Subgroups — #31010335

Order 17152.

650 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₅₃₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#22569864	C₂	1	2	yes	maximal
3	#22562317	C₂	1	4	yes	maximal, complemented
4	#22562300	C₂	1	4	yes	maximal, complemented
5	#22561993	C₂	1	1	yes	maximal, characteristic, complemented
6	#22561930	C₂	1	2	yes	maximal, complemented
7	C₂×#504×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₆₈	C₂²	1	2	yes	—
9	#16863967	—	2	2	no	—
10	C₂×Dic₅₃₆	—	2	4	no	—
11	C₂×Dic₅₃₆	C₂²	1	2	yes	—
12	C₂×Dic₅₃₆	C₂²	1	2	yes	—
13	#16863654	—	2	4	no	—
14	#16863637	C₂²	1	8	yes	—
15	#16863434	C₂²	1	2	yes	—
16	#16863406	—	2	2	no	—
17	#16863224	C₂²	1	2	yes	—
18	#16863222	C₂²	1	2	yes	—
19	#16863201	C₂²	1	8	yes	—
20	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
21	#504×C₆₇	—	2	4	no	—
22	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
23	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
25	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₃₄	—	2	2	no	—
27	C₂×Dic₂₆₈	—	2	4	no	—
28	C₂×Dic₂₆₈	—	2	2	no	—
29	C₂×Dic₂₆₈	—	2	4	no	—
30	C₂×Dic₂₆₈	C₂³	1	4	yes	—
31	#14524041	—	4	2	no	—
32	#14524030	—	4	2	no	—
33	#14524024	—	4	4	no	—
34	#14524022	—	2	8	no	—
35	#14524002	—	2	2	no	—
36	Dic₅₃₆	—	2	4	no	—
37	Dic₅₃₆	—	2	4	no	—
38	#14523977	C₂³	1	4	yes	—
39	#14523975	—	2	2	no	—
40	#14523968	—	2	8	no	—
41	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
42	#131×C₆₇	—	2	2	no	—
43	#131×C₆₇	—	2	2	no	—
44	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
45	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	M₃₂×C₆₇	—	2	2	no	—
49	M₃₂×C₆₇	—	2	4	no	—
50	C₂×C₁₆×C₆₇	—	2	4	no	—
51	C₂×C₁₆×C₆₇	D₄	1	4	yes	—
52	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
53	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
55	C₂×Dic₁₃₄	—	2	4	no	—
56	C₂×Dic₁₃₄	—	4	4	no	—
57	C₂×Dic₁₃₄	—	4	4	no	—
58	C₂×Dic₁₃₄	—	4	2	no	—
59	C₂²×Dic₆₇	—	4	2	no	—
60	Dic₂₆₈	—	2	4	no	—
61	Dic₂₆₈	—	2	4	no	—
62	Dic₂₆₈	—	4	4	no	—
63	C₄⋊Dic₆₇	—	2	4	no	—
64	Dic₆₇⋊C₄	—	4	8	no	—
65	C₂²⋊Dic₆₇	—	4	2	no	—
66	C₂²⋊Dic₆₇	—	4	8	no	—
67	C₄×Dic₆₇	—	4	2	no	—
68	#41×C₆₇	—	2	2	no	—
69	#41×C₆₇	—	2	2	no	—
70	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
71	C₂×D₄×C₆₇	D₈	1	2	yes	—
72	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
75	M₁₆×C₆₇	—	2	2	no	—
76	M₁₆×C₆₇	—	2	4	no	—
77	C₂×C₈×C₆₇	—	2	2	no	—
78	C₂×C₈×C₆₇	—	2	2	no	—
79	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₆₇	#41	1	2	yes	—
83	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
84	C₁₆×C₆₇	—	2	2	no	—
85	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
86	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
87	C₂³×C₆₇	C₂×D₈	1	2	yes	—
88	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
89	Q₈×C₆₇	—	2	2	no	—
90	D₄×C₆₇	—	2	4	no	—
91	D₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	#121	1	2	yes	—
95	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₆₇	#135	1	2	yes	—
98	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
99	C₂×Dic₆₇	—	4	4	no	—
100	C₂×Dic₆₇	—	4	4	no	—
101	C₂×Dic₆₇	—	8	4	no	—
102	Dic₁₃₄	—	4	4	no	—
103	Dic₁₃₄	—	4	4	no	—
104	Dic₁₃₄	—	8	4	no	—
105	C₈×C₆₇	—	2	2	no	—
106	C₈×C₆₇	#121	1	2	yes	—
107	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
108	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
109	C₂²×C₆₇	—	2	2	no	—
110	C₂²×C₆₇	—	2	4	no	—
111	C₂²×C₆₇	—	2	2	no	—
112	C₂²×C₆₇	#466	1	2	yes	—
113	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
114	C₄×C₆₇	—	2	2	no	—
115	C₄×C₆₇	#466	1	2	yes	—
116	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
117	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
118	Dic₆₇	—	8	4	no	—
119	Dic₆₇	—	8	4	no	—
120	#33608	—	67	1	no	maximal, Sylow
121	C₂×C₆₇	—	2	2	no	—
122	C₂×C₆₇	—	4	2	no	—
123	C₂×C₆₇	C₂×#466	1	1	yes	characteristic
124	C₂×C₆₇	#3417	1	2	yes	—
125	C₂²×Q₃₂	—	67	1	no	—
126	C₂×#504	D₆₇	1	1	yes	characteristic
127	#3059	—	67	2	no	—
128	#2219	—	67	4	no	—
129	#2216	—	67	4	no	—
130	#2158	—	67	1	no	—
131	#2141	—	67	2	no	—
132	C₆₇	#33608	1	1	yes	characteristic, complemented, Sylow
133	C₂²×Q₁₆	—	67	2	no	—
134	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
135	#536	—	134	2	no	—
136	C₂×Q₃₂	—	67	2	no	—
137	C₂×Q₃₂	—	67	2	no	—
138	C₂×Q₃₂	—	134	4	no	—
139	#504	—	2	4	no	—
140	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
141	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
142	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
143	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
144	#470	—	134	4	no	—
145	#467	—	67	8	no	—
146	#427	—	67	2	no	—
147	#419	—	134	2	no	—
148	#367	—	67	2	no	—
149	#366	—	67	2	no	—
150	#358	—	67	8	no	—
151	C₂×#41	D₂₆₈	1	1	yes	characteristic
152	C₂²×Q₈	—	134	2	no	—
153	C₂×Q₁₆	—	67	4	no	—
154	C₂×Q₁₆	—	134	4	no	—
155	C₂×Q₁₆	—	134	2	no	—
156	C₂×Q₁₆	—	134	4	no	—
157	#131	—	2	2	no	—
158	#131	—	2	2	no	—
159	C₂×M₁₆	D₂₆₈	1	1	yes	characteristic
160	C₂×M₁₆	D₂₆₈	1	2	yes	—
161	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
162	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
163	C₄⋊Q₈	—	268	2	no	—
164	#124	—	268	2	no	—
165	#123	—	268	4	no	—
166	C₂²⋊Q₈	—	134	8	no	—
167	#117	—	134	2	no	—
168	Q₃₂	—	134	4	no	—
169	Q₃₂	—	134	4	no	—
170	M₃₂	—	2	2	no	—
171	M₃₂	—	2	4	no	—
172	C₂×C₁₆	—	2	4	no	—
173	C₂×C₁₆	C₂²⋊D₆₇	1	4	yes	—
174	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
175	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
176	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
177	#107	—	67	4	no	—
178	#106	—	134	2	no	—
179	#103	—	134	8	no	—
180	#41	—	2	2	no	—
181	#41	—	2	2	no	—
182	C₂×Q₈	—	268	2	no	—
183	C₂×Q₈	—	268	4	no	—
184	C₂×Q₈	—	268	4	no	—
185	C₂×Q₈	—	134	4	no	—
186	C₂×Q₈	D₅₃₆	1	1	yes	characteristic
187	C₂×D₄	D₅₃₆	1	2	yes	—
188	C₂×D₄	D₅₃₆	1	1	yes	characteristic
189	C₂²×C₄	—	268	2	no	—
190	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
191	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
192	Q₁₆	—	268	4	no	—
193	Q₁₆	—	134	4	no	—
194	Q₁₆	—	134	4	no	—
195	M₁₆	—	2	2	no	—
196	M₁₆	—	2	4	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
200	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
201	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
202	C₂×C₈	#11761628	1	2	yes	—
203	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
204	C₄⋊C₄	—	268	8	no	—
205	C₄⋊C₄	—	134	4	no	—
206	C₂²⋊C₄	—	268	2	no	—
207	C₂²⋊C₄	—	268	8	no	—
208	C₄²	—	268	2	no	—
209	C₁₆	—	2	2	no	—
210	C₁₆	C₂×C₂²⋊D₆₇	1	2	yes	—
211	C₁₆	C₂×C₂²⋊D₆₇	1	2	yes	—
212	C₂³	C₂×D₅₃₆	1	2	yes	—
213	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
214	Q₈	—	2	2	no	—
215	Q₈	—	268	4	no	—
216	Q₈	—	268	4	no	—
217	Q₈	—	536	4	no	—
218	D₄	—	2	4	no	—
219	D₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	268	4	no	—
223	C₂×C₄	—	268	4	no	—
224	C₂×C₄	—	536	4	no	—
225	C₂×C₄	#14524017	1	2	yes	—
226	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
227	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
228	C₂×C₄	#14524065	1	2	yes	—
229	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	#14524017	1	2	yes	—
232	C₈	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
233	C₈	C₂×#11761628	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	4	no	—
237	C₂²	#16863632	1	2	yes	—
238	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
239	C₄	—	2	2	no	—
240	C₄	—	536	4	no	—
241	C₄	—	536	4	no	—
242	C₄	#16863632	1	2	yes	—
243	C₄	C₂×#14524017	1	1	yes	characteristic
244	C₄	C₂×#14524065	1	1	yes	characteristic
245	C₂	—	2	2	no	—
246	C₂	—	4	2	no	—
247	C₂	C₂×#16863632	1	1	yes	characteristic
248	C₂	#22573362	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₅₃₆
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#504×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂×#41×C₆₇	D₄
hypercenter	C₂×#504	D₆₇
hypocenter	C₆₇	#33608
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₅₃₆	C₂	semidirect
3. #22562317	C₂	semidirect
4. #22562300	C₂	semidirect
5. #22561993	C₂	semidirect
6. #22561930	C₂	semidirect
132. C₆₇	#33608	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₅₃₆	2
#22569864	2
#22562317	2
#22562300	2
#22561993	2
#22561930	2
C₂×#504×C₆₇	2
#33608	67