Subgroups — #31968313

Order 18176.

544 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597435	C₂	1	2	yes	maximal
2	#22587212	C₂	1	4	yes	maximal, complemented
3	#22587191	C₂	1	4	yes	maximal, complemented
4	#22587103	C₂	1	1	yes	maximal, characteristic
5	#22586813	C₂	1	2	yes	maximal
6	#22586632	C₂	1	1	yes	maximal, characteristic
7	C₇₁×#3170	C₂	1	1	yes	maximal, characteristic
8	#17986130	—	2	4	no	—
9	#17986130	C₂²	1	4	yes	—
10	Q₈×C₇₁⋊C₈	C₂²	1	4	yes	—
11	#17986060	—	2	4	no	—
12	#17986056	C₂²	1	2	yes	—
13	#17986052	C₂²	1	1	yes	characteristic
14	#17986052	C₂²	1	1	yes	characteristic
15	#17985752	C₂²	1	2	yes	—
16	#17985750	C₂²	1	2	yes	—
17	Q₈⋊(C₇₁⋊C₈)	C₂²	1	8	yes	—
18	#17985721	C₂²	1	2	yes	—
19	#17985721	C₂²	1	2	yes	—
20	#533×C₇₁	C₂²	1	2	yes	—
21	#500×C₇₁	C₄	1	1	yes	characteristic
22	#494×C₇₁	C₄	1	1	yes	characteristic
23	#487×C₇₁	C₄	1	2	yes	—
24	#467×C₇₁	C₄	1	2	yes	—
25	#467×C₇₁	C₄	1	2	yes	—
26	C₄×Q₁₆×C₇₁	C₂²	1	2	yes	—
27	C₄×Q₁₆×C₇₁	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
29	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
30	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
31	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
32	C₄⋊(C₇₁⋊C₈)	D₄	1	2	yes	—
33	C₄⋊(C₇₁⋊C₈)	C₂³	1	4	yes	—
34	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
35	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
36	C₂²⋊(C₇₁⋊C₈)	D₄	1	2	yes	—
37	#14556039	—	2	4	no	—
38	#14556039	—	2	2	no	—
39	C₄×C₇₁⋊C₈	—	2	4	no	—
40	C₄×C₇₁⋊C₈	C₂³	1	2	yes	—
41	C₄×C₇₁⋊C₈	C₂³	1	2	yes	—
42	C₂×Q₁₆×C₇₁	C₂×C₄	1	2	yes	—
43	C₂×Q₁₆×C₇₁	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
46	C₄⋊Q₈×C₇₁	C₂×C₄	1	2	yes	—
47	C₄⋊Q₈×C₇₁	C₂×C₄	1	2	yes	—
48	#125×C₇₁	—	2	2	no	—
49	C₂²⋊Q₈×C₇₁	—	2	2	no	—
50	C₂²⋊Q₈×C₇₁	C₂×C₄	1	4	yes	—
51	C₄×Q₈×C₇₁	—	2	2	no	—
52	C₄×Q₈×C₇₁	C₂³	1	4	yes	—
53	#117×C₇₁	—	2	2	no	—
54	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
55	#107×C₇₁	C₂×C₄	1	2	yes	—
56	#107×C₇₁	C₂×C₄	1	2	yes	—
57	#103×C₇₁	C₂×C₄	1	4	yes	—
58	#103×C₇₁	C₂×C₄	1	4	yes	—
59	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
63	C₂×C₇₁⋊C₈	—	2	4	no	—
64	C₂×C₇₁⋊C₈	—	2	2	no	—
65	C₂×C₇₁⋊C₈	C₂×D₄	1	4	yes	—
66	C₂×Q₈×C₇₁	—	2	2	no	—
67	C₂×Q₈×C₇₁	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
71	Q₁₆×C₇₁	—	4	2	no	—
72	Q₁₆×C₇₁	—	4	2	no	—
73	C₂×C₈×C₇₁	—	4	1	no	—
74	C₂×C₈×C₇₁	—	4	1	no	—
75	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
79	C₄⋊C₄×C₇₁	—	2	2	no	—
80	C₄⋊C₄×C₇₁	—	2	2	no	—
81	C₄⋊C₄×C₇₁	—	2	2	no	—
82	C₄⋊C₄×C₇₁	—	2	4	no	—
83	C₄⋊C₄×C₇₁	—	2	4	no	—
84	C₄⋊C₄×C₇₁	C₂²×C₄	1	4	yes	—
85	C₂²⋊C₄×C₇₁	—	2	4	no	—
86	C₄²×C₇₁	—	2	4	no	—
87	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₇₁	#41	1	1	yes	characteristic
90	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
91	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
92	Q₈×C₇₁	—	2	4	no	—
93	Q₈×C₇₁	—	2	4	no	—
94	Q₈×C₇₁	—	8	2	no	—
95	C₂×C₄×C₇₁	—	2	1	no	—
96	C₂×C₄×C₇₁	—	2	1	no	—
97	C₂×C₄×C₇₁	—	2	2	no	—
98	C₂×C₄×C₇₁	—	2	4	no	—
99	C₂×C₄×C₇₁	—	2	4	no	—
100	C₂×C₄×C₇₁	—	4	1	no	—
101	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
107	C₈×C₇₁	—	2	1	no	—
108	C₈×C₇₁	—	2	1	no	—
109	C₈×C₇₁	—	2	1	no	—
110	C₈×C₇₁	—	2	1	no	—
111	C₇₁⋊C₈	—	2	4	no	—
112	C₇₁⋊C₈	—	4	2	no	—
113	C₇₁⋊C₈	—	4	4	no	—
114	C₂²×C₇₁	—	2	1	no	—
115	C₂²×C₇₁	—	2	1	no	—
116	C₂²×C₇₁	—	2	1	no	—
117	C₂²×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
118	C₄×C₇₁	—	2	1	no	—
119	C₄×C₇₁	—	2	1	no	—
120	C₄×C₇₁	—	2	1	no	—
121	C₄×C₇₁	—	2	1	no	—
122	C₄×C₇₁	—	4	4	no	—
123	C₄×C₇₁	—	4	4	no	—
124	C₄×C₇₁	#568	1	1	yes	characteristic
125	C₄×C₇₁	#568	1	1	yes	characteristic
126	C₄×C₇₁	#578	1	2	yes	—
127	#17991	—	71	1	no	maximal, Sylow
128	C₂×C₇₁	—	4	1	no	—
129	C₂×C₇₁	#2932	1	1	yes	characteristic
130	C₂×C₇₁	#2950	1	1	yes	characteristic
131	C₂×C₇₁	#2957	1	1	yes	characteristic
132	#3170	D₇₁	1	1	yes	characteristic
133	#2985	—	71	2	no	—
134	#1587	—	71	4	no	—
135	#1584	—	71	4	no	—
136	#1568	—	71	1	no	—
137	#1497	—	71	2	no	—
138	#1452	—	71	1	no	—
139	C₇₁	#17991	1	1	yes	characteristic, complemented, Sylow
140	#533	C₂×D₇₁	1	2	yes	—
141	#500	Dic₇₁	1	1	yes	characteristic
142	#494	Dic₇₁	1	1	yes	characteristic
143	#487	Dic₇₁	1	2	yes	—
144	#467	Dic₇₁	1	2	yes	—
145	#467	Dic₇₁	1	2	yes	—
146	#446	—	71	4	no	—
147	#446	—	142	4	no	—
148	C₈×Q₈	—	71	4	no	—
149	C₄×Q₁₆	C₂×D₇₁	1	2	yes	—
150	C₄×Q₁₆	C₂×D₇₁	1	2	yes	—
151	#433	—	142	4	no	—
152	#432	—	71	2	no	—
153	#431	—	71	1	no	—
154	#431	—	71	1	no	—
155	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
156	#335	—	71	2	no	—
157	#334	—	71	2	no	—
158	Q₈⋊C₈	—	71	8	no	—
159	#322	—	71	2	no	—
160	#322	—	71	2	no	—
161	C₂×Q₁₆	C₂×Dic₇₁	1	2	yes	—
162	C₂×Q₁₆	C₂×Dic₇₁	1	2	yes	—
163	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
164	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
165	C₄⋊Q₈	C₂×Dic₇₁	1	2	yes	—
166	C₄⋊Q₈	C₂×Dic₇₁	1	2	yes	—
167	#125	—	2	2	no	—
168	C₂²⋊Q₈	—	2	2	no	—
169	C₂²⋊Q₈	C₂×Dic₇₁	1	4	yes	—
170	C₄×Q₈	—	2	2	no	—
171	C₄×Q₈	C₂²×D₇₁	1	4	yes	—
172	#117	—	2	2	no	—
173	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
174	#107	C₂×Dic₇₁	1	2	yes	—
175	#107	C₂×Dic₇₁	1	2	yes	—
176	C₄⋊C₈	—	71	2	no	—
177	C₄⋊C₈	—	71	4	no	—
178	C₄⋊C₈	—	142	4	no	—
179	C₄⋊C₈	—	142	4	no	—
180	C₄⋊C₈	—	142	1	no	—
181	#103	C₂×Dic₇₁	1	4	yes	—
182	#103	C₂×Dic₇₁	1	4	yes	—
183	C₂²⋊C₈	—	71	2	no	—
184	C₂²⋊C₈	—	142	2	no	—
185	C₂²⋊C₈	—	142	1	no	—
186	#97	—	142	4	no	—
187	#97	—	142	2	no	—
188	C₄×C₈	—	71	2	no	—
189	C₄×C₈	—	71	2	no	—
190	C₄×C₈	—	142	4	no	—
191	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
192	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	C₂²×Dic₇₁	1	4	yes	—
197	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
198	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
199	C₂²×C₄	#11765864	1	1	yes	characteristic
200	Q₁₆	—	4	2	no	—
201	Q₁₆	—	4	2	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	71	4	no	—
205	C₂×C₈	—	142	4	no	—
206	C₂×C₈	—	142	2	no	—
207	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
208	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
209	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
210	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	C₂²×Dic₇₁	1	4	yes	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₄²	—	2	4	no	—
219	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
220	C₄²	D₄×D₇₁	1	1	yes	characteristic
221	C₄²	#11765864	1	1	yes	characteristic
222	C₄²	C₂³×D₇₁	1	1	yes	characteristic
223	C₂³	D₄×Dic₇₁	1	1	yes	characteristic
224	Q₈	—	2	4	no	—
225	Q₈	—	2	4	no	—
226	Q₈	—	8	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
234	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
235	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
236	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
237	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
238	C₂×C₄	C₂×#11765864	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	284	2	no	—
244	C₈	—	284	4	no	—
245	C₈	—	142	4	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	C₂×D₄×Dic₇₁	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	4	no	—
255	C₄	—	4	4	no	—
256	C₄	#17986766	1	1	yes	characteristic
257	C₄	#17986766	1	1	yes	characteristic
258	C₄	#17986812	1	2	yes	—
259	C₂	—	4	1	no	—
260	C₂	#22597058	1	1	yes	characteristic
261	C₂	#22597208	1	1	yes	characteristic
262	C₂	#22597257	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂³×C₄
Frattini	C₄²	C₂³×D₇₁
Fitting	C₇₁×#3170	C₂
socle	C₂²×C₇₁	C₂×C₄×D₄
prime radical	C₂³×C₇₁	C₄×D₄
cosocle	C₄²×C₇₁	C₂⁴
hypercenter	#3170	D₇₁
hypocenter	C₇₁	#17991
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #22587212	C₂	semidirect
3. #22587191	C₂	semidirect
139. C₇₁	#17991	semidirect

Maximal subgroups

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

subgroup	index
#22597435	2
#22587212	2
#22587191	2
#22587103	2
#22586813	2
#22586632	2
C₇₁×#3170	2
#17991	71