Subgroups — #30827649

Order 17152.

578 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570795	C₂	1	1	yes	maximal, characteristic
2	#22569156	C₂	1	2	yes	maximal
3	#22559119	C₂	1	2	yes	maximal, complemented
4	#22559082	C₂	1	4	yes	maximal, complemented
5	#22559029	C₂	1	2	yes	maximal, complemented
6	C₂×#16863147	C₂	1	1	yes	maximal, characteristic
7	#22558613	C₂	1	2	yes	maximal
8	C₆₇×#1451	C₂	1	1	yes	maximal, characteristic
9	#16863953	C₂²	1	2	yes	—
10	#16863801	C₄	1	1	yes	characteristic
11	#16863776	C₄	1	1	yes	characteristic
12	#16863747	C₄	1	2	yes	—
13	#16863637	C₄	1	2	yes	—
14	#16863637	C₄	1	2	yes	—
15	#16863526	C₂²	1	4	yes	—
16	#16863524	—	2	4	no	—
17	C₈×Dic₁₃₄	C₂²	1	4	yes	—
18	C₄×Dic₂₆₈	C₂²	1	2	yes	—
19	C₄×Dic₂₆₈	C₂²	1	2	yes	—
20	#16863424	—	2	2	no	—
21	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
22	#16863354	—	2	2	no	—
23	#16863147	C₂²	1	1	yes	characteristic
24	#16863147	C₂²	1	2	yes	—
25	#16863147	C₂²	1	1	yes	characteristic
26	#16863128	C₂²	1	4	yes	—
27	#16863128	C₂²	1	4	yes	—
28	#431×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
31	#322×C₆₇	C₂²	1	2	yes	—
32	#322×C₆₇	C₂²	1	2	yes	—
33	C₂×Dic₂₆₈	C₂×C₄	1	2	yes	—
34	C₂×Dic₂₆₈	C₂×C₄	1	2	yes	—
35	C₂²×C₆₇⋊C₈	—	2	2	no	—
36	#14524043	C₂×C₄	1	2	yes	—
37	#14524043	C₂×C₄	1	2	yes	—
38	#14524035	—	2	2	no	—
39	#14524022	—	2	2	no	—
40	#14524022	C₂×C₄	1	4	yes	—
41	C₄×Dic₁₃₄	—	2	2	no	—
42	C₄×Dic₁₃₄	C₂³	1	4	yes	—
43	#14524002	—	2	2	no	—
44	#14523977	C₂×C₄	1	2	yes	—
45	#14523977	C₂×C₄	1	2	yes	—
46	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
47	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
48	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
49	Dic₆₇⋊C₈	—	2	4	no	—
50	Dic₆₇⋊C₈	—	2	4	no	—
51	#14523968	C₂×C₄	1	4	yes	—
52	#14523968	C₂×C₄	1	4	yes	—
53	#14523953	—	2	4	no	—
54	C₈×Dic₆₇	—	2	4	no	—
55	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
56	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₆₇	—	2	1	no	—
60	C₂²⋊C₈×C₆₇	—	2	1	no	—
61	#97×C₆₇	D₄	1	2	yes	—
62	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₆₇	C₂³	1	2	yes	—
65	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₆₇	C₂³	1	2	yes	—
68	C₂×Dic₁₃₄	—	2	2	no	—
69	C₂×Dic₁₃₄	C₂²×C₄	1	4	yes	—
70	Dic₂₆₈	—	2	4	no	—
71	Dic₂₆₈	—	4	2	no	—
72	C₂×C₆₇⋊C₈	—	2	2	no	—
73	C₂×C₆₇⋊C₈	—	2	2	no	—
74	C₂×C₆₇⋊C₈	—	4	4	no	—
75	C₄⋊Dic₆₇	—	2	2	no	—
76	C₄⋊Dic₆₇	—	2	2	no	—
77	C₄⋊Dic₆₇	—	2	2	no	—
78	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
79	Dic₆₇⋊C₄	—	2	4	no	—
80	Dic₆₇⋊C₄	—	2	4	no	—
81	C₂²⋊Dic₆₇	—	2	4	no	—
82	C₄×Dic₆₇	—	2	4	no	—
83	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
86	M₁₆×C₆₇	—	2	4	no	—
87	C₂×C₈×C₆₇	—	2	2	no	—
88	C₂×C₈×C₆₇	—	2	2	no	—
89	C₂×C₈×C₆₇	—	4	1	no	—
90	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
95	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₆₇	#41	1	1	yes	characteristic
98	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
99	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₆₇	—	2	1	no	—
101	C₂×C₄×C₆₇	—	2	1	no	—
102	C₂×C₄×C₆₇	—	4	1	no	—
103	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	#137	1	4	yes	—
107	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
110	C₂×Dic₆₇	—	2	4	no	—
111	C₂×Dic₆₇	—	2	4	no	—
112	Dic₁₃₄	—	2	4	no	—
113	Dic₁₃₄	—	2	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	4	no	—
119	C₈×C₆₇	—	4	2	no	—
120	C₈×C₆₇	#131	1	2	yes	—
121	C₆₇⋊C₈	—	4	2	no	—
122	C₆₇⋊C₈	—	4	2	no	—
123	C₂²×C₆₇	—	2	1	no	—
124	C₂²×C₆₇	—	2	1	no	—
125	C₂²×C₆₇	#441	1	2	yes	—
126	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
127	C₄×C₆₇	—	2	1	no	—
128	C₄×C₆₇	—	2	1	no	—
129	C₄×C₆₇	—	2	1	no	—
130	C₄×C₆₇	#441	1	2	yes	—
131	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
132	C₄×C₆₇	#568	1	1	yes	characteristic
133	C₄×C₆₇	C₂×#137	1	2	yes	—
134	Dic₆₇	—	4	4	no	—
135	Dic₆₇	—	4	4	no	—
136	#18072	—	67	1	no	maximal, Sylow
137	C₂×C₆₇	—	2	2	no	—
138	C₂×C₆₇	#2935	1	1	yes	characteristic
139	C₂×C₆₇	C₂×#441	1	1	yes	characteristic
140	C₂×C₆₇	#2957	1	1	yes	characteristic
141	#3170	—	67	1	no	—
142	#2962	—	67	2	no	—
143	#1593	—	67	2	no	—
144	#1587	—	67	4	no	—
145	#1579	—	67	2	no	—
146	C₂×#334	—	67	1	no	—
147	#1481	—	67	2	no	—
148	#1451	D₆₇	1	1	yes	characteristic
149	C₆₇	#18072	1	1	yes	characteristic, complemented, Sylow
150	#533	—	67	2	no	—
151	#500	—	67	1	no	—
152	#494	—	67	1	no	—
153	#487	—	67	2	no	—
154	#467	—	67	2	no	—
155	#467	—	67	2	no	—
156	#446	—	67	4	no	—
157	#446	—	134	4	no	—
158	C₈×Q₈	—	67	4	no	—
159	C₄×Q₁₆	—	67	2	no	—
160	C₄×Q₁₆	—	67	2	no	—
161	#431	C₂×D₆₇	1	1	yes	characteristic
162	#424	—	134	2	no	—
163	C₂×C₄⋊C₈	—	67	2	no	—
164	#405	—	134	2	no	—
165	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
167	#334	—	67	1	no	—
168	#334	—	67	2	no	—
169	#334	—	67	1	no	—
170	Q₈⋊C₈	—	67	4	no	—
171	Q₈⋊C₈	—	67	4	no	—
172	#322	C₂×D₆₇	1	2	yes	—
173	#322	C₂×D₆₇	1	2	yes	—
174	C₂×Q₁₆	—	67	2	no	—
175	C₂×Q₁₆	—	67	2	no	—
176	C₂×M₁₆	D₂₆₈	1	2	yes	—
177	C₂²×C₈	—	134	2	no	—
178	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
179	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
180	C₄⋊Q₈	—	67	2	no	—
181	C₄⋊Q₈	—	67	2	no	—
182	#125	—	134	2	no	—
183	C₂²⋊Q₈	—	67	4	no	—
184	C₂²⋊Q₈	—	134	2	no	—
185	C₄×Q₈	—	67	4	no	—
186	C₄×Q₈	—	134	2	no	—
187	#117	—	134	2	no	—
188	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
189	#107	—	67	2	no	—
190	#107	—	67	2	no	—
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	67	2	no	—
193	C₄⋊C₈	—	67	2	no	—
194	C₄⋊C₈	—	134	4	no	—
195	C₄⋊C₈	—	134	4	no	—
196	C₄⋊C₈	—	134	2	no	—
197	#103	—	67	4	no	—
198	#103	—	67	4	no	—
199	C₂²⋊C₈	—	2	1	no	—
200	#97	—	134	4	no	—
201	#97	D₂₆₈	1	2	yes	—
202	C₄×C₈	—	134	4	no	—
203	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
204	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
205	C₄×C₈	C₂²×D₆₇	1	2	yes	—
206	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₆₇	1	2	yes	—
209	C₂×Q₈	—	67	4	no	—
210	C₂×Q₈	—	134	2	no	—
211	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
213	C₂²×C₄	#11761628	1	1	yes	characteristic
214	Q₁₆	—	268	2	no	—
215	Q₁₆	—	134	4	no	—
216	M₁₆	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	268	4	no	—
221	C₂×C₈	—	134	2	no	—
222	C₂×C₈	—	134	2	no	—
223	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
227	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
228	C₄⋊C₄	—	67	4	no	—
229	C₄⋊C₄	—	134	2	no	—
230	C₄⋊C₄	—	134	2	no	—
231	C₄⋊C₄	—	134	4	no	—
232	C₄⋊C₄	—	134	2	no	—
233	C₄⋊C₄	—	134	4	no	—
234	C₂²⋊C₄	—	134	4	no	—
235	C₄²	—	134	4	no	—
236	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
237	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
238	C₄²	#11761628	1	1	yes	characteristic
239	C₄²	C₂³×D₆₇	1	1	yes	characteristic
240	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
241	Q₈	—	536	2	no	—
242	Q₈	—	134	4	no	—
243	Q₈	—	134	4	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	134	4	no	—
248	C₂×C₄	—	134	4	no	—
249	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
250	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
251	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
252	C₂×C₄	#14524076	1	4	yes	—
253	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
254	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
255	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	2	4	no	—
260	C₈	—	4	2	no	—
261	C₈	—	268	2	no	—
262	C₈	—	268	2	no	—
263	C₈	#14524051	1	2	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	#16863504	1	2	yes	—
267	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	268	4	no	—
272	C₄	—	268	4	no	—
273	C₄	#16863504	1	2	yes	—
274	C₄	C₂×#14524051	1	1	yes	characteristic
275	C₄	#16864162	1	1	yes	characteristic
276	C₄	C₂×#14524076	1	2	yes	—
277	C₂	—	2	2	no	—
278	C₂	#22568955	1	1	yes	characteristic
279	C₂	C₂×#16863504	1	1	yes	characteristic
280	C₂	#22569127	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1451	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1451	D₆₇
hypocenter	C₆₇	#18072
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #1451

Complements

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

N	H ≅ G/N	type
3. #22559119	C₂	semidirect
4. #22559082	C₂	semidirect
5. #22559029	C₂	semidirect
149. C₆₇	#18072	semidirect

Maximal subgroups

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

subgroup	index
#22570795	2
#22569156	2
#22559119	2
#22559082	2
#22559029	2
C₂×#16863147	2
#22558613	2
C₆₇×#1451	2
#18072	67