Subgroups — #30754413

Order 17152.

404 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	#22560361	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863146	C₂	1	2	yes	maximal
4	#22558043	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
8	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	1	yes	characteristic
9	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	1	yes	characteristic
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
12	#16863151	C₄	1	2	yes	—
13	#16863146	—	2	4	no	—
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₆₇	—	2	1	no	—
17	C₄³×C₆₇	C₂²	1	1	yes	characteristic
18	#336×C₆₇	—	2	1	no	—
19	C₂²×C₆₇⋊C₈	—	2	2	no	—
20	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
26	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
27	C₄⋊(C₆₇⋊C₈)	C₈	1	4	yes	complemented
28	C₄×C₆₇⋊C₈	—	2	2	no	—
29	C₄×C₆₇⋊C₈	—	2	2	no	—
30	C₂²×C₈×C₆₇	—	2	1	no	—
31	C₂²×C₈×C₆₇	—	2	1	no	—
32	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₆₇	—	2	1	no	—
40	C₄⋊C₈×C₆₇	—	2	1	no	—
41	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
42	C₄⋊C₈×C₆₇	Q₈	1	2	yes	—
43	C₄×C₈×C₆₇	—	4	2	no	—
44	C₂×C₆₇⋊C₈	—	2	2	no	—
45	C₂×C₆₇⋊C₈	—	2	2	no	—
46	C₂×C₆₇⋊C₈	—	2	2	no	—
47	C₂×C₆₇⋊C₈	—	2	2	no	—
48	C₂×C₆₇⋊C₈	—	2	2	no	—
49	C₂×C₆₇⋊C₈	—	4	2	no	—
50	C₂×C₆₇⋊C₈	—	4	2	no	—
51	C₂×C₆₇⋊C₈	—	4	2	no	—
52	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
53	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₆₇	—	2	2	no	—
61	C₂×C₈×C₆₇	—	2	2	no	—
62	C₂×C₈×C₆₇	—	4	1	no	—
63	C₂×C₈×C₆₇	—	4	1	no	—
64	C₂×C₈×C₆₇	—	4	1	no	—
65	C₂×C₈×C₆₇	—	4	1	no	—
66	C₄²×C₆₇	—	2	2	no	—
67	C₄²×C₆₇	—	2	1	no	—
68	C₄²×C₆₇	—	2	1	no	—
69	C₄²×C₆₇	—	2	1	no	—
70	C₄²×C₆₇	—	2	1	no	—
71	C₄²×C₆₇	—	4	1	no	—
72	C₄²×C₆₇	C₂×C₈	1	2	yes	—
73	C₄²×C₆₇	M₁₆	1	2	yes	—
74	C₄²×C₆₇	SD₁₆	1	2	yes	—
75	C₄²×C₆₇	SD₁₆	1	2	yes	—
76	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
78	C₄²×C₆₇	#41	1	1	yes	characteristic
79	C₄²×C₆₇	#41	1	1	yes	characteristic
80	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₆₇	—	2	1	no	—
82	C₂×C₄×C₆₇	—	2	1	no	—
83	C₂×C₄×C₆₇	—	2	1	no	—
84	C₂×C₄×C₆₇	—	2	1	no	—
85	C₂×C₄×C₆₇	—	2	1	no	—
86	C₂×C₄×C₆₇	—	2	1	no	—
87	C₂×C₄×C₆₇	—	2	1	no	—
88	C₂×C₄×C₆₇	—	2	1	no	—
89	C₂×C₄×C₆₇	—	2	1	no	—
90	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
91	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
92	C₂×C₄×C₆₇	#106	1	2	yes	—
93	C₂×C₄×C₆₇	#106	1	2	yes	—
94	C₂×C₄×C₆₇	#108	1	2	yes	—
95	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
104	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
109	C₈×C₆₇	—	4	2	no	—
110	C₈×C₆₇	—	4	2	no	—
111	C₆₇⋊C₈	—	2	4	no	—
112	C₆₇⋊C₈	—	4	2	no	—
113	C₆₇⋊C₈	—	8	2	no	—
114	C₂²×C₆₇	#389	1	1	yes	characteristic
115	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
116	C₂²×C₆₇	#424	1	1	yes	characteristic
117	C₂²×C₆₇	C₂×#106	1	1	yes	characteristic
118	C₂²×C₆₇	#428	1	1	yes	characteristic
119	C₂²×C₆₇	C₂×#108	1	1	yes	characteristic
120	C₂²×C₆₇	#430	1	1	yes	characteristic
121	C₄×C₆₇	—	2	2	no	—
122	C₄×C₆₇	—	2	2	no	—
123	C₄×C₆₇	—	2	1	no	—
124	C₄×C₆₇	—	2	1	no	—
125	C₄×C₆₇	—	4	1	no	—
126	C₄×C₆₇	#334	1	4	yes	complemented
127	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
128	C₄×C₆₇	#436	1	1	yes	characteristic
129	C₄×C₆₇	C₈×Q₈	1	1	yes	characteristic
130	C₄×C₆₇	#446	1	1	yes	characteristic
131	C₄×C₆₇	#460	1	1	yes	characteristic
132	C₄×C₆₇	#461	1	1	yes	characteristic
133	C₄×C₆₇	#475	1	1	yes	characteristic
134	C₄×C₆₇	#476	1	1	yes	characteristic
135	#9602	—	67	1	no	maximal, Sylow
136	C₂×C₆₇	C₂×#334	1	1	yes	characteristic
137	C₂×C₆₇	#1565	1	1	yes	characteristic
138	C₂×C₆₇	#1570	1	1	yes	characteristic
139	C₂×C₆₇	#1574	1	1	yes	characteristic
140	C₂×C₆₇	#1939	1	1	yes	characteristic
141	C₂×C₆₇	#1948	1	1	yes	characteristic
142	C₂×C₆₇	#1950	1	1	yes	characteristic
143	#1842	—	67	1	no	—
144	C₄×C₄⋊C₈	—	67	1	no	—
145	C₄×C₄⋊C₈	D₆₇	1	1	yes	characteristic
146	C₂×#334	—	67	2	no	—
147	#1277	—	67	2	no	—
148	C₆₇	#9602	1	1	yes	characteristic, complemented, Sylow
149	C₂×C₄⋊C₈	—	67	1	no	—
150	C₂×C₄⋊C₈	—	67	1	no	—
151	C₂×C₄⋊C₈	—	67	2	no	—
152	C₂×C₄⋊C₈	—	134	1	no	—
153	C₂×C₄⋊C₈	—	134	1	no	—
154	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
155	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
156	C₂×C₄×C₈	—	2	1	no	—
157	C₂×C₄×C₈	—	67	2	no	—
158	C₄³	C₂×D₆₇	1	1	yes	characteristic
159	#336	—	2	1	no	—
160	#336	—	67	2	no	—
161	#334	—	134	4	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	—	2	1	no	—
164	C₂²×C₈	—	67	2	no	—
165	C₂²×C₈	—	67	2	no	—
166	C₂²×C₈	—	134	2	no	—
167	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
168	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
169	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
170	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
171	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
172	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
173	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
174	C₄⋊C₈	—	2	1	no	—
175	C₄⋊C₈	—	2	1	no	—
176	C₄⋊C₈	—	268	2	no	—
177	C₄⋊C₈	—	268	2	no	—
178	C₄⋊C₈	—	67	4	no	—
179	C₄⋊C₈	—	134	2	no	—
180	C₄⋊C₈	—	134	2	no	—
181	C₄⋊C₈	—	134	2	no	—
182	C₄⋊C₈	Dic₁₃₄	1	2	yes	—
183	C₄⋊C₈	D₂₆₈	1	2	yes	—
184	C₄×C₈	—	4	2	no	—
185	C₄×C₈	—	134	2	no	—
186	C₄×C₈	—	134	2	no	—
187	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
188	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
189	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
190	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
193	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	268	2	no	—
201	C₂×C₈	—	268	2	no	—
202	C₂×C₈	—	268	2	no	—
203	C₂×C₈	—	67	4	no	—
204	C₂×C₈	—	134	2	no	—
205	C₂×C₈	—	134	2	no	—
206	C₂×C₈	—	134	2	no	—
207	C₂×C₈	—	134	2	no	—
208	C₂×C₈	—	134	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	4	1	no	—
215	C₄²	C₈×D₆₇	1	2	yes	—
216	C₄²	#11761611	1	2	yes	—
217	C₄²	#11761615	1	2	yes	—
218	C₄²	#11761616	1	2	yes	—
219	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
220	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
221	C₄²	#11761628	1	1	yes	characteristic
222	C₄²	#11761628	1	1	yes	characteristic
223	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	Dic₆₇⋊C₈	1	2	yes	—
234	C₂×C₄	Dic₆₇⋊C₈	1	2	yes	—
235	C₂×C₄	#14523974	1	2	yes	—
236	C₂×C₄	#14523974	1	2	yes	—
237	C₂×C₄	#14523978	1	2	yes	—
238	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
239	C₂×C₄	C₄×Dic₁₃₄	1	1	yes	characteristic
240	C₂×C₄	#14524017	1	1	yes	characteristic
241	C₂×C₄	#14524022	1	1	yes	characteristic
242	C₂×C₄	#14524035	1	1	yes	characteristic
243	C₂×C₄	#14524043	1	1	yes	characteristic
244	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
245	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
246	C₂×C₄	#14524049	1	1	yes	characteristic
247	C₂×C₄	#14524049	1	1	yes	characteristic
248	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
249	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
250	C₂×C₄	#14524068	1	1	yes	characteristic
251	C₂×C₄	#14524073	1	1	yes	characteristic
252	C₈	—	4	2	no	—
253	C₈	—	4	2	no	—
254	C₈	—	268	2	no	—
255	C₈	—	536	2	no	—
256	C₈	—	134	4	no	—
257	C₂²	#16863294	1	1	yes	characteristic
258	C₂²	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
259	C₂²	#16863422	1	1	yes	characteristic
260	C₂²	C₂×#14523974	1	1	yes	characteristic
261	C₂²	#16863439	1	1	yes	characteristic
262	C₂²	C₂×#14523978	1	1	yes	characteristic
263	C₂²	#16863444	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	2	no	—
268	C₄	—	4	1	no	—
269	C₄	#16863146	1	4	yes	complemented
270	C₄	C₈×D₂₆₈	1	1	yes	characteristic
271	C₄	#16863475	1	1	yes	characteristic
272	C₄	C₈×Dic₁₃₄	1	1	yes	characteristic
273	C₄	#16863526	1	1	yes	characteristic
274	C₄	#16863596	1	1	yes	characteristic
275	C₄	#16863601	1	1	yes	characteristic
276	C₄	#16863681	1	1	yes	characteristic
277	C₄	#16863686	1	1	yes	characteristic
278	C₂	C₂×#16863146	1	1	yes	characteristic
279	C₂	#22558962	1	1	yes	characteristic
280	C₂	#22558979	1	1	yes	characteristic
281	C₂	#22558997	1	1	yes	characteristic
282	C₂	#22560927	1	1	yes	characteristic
283	C₂	#22560972	1	1	yes	characteristic
284	C₂	#22560984	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×C₄⋊C₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×C₄⋊C₈	D₆₇
hypocenter	C₆₇	#9602
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
27. C₄⋊(C₆₇⋊C₈)	C₈	semidirect
126. C₄×C₆₇	#334	semidirect
148. C₆₇	#9602	semidirect
269. C₄	#16863146	semidirect

Maximal subgroups

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

subgroup	index
#22560361	2
C₄×C₄⋊(C₆₇⋊C₈)	2
C₂×#16863146	2
#22558043	2
C₄×C₄⋊C₈×C₆₇	2
#9602	67