	—
146	C₂×#106	—	71	2	no	—
147	C₂×#97	C₂×D₇₁	1	1	yes	characteristic
148	C₂×#97	C₂×D₇₁	1	1	yes	characteristic
149	C₂×#97	C₂×D₇₁	1	1	yes	characteristic
150	C₂×#97	C₂×D₇₁	1	1	yes	characteristic
151	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
153	#389	—	71	4	no	—
154	#387	—	142	4	no	—
155	#382	—	71	2	no	—
156	C₄×C₄⋊C₄	—	71	4	no	—
157	C₄×C₄⋊C₄	—	142	4	no	—
158	C₄³	C₂×D₇₁	1	1	yes	characteristic
159	#340	—	71	4	no	—
160	#340	—	71	4	no	—
161	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
162	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
163	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
165	#125	—	71	8	no	—
166	#125	—	142	4	no	—
167	C₂×C₄⋊C₄	—	71	4	no	—
168	C₂×C₄⋊C₄	—	71	4	no	—
169	C₂×C₄⋊C₄	—	142	4	no	—
170	C₂×C₄⋊C₄	—	142	4	no	—
171	C₂×C₄²	—	142	4	no	—
172	C₂×C₄²	D₂₈₄	1	4	yes	—
173	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
175	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
176	#107	—	71	8	no	—
177	#107	—	142	4	no	—
178	#106	—	71	8	no	—
179	#106	—	142	4	no	—
180	#97	—	2	2	no	—
181	#97	—	2	2	no	—
182	#97	Dic₁₄₂	1	4	yes	—
183	#97	Dic₁₄₂	1	4	yes	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	C₄×D₇₁	1	4	yes	—
186	#95	—	142	4	no	—
187	#95	—	142	4	no	—
188	#95	—	142	4	no	—
189	C₂²×C₄	—	142	4	no	—
190	C₂²×C₄	—	142	4	no	—
191	C₂²×C₄	C₂×D₂₈₄	1	2	yes	—
192	C₂²×C₄	C₂×D₂₈₄	1	2	yes	—
193	C₂²×C₄	C₂×D₂₈₄	1	2	yes	—
194	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
200	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
201	C₂×C₈	C₂×Dic₁₄₂	1	2	yes	—
202	C₂×C₈	C₂×Dic₁₄₂	1	2	yes	—
203	C₂×C₈	C₂×Dic₁₄₂	1	2	yes	—
204	C₂×C₈	C₂×Dic₁₄₂	1	2	yes	—
205	C₂×C₈	#11765865	1	2	yes	—
206	C₂×C₈	#11765865	1	2	yes	—
207	C₄⋊C₄	—	284	8	no	—
208	C₄⋊C₄	—	71	8	no	—
209	C₄⋊C₄	—	142	4	no	—
210	C₄⋊C₄	—	142	4	no	—
211	C₄⋊C₄	—	142	4	no	—
212	C₄⋊C₄	—	142	16	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	4	4	no	—
216	C₄²	—	284	8	no	—
217	C₄²	C₂×C₄×D₇₁	1	2	yes	—
218	C₄²	C₂×Dic₁₄₂	1	2	yes	—
219	C₄²	C₂×Dic₁₄₂	1	2	yes	—
220	C₄²	#11765865	1	2	yes	—
221	C₂³	C₂²×D₂₈₄	1	1	yes	characteristic
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	284	4	no	—
227	C₂×C₄	—	284	4	no	—
228	C₂×C₄	—	284	4	no	—
229	C₂×C₄	—	284	4	no	—
230	C₂×C₄	—	284	4	no	—
231	C₂×C₄	—	142	8	no	—
232	C₂×C₄	C₄×D₂₈₄	1	4	yes	—
233	C₂×C₄	#14556128	1	4	yes	—
234	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
235	C₂×C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
236	C₂×C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
237	C₂×C₄	C₂×#11765865	1	1	yes	characteristic
238	C₂×C₄	C₂×#11765865	1	1	yes	characteristic
239	C₂×C₄	C₂×#11765865	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	C₄×Dic₁₄₂	1	4	yes	—
243	C₈	C₄×Dic₁₄₂	1	4	yes	—
244	C₂²	C₂×C₄×D₂₈₄	1	1	yes	characteristic
245	C₂²	C₂×#14556128	1	1	yes	characteristic
246	C₂²	#17986549	1	1	yes	characteristic
247	C₂²	#17986799	1	4	yes	—
248	C₄	—	2	4	no	—
249	C₄	—	2	4	no	—
250	C₄	—	4	2	no	—
251	C₄	—	284	8	no	—
252	C₄	—	568	4	no	—
253	C₄	C₂×C₄×Dic₁₄₂	1	1	yes	characteristic
254	C₄	C₂×C₄×Dic₁₄₂	1	1	yes	characteristic
255	C₄	#17986556	1	1	yes	characteristic
256	C₄	#17986556	1	1	yes	characteristic
257	C₂	C₄×#14556128	1	1	yes	characteristic
258	C₂	#22597142	1	2	yes	—
259	C₂	#22598892	1	2	yes	—
260	C₂	#22599069	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₁
Fitting	C₄×#97×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂²×D₄
cosocle	C₂²×C₄×C₇₁	C₂⁴
hypercenter	C₄×#97	D₇₁
hypocenter	C₇₁	#21728
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
11. #17986401	C₄	semidirect
140. C₇₁	#21728	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986401	2
C₄×#14556120	2
#22589305	2
#22589083	2
#22589078	2
#22588504	2
#22588237	2
C₄×#97×C₇₁	2
#21728	71