Subgroups — #30874157

Order 17152.

898 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863503	C₂	1	2	yes	maximal
2	#22563130	C₂	1	2	yes	maximal
3	#22560948	C₂	1	2	yes	maximal
4	#22560465	C₂	1	4	yes	maximal
5	#22560373	C₂	1	1	yes	maximal, characteristic
6	#22560224	C₂	1	2	yes	maximal
7	#22560076	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₈×Dic₆₇	C₂²	1	4	yes	—
10	#16863503	—	2	8	no	—
11	C₂×#14523977	C₂²	1	2	yes	—
12	C₂×#14523975	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×#14523967	C₂²	1	4	yes	—
15	C₂×#14523953	C₂²	1	2	yes	—
16	#16863295	—	2	4	no	—
17	#16863295	C₂²	1	4	yes	—
18	#16863262	—	2	2	no	—
19	#16863262	C₂²	1	2	yes	—
20	C₄×C₄⋊Dic₆₇	—	2	2	no	—
21	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
22	#16863158	C₂²	1	4	yes	—
23	#16863150	C₂²	1	2	yes	—
24	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
26	C₄×Q₁₆×C₆₇	C₄	1	8	yes	complemented
27	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#103×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
31	Q₈×Dic₆₇	—	2	8	no	—
32	Q₈×Dic₆₇	—	4	4	no	—
33	C₂×C₄⋊Dic₆₇	—	2	4	no	—
34	C₂×C₄⋊Dic₆₇	—	2	4	no	—
35	C₂×C₄⋊Dic₆₇	D₄	1	2	yes	—
36	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
37	C₂×C₄×Dic₆₇	—	2	4	no	—
38	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
39	#14523977	—	4	2	no	—
40	#14523975	—	2	4	no	—
41	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
42	#14523967	—	2	8	no	—
43	#14523953	—	2	4	no	—
44	C₆₇⋊#95	—	2	4	no	—
45	C₆₇⋊#95	—	2	2	no	—
46	C₆₇⋊#95	—	2	2	no	—
47	C₆₇⋊#95	D₄	1	2	yes	—
48	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
49	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	—
50	C₂×Q₁₆×C₆₇	D₄	1	4	yes	—
51	C₂×Q₁₆×C₆₇	Q₈	1	4	yes	—
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₄×Q₈×C₆₇	—	2	4	no	—
55	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	—
56	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
58	C₂×C₄²×C₆₇	—	2	2	no	—
59	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	#107×C₆₇	C₂×C₄	1	4	yes	—
61	#103×C₆₇	C₂×C₄	1	8	yes	—
62	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
63	C₂²×Dic₆₇	—	2	4	no	—
64	C₂²×Dic₆₇	C₂×D₄	1	4	yes	—
65	C₂×C₆₇⋊C₈	—	2	2	no	—
66	C₂×C₆₇⋊C₈	—	2	2	no	—
67	C₂×C₆₇⋊C₈	—	2	2	no	—
68	C₄⋊Dic₆₇	—	2	4	no	—
69	C₄⋊Dic₆₇	—	2	4	no	—
70	C₄⋊Dic₆₇	—	4	2	no	—
71	C₄⋊Dic₆₇	—	4	8	no	—
72	C₄⋊Dic₆₇	—	4	8	no	—
73	C₄×Dic₆₇	—	2	2	no	—
74	C₄×Dic₆₇	—	2	2	no	—
75	C₄×Dic₆₇	—	4	8	no	—
76	C₂×Q₈×C₆₇	—	2	2	no	—
77	C₂×Q₈×C₆₇	—	2	2	no	—
78	C₂×Q₈×C₆₇	—	2	2	no	—
79	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
80	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
81	C₂×Q₈×C₆₇	C₂×Q₈	1	4	yes	—
82	C₂²×C₄×C₆₇	—	2	2	no	—
83	C₂²×C₄×C₆₇	—	2	2	no	—
84	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
87	Q₁₆×C₆₇	C₄⋊C₄	1	16	yes	complemented
88	C₂×C₈×C₆₇	—	2	1	no	—
89	C₂×C₈×C₆₇	—	2	1	no	—
90	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
94	C₄⋊C₄×C₆₇	—	2	8	no	—
95	C₄⋊C₄×C₆₇	—	2	2	no	—
96	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
97	C₄²×C₆₇	—	2	8	no	—
98	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
99	C₄²×C₆₇	#41	1	2	yes	—
100	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
101	Q₈×C₆₇	—	2	4	no	—
102	Q₈×C₆₇	—	2	4	no	—
103	Q₈×C₆₇	—	2	4	no	—
104	Q₈×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
105	C₂×C₄×C₆₇	—	2	4	no	—
106	C₂×C₄×C₆₇	—	2	4	no	—
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	2	4	no	—
111	C₂×C₄×C₆₇	—	4	4	no	—
112	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
119	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
120	C₂×Dic₆₇	—	2	4	no	—
121	C₂×Dic₆₇	—	2	4	no	—
122	C₂×Dic₆₇	—	2	4	no	—
123	C₂×Dic₆₇	—	4	4	no	—
124	C₂×Dic₆₇	—	4	4	no	—
125	C₂×Dic₆₇	—	4	4	no	—
126	C₈×C₆₇	—	2	2	no	—
127	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
128	C₆₇⋊C₈	—	4	2	no	—
129	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
130	C₂²×C₆₇	D₄²	1	1	yes	characteristic
131	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
132	C₂²×C₆₇	#575	1	2	yes	—
133	C₂²×C₆₇	#578	1	2	yes	—
134	C₄×C₆₇	—	2	8	no	—
135	C₄×C₆₇	—	2	2	no	—
136	C₄×C₆₇	—	2	2	no	—
137	C₄×C₆₇	—	4	4	no	—
138	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
139	C₄×C₆₇	#520	1	1	yes	characteristic
140	C₄×C₆₇	#547	1	1	yes	characteristic
141	C₄×C₆₇	#554	1	1	yes	characteristic
142	Dic₆₇	—	4	4	no	—
143	Dic₆₇	—	8	4	no	—
144	#22675	—	67	1	no	maximal, Sylow
145	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₆₇	#2943	1	1	yes	characteristic
147	C₂×C₆₇	#2950	1	1	yes	characteristic
148	C₂×C₆₇	#3277	1	2	yes	—
149	C₂×C₆₇	#3390	1	2	yes	—
150	C₂×#441	—	67	2	no	—
151	C₂×C₄×Q₁₆	D₆₇	1	1	yes	characteristic
152	#2351	—	67	2	no	—
153	#1943	—	67	2	no	—
154	#1863	—	67	4	no	—
155	#1846	—	67	1	no	—
156	#1805	—	67	2	no	—
157	#1770	—	67	1	no	—
158	C₆₇	#22675	1	1	yes	characteristic, complemented, Sylow
159	C₂²×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
160	C₂×C₄×Q₈	—	67	4	no	—
161	C₂×C₄×Q₈	C₂×D₆₇	1	2	yes	—
162	#441	—	134	8	no	—
163	C₄×Q₁₆	Dic₆₇	1	8	yes	complemented
164	C₂×#107	—	67	2	no	—
165	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
166	C₂×#106	—	67	2	no	—
167	C₂×C₄⋊C₈	—	67	1	no	—
168	C₂×#103	—	67	4	no	—
169	C₂×#103	C₂×D₆₇	1	2	yes	—
170	C₂×#97	—	67	2	no	—
171	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
172	#389	—	67	4	no	—
173	#389	—	134	4	no	—
174	#382	—	67	2	no	—
175	#382	—	134	2	no	—
176	C₄×C₄⋊C₄	—	67	1	no	—
177	C₄×C₄⋊C₄	—	134	2	no	—
178	#340	—	67	4	no	—
179	#336	—	67	2	no	—
180	C₂²×Q₈	C₂²×D₆₇	1	2	yes	—
181	C₂×Q₁₆	Dic₁₃₄	1	4	yes	—
182	C₂×Q₁₆	D₂₆₈	1	4	yes	—
183	C₂×Q₁₆	C₂×Dic₆₇	1	4	yes	—
184	C₂²×C₈	—	67	2	no	—
185	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
187	C₄×Q₈	—	2	4	no	—
188	C₄×Q₈	—	268	4	no	—
189	C₄×Q₈	—	134	8	no	—
190	C₄×Q₈	C₂×Dic₆₇	1	8	yes	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	67	2	no	—
193	C₂×C₄⋊C₄	—	67	4	no	—
194	C₂×C₄⋊C₄	—	134	4	no	—
195	C₂×C₄⋊C₄	—	134	4	no	—
196	C₂×C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	—	67	2	no	—
199	C₂×C₄²	—	134	4	no	—
200	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
201	#107	—	268	2	no	—
202	#107	C₂×Dic₆₇	1	4	yes	—
203	#106	—	134	4	no	—
204	C₄⋊C₈	—	134	2	no	—
205	#103	—	134	8	no	—
206	#103	C₂×Dic₆₇	1	8	yes	—
207	#97	—	134	4	no	—
208	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
209	#95	—	67	2	no	—
210	#95	—	134	4	no	—
211	#95	—	134	2	no	—
212	#95	—	134	2	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	C₂²×Dic₆₇	1	4	yes	—
217	C₂×Q₈	C₂×D₂₆₈	1	4	yes	—
218	C₂×Q₈	C₂×Dic₁₃₄	1	4	yes	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	67	4	no	—
222	C₂²×C₄	—	134	4	no	—
223	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
224	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
225	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
226	Q₁₆	C₄⋊Dic₆₇	1	16	yes	complemented
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	134	2	no	—
230	C₂×C₈	—	134	2	no	—
231	C₂×C₈	—	134	2	no	—
232	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
233	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
234	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
235	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
236	C₄⋊C₄	—	2	8	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	268	8	no	—
239	C₄⋊C₄	—	268	2	no	—
240	C₄⋊C₄	—	268	8	no	—
241	C₄⋊C₄	—	134	4	no	—
242	C₄⋊C₄	—	134	4	no	—
243	C₄⋊C₄	C₂²×Dic₆₇	1	4	yes	—
244	C₄²	—	2	8	no	—
245	C₄²	—	268	8	no	—
246	C₄²	—	134	2	no	—
247	C₄²	—	134	2	no	—
248	C₄²	C₂²×Dic₆₇	1	2	yes	—
249	C₄²	#11761627	1	2	yes	—
250	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
251	Q₈	—	2	4	no	—
252	Q₈	—	2	4	no	—
253	Q₈	—	2	4	no	—
254	Q₈	C₂×C₄⋊Dic₆₇	1	8	yes	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	268	4	no	—
263	C₂×C₄	—	268	4	no	—
264	C₂×C₄	—	268	4	no	—
265	C₂×C₄	—	134	4	no	—
266	C₂×C₄	—	134	4	no	—
267	C₂×C₄	—	134	4	no	—
268	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
269	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
270	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
271	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
272	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
273	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
274	C₂×C₄	#14524089	1	1	yes	characteristic
275	C₂×C₄	#14524090	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	268	2	no	—
278	C₈	C₂×C₄⋊Dic₆₇	1	4	yes	—
279	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
280	C₂²	D₄×D₂₆₈	1	1	yes	characteristic
281	C₂²	D₄×Dic₁₃₄	1	1	yes	characteristic
282	C₂²	#16864193	1	2	yes	—
283	C₂²	#16864209	1	2	yes	—
284	C₄	—	2	8	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	4	4	no	—
288	C₄	—	268	4	no	—
289	C₄	—	536	4	no	—
290	C₄	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
291	C₄	#16863879	1	1	yes	characteristic
292	C₄	#16864050	1	1	yes	characteristic
293	C₄	#16864103	1	1	yes	characteristic
294	C₂	D₄×C₄⋊Dic₆₇	1	1	yes	characteristic
295	C₂	#22569010	1	1	yes	characteristic
296	C₂	#22569077	1	1	yes	characteristic
297	C₂	#22571865	1	2	yes	—
298	C₂	#22573174	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×C₄×Q₁₆×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₆₇
hypocenter	C₆₇	#22675
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
26. C₄×Q₁₆×C₆₇	C₄	semidirect
87. Q₁₆×C₆₇	C₄⋊C₄	semidirect
158. C₆₇	#22675	semidirect
163. C₄×Q₁₆	Dic₆₇	semidirect
226. Q₁₆	C₄⋊Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863503	2
#22563130	2
#22560948	2
#22560465	2
#22560373	2
#22560224	2
#22560076	2
C₂×C₄×Q₁₆×C₆₇	2
#22675	67