Subgroups — #31009080

Order 17152.

642 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573311	C₂	1	2	yes	maximal, complemented
2	#22569113	C₂	1	1	yes	maximal, characteristic
3	#22562052	C₂	1	4	yes	maximal, complemented
4	#22562047	C₂	1	4	yes	maximal, complemented
5	#22561975	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16863179	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	#16864196	C₄	1	1	yes	characteristic
9	C₂×#14524049	C₂²	1	2	yes	—
10	#16863814	—	2	4	no	—
11	#16863812	C₂²	1	8	yes	complemented
12	C₂×#14523983	C₂²	1	2	yes	—
13	M₃₂×D₆₇	—	2	4	no	—
14	#16863519	C₄	1	4	yes	—
15	#16863514	C₂²	1	4	yes	—
16	#16863445	C₄	1	1	yes	characteristic
17	#16863414	C₄	1	2	yes	—
18	#16863217	C₂²	1	4	yes	complemented
19	#16863179	C₂²	1	4	yes	complemented
20	#16863179	C₂²	1	4	yes	complemented
21	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
23	#405×C₆₇	C₂²	1	1	yes	characteristic
24	#346×C₆₇	C₂²	1	2	yes	complemented
25	C₄×C₁₆×C₆₇	C₂²	1	2	yes	complemented
26	C₂×#11761628	C₂×C₄	1	2	yes	—
27	#14524076	C₈	1	4	yes	—
28	#14524071	C₈	1	4	yes	—
29	#14524065	C₂×C₄	1	4	yes	—
30	#14524049	—	2	2	no	—
31	#14524049	—	2	2	no	—
32	#14524049	C₂×C₄	1	4	yes	—
33	#14524049	C₂³	1	4	yes	—
34	C₂×#11761612	C₂×C₄	1	2	yes	—
35	C₂×#11761611	—	2	2	no	—
36	C₂×C₈×D₆₇	—	2	2	no	—
37	#14523983	—	2	2	no	—
38	#14523983	C₂³	1	4	yes	—
39	#14523982	—	2	8	no	—
40	C₂×C₆₇⋊C₁₆	—	2	2	no	—
41	C₁₆×D₆₇	—	2	8	no	—
42	#14523979	C₂×C₄	1	4	yes	—
43	#14523971	C₂×C₄	1	8	yes	—
44	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
47	M₃₂×C₆₇	D₄	1	4	yes	—
48	C₂×C₁₆×C₆₇	—	2	2	no	—
49	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
50	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
51	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
52	#97×C₆₇	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₆₇	C₂³	1	2	yes	—
54	#11761628	—	2	2	no	—
55	#11761628	—	2	2	no	—
56	#11761628	C₂×C₈	1	4	yes	—
57	#11761628	C₂²×C₄	1	4	yes	—
58	C₂×Dic₁₃₄	C₂×C₈	1	2	yes	—
59	C₂×D₂₆₈	C₂×C₈	1	2	yes	—
60	C₂×C₂²⋊D₆₇	—	2	2	no	—
61	C₂×C₄×D₆₇	—	2	2	no	—
62	Dic₂₆₈	C₂×C₈	1	4	yes	—
63	#11761617	C₂×C₈	1	8	yes	—
64	D₅₃₆	C₂×C₈	1	4	yes	—
65	#11761612	—	2	2	no	—
66	#11761612	C₂²×C₄	1	4	yes	—
67	#11761611	—	2	4	no	—
68	#11761611	—	2	4	no	—
69	C₂×C₆₇⋊C₈	—	2	2	no	—
70	C₈×D₆₇	—	2	4	no	—
71	C₈×D₆₇	—	2	4	no	—
72	C₆₇⋊C₁₆	—	2	4	no	—
73	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	M₁₆×C₆₇	C₂×C₈	1	4	yes	—
75	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₆₇	#41	1	2	yes	—
79	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
80	C₄⋊C₄×C₆₇	—	2	1	no	—
81	C₂²⋊C₄×C₆₇	—	2	1	no	—
82	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
83	C₁₆×C₆₇	—	2	1	no	—
84	C₁₆×C₆₇	—	2	1	no	—
85	C₁₆×C₆₇	C₂×D₄	1	4	yes	—
86	C₂³×C₆₇	#131	1	1	yes	characteristic
87	C₂²×D₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	2	2	no	—
89	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
90	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
91	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
93	C₂²⋊D₆₇	—	2	4	no	—
94	C₂²⋊D₆₇	—	2	4	no	—
95	C₂×Dic₆₇	—	2	2	no	—
96	D₂₆₈	—	2	2	no	—
97	D₂₆₈	C₂²×C₈	1	4	yes	—
98	C₄×D₆₇	—	2	4	no	—
99	C₄×D₆₇	—	2	4	no	—
100	Dic₁₃₄	—	2	2	no	—
101	Dic₁₃₄	C₂²×C₈	1	4	yes	—
102	C₈×C₆₇	C₄×D₄	1	2	yes	—
103	C₈×C₆₇	C₂²×C₈	1	4	yes	—
104	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
105	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
106	C₆₇⋊C₈	—	2	4	no	—
107	C₂²×C₆₇	—	4	1	no	—
108	C₂²×C₆₇	C₈×D₄	1	2	yes	—
109	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
110	C₂×D₆₇	—	2	4	no	—
111	C₂×D₆₇	—	8	2	no	—
112	C₄×C₆₇	—	4	2	no	—
113	C₄×C₆₇	C₈×D₄	1	2	yes	—
114	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
115	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
116	Dic₆₇	—	2	4	no	—
117	#33442	—	67	1	no	maximal, Sylow
118	C₂×C₆₇	—	2	1	no	—
119	C₂×C₆₇	—	2	2	no	—
120	C₂×C₆₇	C₂×C₈×D₄	1	1	yes	characteristic
121	D₆₇	—	4	4	no	—
122	#3407	—	67	2	no	—
123	#2955	—	67	1	no	—
124	#2171	—	67	4	no	—
125	#2170	—	67	4	no	—
126	#2153	—	67	1	no	—
127	C₂×#350	—	67	2	no	—
128	#2108	D₆₇	1	1	yes	characteristic, complemented
129	C₆₇	#33442	1	1	yes	characteristic, complemented, Sylow
130	#575	—	67	1	no	—
131	C₂×#131	—	67	2	no	—
132	#504	—	67	8	no	—
133	#504	—	134	4	no	—
134	C₂×M₃₂	—	67	2	no	—
135	C₂×M₃₂	—	134	4	no	—
136	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
137	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
138	#444	—	67	4	no	—
139	#443	—	67	4	no	—
140	#430	—	67	1	no	—
141	#421	—	67	2	no	—
142	#405	C₂×D₆₇	1	1	yes	characteristic
143	#364	—	67	4	no	—
144	#350	—	67	4	no	—
145	#350	—	67	4	no	—
146	#346	C₂×D₆₇	1	2	yes	complemented
147	C₄×C₁₆	C₂×D₆₇	1	2	yes	complemented
148	C₂×#41	—	67	2	no	—
149	#137	—	67	4	no	—
150	AGL(1,ℤ/8)	—	67	4	no	—
151	#135	—	67	4	no	—
152	#131	—	67	4	no	—
153	#131	—	67	4	no	—
154	#131	—	134	2	no	—
155	#131	—	134	2	no	—
156	C₂×M₁₆	—	67	2	no	—
157	C₂×M₁₆	—	134	2	no	—
158	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
159	C₂²×C₈	—	134	2	no	—
160	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
161	#117	C₄×D₆₇	1	1	yes	characteristic
162	M₃₂	—	67	4	no	—
163	M₃₂	—	134	2	no	—
164	M₃₂	—	134	8	no	—
165	M₃₂	D₂₆₈	1	4	yes	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	134	2	no	—
168	C₂×C₁₆	—	134	8	no	—
169	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
170	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
171	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
172	#108	—	67	4	no	—
173	#104	—	67	8	no	—
174	#97	C₄×D₆₇	1	2	yes	—
175	C₄×C₈	C₂²×D₆₇	1	2	yes	—
176	#41	—	67	4	no	—
177	#41	—	67	4	no	—
178	#41	—	134	2	no	—
179	#41	—	134	2	no	—
180	C₂×Q₈	—	67	2	no	—
181	C₂×D₄	—	67	2	no	—
182	C₂×D₄	—	134	2	no	—
183	C₂²×C₄	—	134	2	no	—
184	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
185	Q₁₆	—	67	4	no	—
186	SD₁₆	—	67	8	no	—
187	D₈	—	67	4	no	—
188	M₁₆	—	67	4	no	—
189	M₁₆	—	134	2	no	—
190	M₁₆	—	134	4	no	—
191	M₁₆	—	134	4	no	—
192	M₁₆	C₈×D₆₇	1	4	yes	—
193	C₂×C₈	—	134	2	no	—
194	C₂×C₈	—	134	4	no	—
195	C₂×C₈	—	134	4	no	—
196	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
197	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
198	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
199	C₂×C₈	#11761628	1	2	yes	—
200	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
201	C₄⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₄²	C₂×C₄×D₆₇	1	2	yes	—
204	C₁₆	—	2	1	no	—
205	C₁₆	—	2	1	no	—
206	C₁₆	—	134	4	no	—
207	C₁₆	C₂×D₂₆₈	1	4	yes	—
208	C₂³	—	134	2	no	—
209	C₂³	#14524049	1	1	yes	characteristic
210	Q₈	—	67	4	no	—
211	Q₈	—	134	2	no	—
212	D₄	—	67	4	no	—
213	D₄	—	134	4	no	—
214	D₄	—	134	4	no	—
215	D₄	—	134	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	134	2	no	—
218	C₂×C₄	—	134	4	no	—
219	C₂×C₄	—	134	4	no	—
220	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
221	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
222	C₂×C₄	#14524049	1	1	yes	characteristic
223	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
224	C₈	—	134	4	no	—
225	C₈	C₄×D₂₆₈	1	2	yes	—
226	C₈	C₂×C₈×D₆₇	1	4	yes	—
227	C₈	C₂²×D₂₆₈	1	1	yes	characteristic
228	C₈	C₂×#11761628	1	1	yes	characteristic
229	C₂²	—	4	1	no	—
230	C₂²	—	536	2	no	—
231	C₂²	—	134	4	no	—
232	C₂²	C₈×D₂₆₈	1	2	yes	—
233	C₂²	C₂×#14524049	1	1	yes	characteristic
234	C₄	—	4	2	no	—
235	C₄	—	134	4	no	—
236	C₄	C₈×D₂₆₈	1	2	yes	—
237	C₄	C₂×C₄×D₂₆₈	1	1	yes	characteristic
238	C₄	C₂²×C₈×D₆₇	1	1	yes	characteristic
239	C₂	—	2	1	no	—
240	C₂	—	2	2	no	—
241	C₂	—	268	4	no	—
242	C₂	C₂×C₈×D₂₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×D₂₆₈
commutator	C₄×C₆₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2108	C₂
socle	C₂×C₆₇	C₂×C₈×D₄
prime radical	#16864196	C₄
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#2108	D₆₇
hypocenter	C₆₇	#33442
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ #2108

Complements

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

N	H ≅ G/N	type
1. #22573311	C₂	semidirect
3. #22562052	C₂	semidirect
4. #22562047	C₂	semidirect
5. #22561975	C₂	semidirect
6. C₂×#16863179	C₂	semidirect
7. C₆₇×#2108	C₂	semidirect
11. #16863812	C₂²	semidirect
18. #16863217	C₂²	semidirect
19. #16863179	C₂²	semidirect
20. #16863179	C₂²	semidirect
24. #346×C₆₇	C₂²	semidirect
25. C₄×C₁₆×C₆₇	C₂²	semidirect
128. #2108	D₆₇	semidirect
129. C₆₇	#33442	semidirect
146. #346	C₂×D₆₇	semidirect
147. C₄×C₁₆	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22573311	2
#22569113	2
#22562052	2
#22562047	2
#22561975	2
C₂×#16863179	2
C₆₇×#2108	2
#33442	67