Subgroups — #30826673

Order 17152.

624 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570788	C₂	1	1	yes	maximal, characteristic
2	#22569287	C₂	1	2	yes	maximal, complemented
3	#22559072	C₂	1	4	yes	maximal, complemented
4	#22559037	C₂	1	4	yes	maximal, complemented
5	#22558677	C₂	1	2	yes	maximal, complemented
6	#22558501	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	#16863946	C₂²	1	2	yes	—
9	#16863801	C₄	1	1	yes	characteristic
10	#16863773	C₄	1	1	yes	characteristic
11	#16863743	C₄	1	2	yes	—
12	#16863632	C₄	1	2	yes	—
13	#16863632	C₄	1	2	yes	—
14	C₄×D₅₃₆	C₂²	1	2	yes	—
15	C₄×D₅₃₆	C₂²	1	2	yes	—
16	#16863473	C₂²	1	4	yes	—
17	#16863469	—	2	4	no	—
18	#16863461	C₂²	1	4	yes	—
19	#16863449	C₂²	1	1	yes	characteristic
20	#16863449	C₂²	1	1	yes	characteristic
21	#16863361	—	2	4	no	—
22	#16863123	C₂²	1	8	yes	—
23	#16863118	C₂²	1	2	yes	—
24	#16863118	C₂²	1	2	yes	—
25	#432×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	#335×C₆₇	C₂²	1	2	yes	—
28	#334×C₆₇	C₂²	1	2	yes	—
29	C₂×#11761628	—	2	2	no	—
30	C₂×D₅₃₆	C₂×C₄	1	2	yes	—
31	C₂×D₅₃₆	C₂×C₄	1	2	yes	—
32	C₂×#11761611	—	2	4	no	—
33	C₂×C₈×D₆₇	—	2	4	no	—
34	#14524043	C₂×C₄	1	2	yes	—
35	#14524040	C₂×C₄	1	2	yes	—
36	#14524031	—	2	2	no	—
37	#14524017	—	2	2	no	—
38	#14524017	C₂×C₄	1	4	yes	—
39	C₄×D₂₆₈	—	2	2	no	—
40	C₄×D₂₆₈	C₂³	1	4	yes	—
41	#14523977	C₂×C₄	1	2	yes	—
42	#14523977	C₂×C₄	1	2	yes	—
43	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
44	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
45	#14523965	C₂×C₄	1	4	yes	—
46	#14523965	C₂×C₄	1	4	yes	—
47	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
48	C₂²⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
49	(C₂×D₆₇)⋊C₈	—	2	4	no	—
50	(C₂×D₆₇)⋊C₈	—	2	4	no	—
51	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
52	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
53	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
57	C₂²⋊C₈×C₆₇	—	2	2	no	—
58	#97×C₆₇	—	2	2	no	—
59	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	#11761628	—	8	4	no	—
64	C₂×Dic₁₃₄	—	2	2	no	—
65	C₂×D₂₆₈	—	2	2	no	—
66	C₂×D₂₆₈	—	2	2	no	—
67	C₂×D₂₆₈	C₂²×C₄	1	4	yes	—
68	C₂×C₂²⋊D₆₇	—	2	4	no	—
69	C₂×C₄×D₆₇	—	2	4	no	—
70	D₅₃₆	—	4	2	no	—
71	D₅₃₆	—	4	2	no	—
72	#11761611	—	8	4	no	—
73	C₂×C₆₇⋊C₈	—	2	2	no	—
74	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
75	C₈×D₆₇	—	8	4	no	—
76	C₄⋊Dic₆₇	—	2	2	no	—
77	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
78	(C₂×D₆₇)⋊C₄	—	2	4	no	—
79	(C₂×D₆₇)⋊C₄	—	2	4	no	—
80	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
83	C₂×C₈×C₆₇	—	2	4	no	—
84	C₂×C₈×C₆₇	—	4	1	no	—
85	C₂×C₈×C₆₇	—	4	1	no	—
86	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₆₇	#41	1	1	yes	characteristic
93	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
94	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
95	C₂²×D₆₇	—	2	4	no	—
96	C₂×C₄×C₆₇	—	2	1	no	—
97	C₂×C₄×C₆₇	—	2	1	no	—
98	C₂×C₄×C₆₇	—	2	2	no	—
99	C₂×C₄×C₆₇	—	4	1	no	—
100	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
106	C₂²⋊D₆₇	—	8	4	no	—
107	C₂×Dic₆₇	—	2	4	no	—
108	D₂₆₈	—	2	4	no	—
109	D₂₆₈	—	2	4	no	—
110	D₂₆₈	—	8	2	no	—
111	D₂₆₈	—	8	2	no	—
112	C₄×D₆₇	—	4	4	no	—
113	C₄×D₆₇	—	4	4	no	—
114	Dic₁₃₄	—	8	2	no	—
115	C₈×C₆₇	—	2	1	no	—
116	C₈×C₆₇	—	2	1	no	—
117	C₈×C₆₇	—	2	1	no	—
118	C₈×C₆₇	—	2	1	no	—
119	C₈×C₆₇	—	4	4	no	—
120	C₆₇⋊C₈	—	2	4	no	—
121	C₆₇⋊C₈	—	4	2	no	—
122	C₂²×C₆₇	—	2	1	no	—
123	C₂²×C₆₇	—	2	1	no	—
124	C₂²×C₆₇	—	2	1	no	—
125	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
126	C₂×D₆₇	—	4	4	no	—
127	C₂×D₆₇	—	4	4	no	—
128	C₂×D₆₇	—	4	4	no	—
129	C₄×C₆₇	—	2	1	no	—
130	C₄×C₆₇	—	2	1	no	—
131	C₄×C₆₇	—	2	1	no	—
132	C₄×C₆₇	—	2	1	no	—
133	C₄×C₆₇	#568	1	1	yes	characteristic
134	C₄×C₆₇	#568	1	1	yes	characteristic
135	C₄×C₆₇	#576	1	2	yes	—
136	Dic₆₇	—	4	4	no	—
137	#17989	—	67	1	no	maximal, Sylow
138	C₂×C₆₇	—	4	1	no	—
139	C₂×C₆₇	#2932	1	1	yes	characteristic
140	C₂×C₆₇	#2948	1	1	yes	characteristic
141	C₂×C₆₇	#2957	1	1	yes	characteristic
142	D₆₇	—	8	4	no	—
143	#3169	—	67	1	no	—
144	#2981	—	67	2	no	—
145	#1586	—	67	4	no	—
146	#1581	—	67	4	no	—
147	#1568	D₆₇	1	1	yes	characteristic, complemented
148	#1496	—	67	2	no	—
149	#1452	—	67	1	no	—
150	C₆₇	#17989	1	1	yes	characteristic, complemented, Sylow
151	#532	—	67	2	no	—
152	#500	—	67	1	no	—
153	#493	—	67	1	no	—
154	#486	—	67	2	no	—
155	#466	—	67	2	no	—
156	#466	—	67	2	no	—
157	C₄×D₈	—	67	2	no	—
158	C₄×D₈	—	67	2	no	—
159	#436	—	67	4	no	—
160	#435	—	134	4	no	—
161	C₈×D₄	—	67	4	no	—
162	#432	C₂×D₆₇	1	2	yes	—
163	#431	—	67	1	no	—
164	#431	—	67	1	no	—
165	#408	—	134	4	no	—
166	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
167	#335	C₂×D₆₇	1	2	yes	—
168	#334	C₂×D₆₇	1	2	yes	—
169	D₄⋊C₈	—	67	8	no	—
170	#322	—	67	2	no	—
171	#322	—	67	2	no	—
172	C₂×#41	—	134	2	no	—
173	C₂×D₈	—	67	2	no	—
174	C₂×D₈	—	67	2	no	—
175	C₂×M₁₆	—	134	4	no	—
176	C₂²×C₈	—	134	4	no	—
177	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
178	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
179	C₄⋊Q₈	—	67	2	no	—
180	#127	—	67	2	no	—
181	#124	—	134	2	no	—
182	#121	—	67	4	no	—
183	#121	—	134	2	no	—
184	C₄×D₄	—	67	4	no	—
185	C₄×D₄	—	134	2	no	—
186	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
187	#107	—	67	2	no	—
188	#107	—	67	2	no	—
189	C₄⋊C₈	—	67	2	no	—
190	C₄⋊C₈	—	134	1	no	—
191	C₄⋊C₈	C₂²×D₆₇	1	4	yes	—
192	#102	—	67	4	no	—
193	#102	—	67	4	no	—
194	C₂²⋊C₈	—	2	2	no	—
195	C₂²⋊C₈	—	67	2	no	—
196	C₂²⋊C₈	—	134	4	no	—
197	C₂²⋊C₈	—	134	4	no	—
198	C₂²⋊C₈	—	134	1	no	—
199	#97	—	2	2	no	—
200	C₄×C₈	—	67	2	no	—
201	C₄×C₈	—	67	2	no	—
202	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
203	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
204	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
205	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
206	#41	—	536	4	no	—
207	C₂×Q₈	—	134	2	no	—
208	C₂×D₄	—	67	4	no	—
209	C₂×D₄	—	134	4	no	—
210	C₂×D₄	—	134	2	no	—
211	C₂×D₄	—	134	2	no	—
212	C₂²×C₄	—	134	4	no	—
213	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
214	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
215	C₂²×C₄	#11761629	1	1	yes	characteristic
216	D₈	—	268	2	no	—
217	D₈	—	268	2	no	—
218	M₁₆	—	536	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	536	4	no	—
223	C₂×C₈	—	67	4	no	—
224	C₂×C₈	—	134	2	no	—
225	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
229	C₄⋊C₄	—	67	4	no	—
230	C₄⋊C₄	—	134	2	no	—
231	C₂²⋊C₄	—	134	4	no	—
232	C₂²⋊C₄	—	134	4	no	—
233	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
234	C₄²	D₄×D₆₇	1	1	yes	characteristic
235	C₄²	#11761629	1	1	yes	characteristic
236	C₄²	C₂³×D₆₇	1	1	yes	characteristic
237	C₂³	—	134	4	no	—
238	C₂³	#14524005	1	1	yes	characteristic
239	Q₈	—	536	2	no	—
240	D₄	—	536	2	no	—
241	D₄	—	536	2	no	—
242	D₄	—	536	4	no	—
243	D₄	—	134	4	no	—
244	D₄	—	134	4	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	268	4	no	—
250	C₂×C₄	—	268	4	no	—
251	C₂×C₄	—	134	4	no	—
252	C₂×C₄	#14524005	1	1	yes	characteristic
253	C₂×C₄	#14524005	1	1	yes	characteristic
254	C₂×C₄	#14524005	1	1	yes	characteristic
255	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
257	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	4	4	no	—
263	C₈	—	268	2	no	—
264	C₈	—	134	4	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	268	4	no	—
269	C₂²	—	268	4	no	—
270	C₂²	—	268	4	no	—
271	C₂²	C₂×#14524005	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	268	4	no	—
277	C₄	#16864162	1	1	yes	characteristic
278	C₄	#16864162	1	1	yes	characteristic
279	C₄	#16864198	1	2	yes	—
280	C₂	—	4	1	no	—
281	C₂	—	536	4	no	—
282	C₂	#22568926	1	1	yes	characteristic
283	C₂	#22569060	1	1	yes	characteristic
284	C₂	#22569125	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1568	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#22570788	C₂
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1568	D₆₇
hypocenter	C₆₇	#17989
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
2. #22569287	C₂	semidirect
3. #22559072	C₂	semidirect
4. #22559037	C₂	semidirect
5. #22558677	C₂	semidirect
6. #22558501	C₂	semidirect
7. C₆₇×#1568	C₂	semidirect
147. #1568	D₆₇	semidirect
150. C₆₇	#17989	semidirect

Maximal subgroups

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

subgroup	index
#22570788	2
#22569287	2
#22559072	2
#22559037	2
#22558677	2
#22558501	2
C₆₇×#1568	2
#17989	67