Subgroups — #30754768

Order 17152.

498 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22558957	C₂	1	1	yes	maximal, characteristic, complemented
2	#22558600	C₂	1	2	yes	maximal, complemented
3	#22558530	C₂	1	2	yes	maximal, complemented
4	#22558501	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	2	yes	—
7	#16863449	C₂²	1	1	yes	characteristic
8	#16863449	C₂²	1	1	yes	characteristic
9	#16863149	—	2	1	no	—
10	#16863147	—	2	1	no	—
11	D₄⋊(C₆₇⋊C₈)	—	2	4	no	—
12	#16863120	—	2	4	no	—
13	#16863118	—	2	1	no	—
14	#16863118	—	2	1	no	—
15	C₂²×D₈×C₆₇	C₄	1	1	yes	characteristic
16	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
17	C₄×D₈×C₆₇	—	2	1	no	—
18	C₄×D₈×C₆₇	—	2	1	no	—
19	C₄×D₈×C₆₇	—	2	1	no	—
20	C₄×D₈×C₆₇	—	2	1	no	—
21	C₂×#107×C₆₇	C₄	1	1	yes	characteristic
22	C₂×#102×C₆₇	C₄	1	2	yes	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
26	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
27	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
28	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
29	C₂²⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
30	C₄×C₆₇⋊C₈	—	2	1	no	—
31	C₄×C₆₇⋊C₈	—	2	1	no	—
32	C₄×C₆₇⋊C₈	—	2	2	no	—
33	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₆₇	—	2	2	no	—
35	C₂×D₈×C₆₇	—	2	4	no	—
36	C₂×D₈×C₆₇	—	2	1	no	—
37	C₂×D₈×C₆₇	—	2	1	no	—
38	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₄×D₄×C₆₇	—	2	2	no	—
41	C₄×D₄×C₆₇	—	2	2	no	—
42	C₄×D₄×C₆₇	—	2	2	no	—
43	C₄×D₄×C₆₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
46	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	#107×C₆₇	—	2	1	no	—
48	#107×C₆₇	—	2	1	no	—
49	#102×C₆₇	—	2	2	no	—
50	#102×C₆₇	—	2	2	no	—
51	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
52	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
53	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
54	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
55	C₂×C₆₇⋊C₈	—	2	2	no	—
56	C₂×C₆₇⋊C₈	—	2	2	no	—
57	C₂×C₆₇⋊C₈	—	4	2	no	—
58	C₂⁴×C₆₇	—	2	2	no	—
59	C₂×D₄×C₆₇	—	2	2	no	—
60	C₂×D₄×C₆₇	—	2	2	no	—
61	C₂×D₄×C₆₇	—	2	2	no	—
62	C₂×D₄×C₆₇	—	2	2	no	—
63	C₂×D₄×C₆₇	—	2	2	no	—
64	C₂×D₄×C₆₇	M₁₆	1	4	yes	—
65	C₂²×C₄×C₆₇	—	2	2	no	—
66	C₂²×C₄×C₆₇	—	4	2	no	—
67	C₂²×C₄×C₆₇	—	4	2	no	—
68	C₂²×C₄×C₆₇	—	8	4	no	—
69	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
72	D₈×C₆₇	—	2	4	no	—
73	D₈×C₆₇	—	2	4	no	—
74	C₂×C₈×C₆₇	—	2	2	no	—
75	C₂×C₈×C₆₇	—	2	2	no	—
76	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₄⋊C₄×C₆₇	—	2	2	no	—
81	C₄⋊C₄×C₆₇	—	2	2	no	—
82	C₂²⋊C₄×C₆₇	—	4	2	no	—
83	C₂²⋊C₄×C₆₇	—	4	2	no	—
84	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₆₇	#41	1	1	yes	characteristic
88	C₂³×C₆₇	—	2	4	no	—
89	C₂³×C₆₇	—	4	2	no	—
90	C₂³×C₆₇	—	4	2	no	—
91	C₂³×C₆₇	—	8	2	no	—
92	C₂³×C₆₇	—	8	2	no	—
93	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
94	D₄×C₆₇	—	2	4	no	—
95	D₄×C₆₇	—	2	4	no	—
96	D₄×C₆₇	—	4	2	no	—
97	D₄×C₆₇	—	4	2	no	—
98	C₂×C₄×C₆₇	—	2	2	no	—
99	C₂×C₄×C₆₇	—	4	1	no	—
100	C₂×C₄×C₆₇	—	4	2	no	—
101	C₂×C₄×C₆₇	—	8	4	no	—
102	C₂×C₄×C₆₇	—	8	4	no	—
103	C₂×C₄×C₆₇	—	8	2	no	—
104	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
107	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
108	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
111	C₂×C₄×C₆₇	#131	1	2	yes	—
112	C₈×C₆₇	—	2	1	no	—
113	C₈×C₆₇	—	2	1	no	—
114	C₈×C₆₇	—	2	1	no	—
115	C₈×C₆₇	—	2	1	no	—
116	C₆₇⋊C₈	—	4	2	no	—
117	C₆₇⋊C₈	—	4	2	no	—
118	C₆₇⋊C₈	—	8	2	no	—
119	C₂²×C₆₇	—	2	1	no	—
120	C₂²×C₆₇	—	2	1	no	—
121	C₂²×C₆₇	—	4	4	no	—
122	C₂²×C₆₇	—	8	4	no	—
123	C₂²×C₆₇	—	8	4	no	—
124	C₂²×C₆₇	—	8	2	no	—
125	C₂²×C₆₇	—	8	2	no	—
126	C₂²×C₆₇	#390	1	1	yes	characteristic
127	C₂²×C₆₇	#435	1	2	yes	—
128	C₄×C₆₇	—	2	1	no	—
129	C₄×C₆₇	—	2	1	no	—
130	C₄×C₆₇	—	2	1	no	—
131	C₄×C₆₇	—	8	2	no	—
132	C₄×C₆₇	#407	1	1	yes	characteristic
133	C₄×C₆₇	#408	1	1	yes	characteristic
134	C₄×C₆₇	#435	1	2	yes	—
135	C₄×C₆₇	#472	1	2	yes	—
136	#9670	—	67	1	no	maximal, Sylow
137	C₂×C₆₇	—	2	2	no	—
138	C₂×C₆₇	—	8	4	no	—
139	C₂×C₆₇	#1955	1	1	yes	characteristic
140	C₂×C₆₇	#1976	1	1	yes	characteristic
141	C₂×C₆₇	#1978	1	1	yes	characteristic
142	C₂×C₄×D₈	D₆₇	1	1	yes	characteristic
143	#1564	—	67	1	no	—
144	#1478	—	67	2	no	—
145	#1460	—	67	2	no	—
146	#1452	—	67	1	no	—
147	C₆₇	#9670	1	1	yes	characteristic, complemented, Sylow
148	C₂²×D₈	Dic₆₇	1	1	yes	characteristic
149	C₂×C₄×D₄	C₂×D₆₇	1	2	yes	—
150	C₄×D₈	—	2	1	no	—
151	C₄×D₈	—	2	1	no	—
152	C₄×D₈	—	2	1	no	—
153	C₄×D₈	—	2	1	no	—
154	#431	—	67	1	no	—
155	#431	—	67	1	no	—
156	#431	—	67	2	no	—
157	C₂×#107	Dic₆₇	1	1	yes	characteristic
158	C₂×#102	Dic₆₇	1	2	yes	—
159	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
160	#335	—	134	1	no	—
161	#334	—	134	1	no	—
162	D₄⋊C₈	—	134	4	no	—
163	#323	—	134	4	no	—
164	#322	—	134	1	no	—
165	#322	—	134	1	no	—
166	C₂²×D₄	C₂×Dic₆₇	1	2	yes	—
167	C₂³×C₄	—	2	2	no	—
168	C₂×D₈	—	2	4	no	—
169	C₂×D₈	—	2	1	no	—
170	C₂×D₈	—	2	1	no	—
171	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
172	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
173	C₄×D₄	—	2	2	no	—
174	C₄×D₄	—	2	2	no	—
175	C₄×D₄	—	2	2	no	—
176	C₄×D₄	—	2	2	no	—
177	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
178	C₂×C₂²⋊C₄	—	2	2	no	—
179	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
180	#107	—	2	1	no	—
181	#107	—	2	1	no	—
182	C₄⋊C₈	—	67	2	no	—
183	C₄⋊C₈	—	134	2	no	—
184	C₄⋊C₈	—	134	1	no	—
185	#102	—	2	2	no	—
186	#102	—	2	2	no	—
187	C₂²⋊C₈	—	67	2	no	—
188	C₂²⋊C₈	—	134	2	no	—
189	C₂²⋊C₈	—	134	1	no	—
190	C₄×C₈	—	134	2	no	—
191	C₄×C₈	—	134	1	no	—
192	C₄×C₈	—	134	1	no	—
193	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
194	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
195	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
196	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
197	C₂⁴	—	2	2	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	#11761612	1	4	yes	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	—	8	4	no	—
208	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
209	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
210	C₂²×C₄	#11761627	1	1	yes	characteristic
211	D₈	—	2	4	no	—
212	D₈	—	2	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	268	2	no	—
216	C₂×C₈	—	134	2	no	—
217	C₂×C₈	—	134	2	no	—
218	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
219	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
220	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
221	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	—	4	2	no	—
225	C₂²⋊C₄	—	4	2	no	—
226	C₄²	D₄×D₆₇	1	1	yes	characteristic
227	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
228	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
229	C₄²	#11761627	1	1	yes	characteristic
230	C₂³	—	2	4	no	—
231	C₂³	—	4	2	no	—
232	C₂³	—	4	2	no	—
233	C₂³	—	8	2	no	—
234	C₂³	—	8	2	no	—
235	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
236	D₄	—	2	4	no	—
237	D₄	—	2	4	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	8	4	no	—
244	C₂×C₄	—	8	4	no	—
245	C₂×C₄	—	8	2	no	—
246	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
247	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
248	C₂×C₄	#14524014	1	1	yes	characteristic
249	C₂×C₄	#14524014	1	1	yes	characteristic
250	C₂×C₄	#14524030	1	1	yes	characteristic
251	C₂×C₄	#14524039	1	1	yes	characteristic
252	C₂×C₄	C₂×#11761612	1	2	yes	—
253	C₂×C₄	#14524050	1	2	yes	—
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	268	2	no	—
259	C₈	—	268	2	no	—
260	C₈	—	536	2	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	4	4	no	—
264	C₂²	—	8	4	no	—
265	C₂²	—	8	4	no	—
266	C₂²	—	8	2	no	—
267	C₂²	—	8	2	no	—
268	C₂²	#16863299	1	1	yes	characteristic
269	C₂²	#16863468	1	2	yes	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	8	2	no	—
274	C₄	#16863360	1	1	yes	characteristic
275	C₄	#16863363	1	1	yes	characteristic
276	C₄	#16863468	1	2	yes	—
277	C₄	#16863663	1	2	yes	—
278	C₂	—	2	2	no	—
279	C₂	—	8	4	no	—
280	C₂	#22561008	1	1	yes	characteristic
281	C₂	#22561108	1	1	yes	characteristic
282	C₂	#22561120	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524014
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×C₄×D₈×C₆₇	C₂
socle	C₂²×C₆₇	#390
prime radical	C₂²×D₈×C₆₇	C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×C₄×D₈	D₆₇
hypocenter	C₆₇	#9670
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #22558957	C₂	semidirect
2. #22558600	C₂	semidirect
3. #22558530	C₂	semidirect
4. #22558501	C₂	semidirect
147. C₆₇	#9670	semidirect

Maximal subgroups

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

subgroup	index
#22558957	2
#22558600	2
#22558530	2
#22558501	2
C₂×C₄×D₈×C₆₇	2
#9670	67