Subgroups — #30864452

Order 17152.

1022 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	#22570637	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863519	C₂	1	4	yes	maximal, complemented
3	#22568922	C₂	1	2	yes	maximal, complemented
4	#22561102	C₂	1	4	yes	maximal, complemented
5	#22561037	C₂	1	2	yes	maximal, complemented
6	#22560381	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#97×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14524065	C₂²	1	4	yes	—
9	C₂×#14524049	C₂²	1	4	yes	—
10	#16863946	C₂²	1	2	yes	—
11	#16863780	C₄	1	4	yes	complemented
12	#16863776	C₄	1	1	yes	characteristic, complemented
13	#16863773	C₄	1	1	yes	characteristic, complemented
14	#16863769	C₄	1	2	yes	complemented
15	#16863519	—	2	16	no	—
16	#16863475	—	2	4	no	—
17	C₈×D₂₆₈	—	2	4	no	—
18	C₂×#14523979	C₂²	1	4	yes	—
19	#16863420	C₂²	1	2	yes	—
20	C₂×#14523971	C₂²	1	8	yes	—
21	#16863362	C₂²	1	4	yes	—
22	C₂×#97×C₆₇	C₂²	1	2	yes	—
23	C₂×#97×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₄³×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#11761628	C₂³	1	4	yes	—
28	#14524065	—	2	8	no	—
29	#14524065	—	2	8	no	—
30	C₂×Dic₂₆₈	C₂×C₄	1	4	yes	—
31	C₂×#11761617	C₂×C₄	1	8	yes	—
32	C₂×D₅₃₆	C₂×C₄	1	4	yes	—
33	#14524049	—	2	16	no	—
34	C₂×#11761612	C₂³	1	4	yes	—
35	C₂×#11761611	—	2	4	no	—
36	C₂×C₈×D₆₇	—	2	4	no	—
37	#14524043	C₂×C₄	1	2	yes	—
38	#14524040	C₂×C₄	1	2	yes	—
39	#14524031	C₂×C₄	1	4	yes	—
40	#14524017	—	2	4	no	—
41	C₄×D₂₆₈	—	2	4	no	—
42	#14523979	—	2	8	no	—
43	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
44	#14523971	—	2	16	no	—
45	(C₂×D₆₇)⋊C₈	—	2	8	no	—
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	—	2	2	no	—
50	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
51	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
52	#97×C₆₇	—	2	4	no	—
53	#97×C₆₇	D₄	1	8	yes	complemented
54	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₆₇	D₄	1	4	yes	—
58	#11761628	—	2	8	no	—
59	#11761628	—	2	8	no	—
60	C₂×Dic₁₃₄	C₂²×C₄	1	4	yes	—
61	C₂×D₂₆₈	—	2	4	no	—
62	C₂×D₂₆₈	C₂²×C₄	1	4	yes	—
63	C₂×C₂²⋊D₆₇	—	2	4	no	—
64	C₂×C₄×D₆₇	—	2	4	no	—
65	Dic₂₆₈	—	2	8	no	—
66	#11761617	—	2	16	no	—
67	D₅₃₆	—	2	8	no	—
68	#11761612	—	2	8	no	—
69	#11761611	—	4	8	no	—
70	C₂×C₆₇⋊C₈	—	2	4	no	—
71	C₈×D₆₇	—	4	8	no	—
72	C₄⋊Dic₆₇	—	2	4	no	—
73	(C₂×D₆₇)⋊C₄	—	2	8	no	—
74	C₂²×C₄×C₆₇	—	2	2	no	—
75	C₂²×C₄×C₆₇	#41	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₆₇	—	2	2	no	—
78	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
82	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
84	C₂×C₈×C₆₇	#41	1	4	yes	—
85	C₄²×C₆₇	—	2	2	no	—
86	C₄²×C₆₇	—	4	4	no	—
87	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₆₇	C₂×D₄	1	4	yes	—
90	C₄²×C₆₇	C₂×D₄	1	2	yes	—
91	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
92	C₂²×D₆₇	—	2	4	no	—
93	C₂×C₄×C₆₇	—	2	4	no	—
94	C₂×C₄×C₆₇	—	2	4	no	—
95	C₂×C₄×C₆₇	—	4	4	no	—
96	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
100	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
102	C₂²⋊D₆₇	—	4	8	no	—
103	C₂×Dic₆₇	—	2	4	no	—
104	D₂₆₈	—	2	8	no	—
105	D₂₆₈	—	4	8	no	—
106	C₄×D₆₇	—	4	4	no	—
107	C₄×D₆₇	—	4	4	no	—
108	Dic₁₃₄	—	2	8	no	—
109	C₈×C₆₇	—	2	2	no	—
110	C₈×C₆₇	C₄×D₄	1	4	yes	—
111	C₈×C₆₇	C₄×D₄	1	4	yes	—
112	C₈×C₆₇	#127	1	4	yes	—
113	C₆₇⋊C₈	—	4	4	no	—
114	C₂²×C₆₇	—	2	2	no	—
115	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
116	C₂²×C₆₇	#532	1	2	yes	—
117	C₂×D₆₇	—	4	4	no	—
118	C₂×D₆₇	—	4	8	no	—
119	C₄×C₆₇	—	2	2	no	—
120	C₄×C₆₇	—	4	4	no	—
121	C₄×C₆₇	#444	1	4	yes	complemented
122	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
123	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
124	C₄×C₆₇	#532	1	1	yes	characteristic
125	Dic₆₇	—	4	4	no	—
126	#21706	—	67	1	no	maximal, Sylow
127	C₂×C₆₇	—	2	2	no	—
128	C₂×C₆₇	C₄×#127	1	1	yes	characteristic
129	C₂×C₆₇	C₂×#444	1	2	yes	—
130	D₆₇	—	8	4	no	—
131	#3147	—	67	1	no	—
132	C₂×#444	—	67	4	no	—
133	#2931	—	67	2	no	—
134	#1975	—	67	4	no	—
135	#1962	—	67	2	no	—
136	#1848	—	67	1	no	—
137	C₄×#97	D₆₇	1	1	yes	characteristic, complemented
138	C₆₇	#21706	1	1	yes	characteristic, complemented, Sylow
139	C₂×#135	—	67	4	no	—
140	C₂×#131	—	67	4	no	—
141	#532	—	67	2	no	—
142	#495	—	67	4	no	—
143	#494	—	67	1	no	—
144	#493	—	67	1	no	—
145	#492	—	67	2	no	—
146	#444	—	134	16	no	—
147	#436	—	134	4	no	—
148	C₈×D₄	—	134	4	no	—
149	C₂×#108	—	67	4	no	—
150	#423	—	67	2	no	—
151	C₂×#104	—	67	8	no	—
152	#408	—	67	4	no	—
153	C₂×#97	C₂×D₆₇	1	2	yes	—
154	C₂×#97	C₂×D₆₇	1	2	yes	—
155	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
156	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
157	C₄³	C₂×D₆₇	1	1	yes	characteristic
158	C₂×#41	—	67	4	no	—
159	#135	—	134	8	no	—
160	#135	—	134	8	no	—
161	C₂×Q₁₆	—	67	4	no	—
162	C₂×SD₁₆	—	67	8	no	—
163	C₂×D₈	—	67	4	no	—
164	#131	—	134	16	no	—
165	C₂×M₁₆	—	67	4	no	—
166	C₂×M₁₆	—	134	4	no	—
167	C₂²×C₈	—	134	4	no	—
168	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
169	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
170	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
171	C₄⋊Q₈	—	67	2	no	—
172	#127	—	67	2	no	—
173	#124	—	67	4	no	—
174	#121	—	134	4	no	—
175	C₄×D₄	—	134	4	no	—
176	C₂×C₄²	—	2	2	no	—
177	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
178	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
179	#108	—	134	8	no	—
180	C₄⋊C₈	—	134	4	no	—
181	#104	—	134	16	no	—
182	C₂²⋊C₈	—	134	8	no	—
183	#97	—	2	4	no	—
184	#97	D₂₆₈	1	8	yes	complemented
185	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
186	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
187	C₄×C₈	C₄×D₆₇	1	2	yes	—
188	C₄×C₈	D₂₆₈	1	4	yes	—
189	#41	—	134	8	no	—
190	#41	—	134	8	no	—
191	C₂×Q₈	—	67	4	no	—
192	C₂×D₄	—	67	4	no	—
193	C₂×D₄	—	134	4	no	—
194	C₂×D₄	—	134	4	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	134	4	no	—
197	C₂²×C₄	#11761628	1	2	yes	—
198	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
199	Q₁₆	—	134	8	no	—
200	SD₁₆	—	134	16	no	—
201	D₈	—	134	8	no	—
202	M₁₆	—	268	8	no	—
203	M₁₆	—	134	8	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	268	8	no	—
206	C₂×C₈	—	134	4	no	—
207	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
208	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
209	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
210	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
211	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
212	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
213	C₂×C₈	#11761628	1	4	yes	—
214	C₄⋊C₄	—	134	4	no	—
215	C₂²⋊C₄	—	134	8	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	4	4	no	—
218	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
219	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
220	C₄²	C₂×D₂₆₈	1	4	yes	—
221	C₄²	C₂×D₂₆₈	1	2	yes	—
222	C₂³	—	134	4	no	—
223	C₂³	C₂×#11761628	1	1	yes	characteristic
224	Q₈	—	134	8	no	—
225	D₄	—	268	8	no	—
226	D₄	—	268	8	no	—
227	D₄	—	134	8	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	268	4	no	—
232	C₂×C₄	—	268	4	no	—
233	C₂×C₄	—	134	4	no	—
234	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
235	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
236	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
237	C₂×C₄	C₂²×D₂₆₈	1	2	yes	—
238	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
239	C₂×C₄	C₂×#11761628	1	2	yes	—
240	C₈	—	2	2	no	—
241	C₈	—	268	4	no	—
242	C₈	C₄×D₂₆₈	1	4	yes	—
243	C₈	C₄×D₂₆₈	1	4	yes	—
244	C₈	#14524040	1	4	yes	—
245	C₂²	—	2	2	no	—
246	C₂²	—	268	8	no	—
247	C₂²	—	268	4	no	—
248	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
249	C₂²	#16863946	1	2	yes	—
250	C₄	—	2	2	no	—
251	C₄	—	4	4	no	—
252	C₄	—	268	4	no	—
253	C₄	#16863519	1	4	yes	complemented
254	C₄	C₂×C₄×D₂₆₈	1	2	yes	—
255	C₄	C₂×#14524040	1	1	yes	characteristic
256	C₄	#16863946	1	1	yes	characteristic
257	C₂	—	2	2	no	—
258	C₂	—	536	4	no	—
259	C₂	C₄×#14524040	1	1	yes	characteristic
260	C₂	C₂×#16863519	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₄×#97×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#22570637	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×#97	D₆₇
hypocenter	C₆₇	#21706
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
2. C₂×#16863519	C₂	semidirect
3. #22568922	C₂	semidirect
4. #22561102	C₂	semidirect
5. #22561037	C₂	semidirect
6. #22560381	C₂	semidirect
7. C₄×#97×C₆₇	C₂	semidirect
11. #16863780	C₄	semidirect
12. #16863776	C₄	semidirect
13. #16863773	C₄	semidirect
14. #16863769	C₄	semidirect
53. #97×C₆₇	D₄	semidirect
121. C₄×C₆₇	#444	semidirect
137. C₄×#97	D₆₇	semidirect
138. C₆₇	#21706	semidirect
184. #97	D₂₆₈	semidirect
253. C₄	#16863519	semidirect

Maximal subgroups

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

subgroup	index
#22570637	2
C₂×#16863519	2
#22568922	2
#22561102	2
#22561037	2
#22560381	2
C₄×#97×C₆₇	2
#21706	67