Subgroups — C₆₇×#33583 #30712834

Order 17152.

702 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#510×C₆₇	C₂	1	2	yes	maximal
2	C₂×#494×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2209	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2208	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2150	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2141	C₂	1	2	yes	maximal, complemented
8	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
9	C₂²×Q₁₆×C₆₇	—	2	1	no	—
10	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
12	C₄×#41×C₆₇	—	2	1	no	—
13	#510×C₆₇	—	2	8	no	—
14	C₂×Q₃₂×C₆₇	C₂²	1	4	yes	—
15	C₂×SD₃₂×C₆₇	C₂²	1	4	yes	—
16	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
17	#494×C₆₇	—	2	2	no	—
18	#494×C₆₇	C₂²	1	4	yes	—
19	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
20	C₄×SD₁₆×C₆₇	—	2	2	no	—
21	C₄×D₈×C₆₇	C₂²	1	2	yes	—
22	#427×C₆₇	C₂²	1	1	yes	characteristic
23	#416×C₆₇	—	2	1	no	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₄⋊C₁₆×C₆₇	C₂²	1	4	yes	—
26	#358×C₆₇	C₂²	1	8	yes	—
27	C₂×#41×C₆₇	—	2	1	no	—
28	C₂²×Q₈×C₆₇	—	2	2	no	—
29	#135×C₆₇	—	2	2	no	—
30	#135×C₆₇	D₄	1	4	yes	—
31	C₂×Q₁₆×C₆₇	—	2	4	no	—
32	C₂×Q₁₆×C₆₇	—	2	2	no	—
33	C₂×Q₁₆×C₆₇	—	2	4	no	—
34	C₂×Q₁₆×C₆₇	—	2	4	no	—
35	C₂×Q₁₆×C₆₇	—	4	4	no	—
36	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
37	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
38	C₂×SD₁₆×C₆₇	—	2	1	no	—
39	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
42	C₄⋊Q₈×C₆₇	—	2	4	no	—
43	C₄⋊Q₈×C₆₇	—	4	1	no	—
44	C₄⋊Q₈×C₆₇	—	4	1	no	—
45	C₄×Q₈×C₆₇	—	2	2	no	—
46	C₄×D₄×C₆₇	—	2	2	no	—
47	C₄×D₄×C₆₇	—	4	2	no	—
48	#117×C₆₇	—	2	1	no	—
49	#117×C₆₇	—	4	1	no	—
50	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
51	C₂×C₄²×C₆₇	—	4	1	no	—
52	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
53	Q₃₂×C₆₇	—	2	8	no	—
54	SD₃₂×C₆₇	—	2	8	no	—
55	M₃₂×C₆₇	—	2	4	no	—
56	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
57	#107×C₆₇	C₂³	1	2	yes	—
58	#106×C₆₇	—	2	1	no	—
59	#103×C₆₇	—	2	2	no	—
60	#102×C₆₇	—	2	2	no	—
61	C₄×C₈×C₆₇	D₄	1	2	yes	—
62	C₄×C₈×C₆₇	C₂³	1	2	yes	—
63	#41×C₆₇	—	2	4	no	—
64	#41×C₆₇	—	4	2	no	—
65	C₂×Q₈×C₆₇	—	2	1	no	—
66	C₂×Q₈×C₆₇	—	2	4	no	—
67	C₂×Q₈×C₆₇	—	4	2	no	—
68	C₂×Q₈×C₆₇	—	4	4	no	—
69	C₂×Q₈×C₆₇	—	4	4	no	—
70	C₂×D₄×C₆₇	—	2	1	no	—
71	C₂×D₄×C₆₇	—	4	1	no	—
72	C₂²×C₄×C₆₇	—	4	1	no	—
73	C₂²×C₄×C₆₇	—	4	2	no	—
74	C₂²×C₄×C₆₇	—	4	2	no	—
75	C₂²×C₄×C₆₇	D₈	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
77	Q₁₆×C₆₇	—	2	4	no	—
78	Q₁₆×C₆₇	—	2	1	no	—
79	Q₁₆×C₆₇	—	2	4	no	—
80	Q₁₆×C₆₇	—	4	8	no	—
81	Q₁₆×C₆₇	—	4	4	no	—
82	Q₁₆×C₆₇	C₂×D₄	1	2	yes	—
83	SD₁₆×C₆₇	—	2	4	no	—
84	D₈×C₆₇	—	2	1	no	—
85	D₈×C₆₇	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₆₇	—	2	2	no	—
87	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₆₇	#41	1	2	yes	—
91	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₆₇	—	2	2	no	—
93	C₄⋊C₄×C₆₇	—	4	4	no	—
94	C₄⋊C₄×C₆₇	—	4	4	no	—
95	C₄⋊C₄×C₆₇	—	4	2	no	—
96	C₂²⋊C₄×C₆₇	—	4	1	no	—
97	C₂²⋊C₄×C₆₇	—	4	2	no	—
98	C₄²×C₆₇	—	4	1	no	—
99	C₄²×C₆₇	—	4	2	no	—
100	C₄²×C₆₇	D₈	1	1	yes	characteristic
101	C₄²×C₆₇	D₈	1	1	yes	characteristic
102	C₄²×C₆₇	C₂×D₄	1	2	yes	—
103	C₁₆×C₆₇	—	2	4	no	—
104	C₂³×C₆₇	—	4	1	no	—
105	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
106	Q₈×C₆₇	—	2	2	no	—
107	Q₈×C₆₇	—	4	1	no	—
108	Q₈×C₆₇	—	4	4	no	—
109	Q₈×C₆₇	—	4	4	no	—
110	Q₈×C₆₇	—	8	4	no	—
111	D₄×C₆₇	—	2	2	no	—
112	D₄×C₆₇	—	4	4	no	—
113	D₄×C₆₇	—	4	1	no	—
114	C₂×C₄×C₆₇	—	2	4	no	—
115	C₂×C₄×C₆₇	—	4	1	no	—
116	C₂×C₄×C₆₇	—	4	4	no	—
117	C₂×C₄×C₆₇	—	4	2	no	—
118	C₂×C₄×C₆₇	—	4	4	no	—
119	C₂×C₄×C₆₇	—	8	4	no	—
120	C₂×C₄×C₆₇	—	8	4	no	—
121	C₂×C₄×C₆₇	#121	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
126	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
127	C₈×C₆₇	—	2	2	no	—
128	C₈×C₆₇	#121	1	2	yes	—
129	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
131	C₂²×C₆₇	—	2	2	no	—
132	C₂²×C₆₇	—	4	2	no	—
133	C₂²×C₆₇	—	8	2	no	—
134	C₂²×C₆₇	#459	1	2	yes	—
135	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
136	C₄×C₆₇	—	2	2	no	—
137	C₄×C₆₇	—	4	2	no	—
138	C₄×C₆₇	—	8	2	no	—
139	C₄×C₆₇	—	8	4	no	—
140	C₄×C₆₇	#459	1	2	yes	—
141	C₄×C₆₇	#510	1	4	yes	complemented
142	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
143	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
144	#33583	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂×C₆₇	—	2	2	no	—
146	C₂×C₆₇	—	8	2	no	—
147	C₂×C₆₇	C₂×#459	1	1	yes	characteristic
148	C₂×C₆₇	C₂×#510	1	2	yes	—
149	C₂×#510	C₂×C₆₇	1	2	yes	—
150	C₂×#494	C₂×C₆₇	1	1	yes	characteristic, complemented
151	C₄×#135	C₂×C₆₇	1	1	yes	characteristic
152	#2209	C₂×C₆₇	1	4	yes	complemented
153	#2208	C₂×C₆₇	1	4	yes	complemented
154	#2150	C₂×C₆₇	1	1	yes	characteristic, complemented
155	#2141	C₂×C₆₇	1	2	yes	complemented
156	C₆₇	#33583	1	1	yes	characteristic, direct product factor, Sylow
157	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
158	C₂²×Q₁₆	—	2	1	no	—
159	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
160	C₂×C₄⋊Q₈	—	2	1	no	—
161	C₄×#41	—	2	1	no	—
162	#510	—	2	8	no	—
163	C₂×Q₃₂	C₂²×C₆₇	1	4	yes	—
164	C₂×SD₃₂	C₂²×C₆₇	1	4	yes	—
165	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
166	#494	—	2	2	no	—
167	#494	C₂²×C₆₇	1	4	yes	—
168	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
169	C₄×SD₁₆	—	2	2	no	—
170	C₄×D₈	C₂²×C₆₇	1	2	yes	—
171	#427	C₂²×C₆₇	1	1	yes	characteristic
172	#416	—	2	1	no	—
173	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
174	C₄⋊C₁₆	C₂²×C₆₇	1	4	yes	—
175	#358	C₂²×C₆₇	1	8	yes	—
176	C₂×#41	—	2	1	no	—
177	C₂²×Q₈	—	2	2	no	—
178	#135	—	2	2	no	—
179	#135	D₄×C₆₇	1	4	yes	—
180	C₂×Q₁₆	—	2	4	no	—
181	C₂×Q₁₆	—	2	4	no	—
182	C₂×Q₁₆	—	2	2	no	—
183	C₂×Q₁₆	—	2	4	no	—
184	C₂×Q₁₆	—	4	4	no	—
185	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
186	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
187	C₂×SD₁₆	—	2	1	no	—
188	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
189	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
190	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
191	C₄⋊Q₈	—	2	4	no	—
192	C₄⋊Q₈	—	4	1	no	—
193	C₄⋊Q₈	—	4	1	no	—
194	C₄×Q₈	—	2	2	no	—
195	C₄×D₄	—	2	2	no	—
196	C₄×D₄	—	4	2	no	—
197	#117	—	2	1	no	—
198	#117	—	4	1	no	—
199	C₂×C₄⋊C₄	—	4	2	no	—
200	C₂×C₄²	—	4	1	no	—
201	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
202	Q₃₂	—	2	8	no	—
203	SD₃₂	—	2	8	no	—
204	M₃₂	—	2	4	no	—
205	C₂×C₁₆	C₂³×C₆₇	1	4	yes	—
206	#107	C₂³×C₆₇	1	2	yes	—
207	#106	—	2	1	no	—
208	#103	—	2	2	no	—
209	#102	—	2	2	no	—
210	C₄×C₈	D₄×C₆₇	1	2	yes	—
211	C₄×C₈	C₂³×C₆₇	1	2	yes	—
212	#41	—	2	4	no	—
213	#41	—	4	2	no	—
214	C₂×Q₈	—	2	1	no	—
215	C₂×Q₈	—	2	4	no	—
216	C₂×Q₈	—	4	2	no	—
217	C₂×Q₈	—	4	4	no	—
218	C₂×Q₈	—	4	4	no	—
219	C₂×D₄	—	2	1	no	—
220	C₂×D₄	—	4	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	D₈×C₆₇	1	2	yes	—
225	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
226	Q₁₆	—	2	4	no	—
227	Q₁₆	—	2	1	no	—
228	Q₁₆	—	2	4	no	—
229	Q₁₆	—	4	8	no	—
230	Q₁₆	—	4	4	no	—
231	Q₁₆	C₂×D₄×C₆₇	1	2	yes	—
232	SD₁₆	—	2	4	no	—
233	D₈	—	2	1	no	—
234	D₈	C₂×D₄×C₆₇	1	2	yes	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
237	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
238	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
239	C₂×C₈	#41×C₆₇	1	2	yes	—
240	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	—	4	4	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₂²⋊C₄	—	4	1	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₄²	—	4	1	no	—
248	C₄²	—	4	2	no	—
249	C₄²	D₈×C₆₇	1	1	yes	characteristic
250	C₄²	D₈×C₆₇	1	1	yes	characteristic
251	C₄²	C₂×D₄×C₆₇	1	2	yes	—
252	C₁₆	—	2	4	no	—
253	C₂³	—	4	1	no	—
254	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
255	Q₈	—	2	2	no	—
256	Q₈	—	4	1	no	—
257	Q₈	—	4	4	no	—
258	Q₈	—	4	4	no	—
259	Q₈	—	8	4	no	—
260	D₄	—	2	2	no	—
261	D₄	—	4	4	no	—
262	D₄	—	4	1	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	8	4	no	—
269	C₂×C₄	—	8	4	no	—
270	C₂×C₄	#121×C₆₇	1	2	yes	—
271	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
272	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
273	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
274	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
275	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	#121×C₆₇	1	2	yes	—
278	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
279	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	8	2	no	—
283	C₂²	#459×C₆₇	1	2	yes	—
284	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	4	2	no	—
287	C₄	—	8	4	no	—
288	C₄	—	8	2	no	—
289	C₄	#459×C₆₇	1	2	yes	—
290	C₄	#510×C₆₇	1	4	yes	complemented
291	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
292	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	8	2	no	—
295	C₂	C₂×#459×C₆₇	1	1	yes	characteristic
296	C₂	C₂×#510×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂×#135×C₆₇	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#494×C₆₇	C₂	semidirect
4. C₆₇×#2209	C₂	semidirect
5. C₆₇×#2208	C₂	semidirect
6. C₆₇×#2150	C₂	semidirect
7. C₆₇×#2141	C₂	semidirect
141. C₄×C₆₇	#510	semidirect
144. #33583	C₆₇	direct (N × H)
150. C₂×#494	C₂×C₆₇	semidirect
152. #2209	C₂×C₆₇	semidirect
153. #2208	C₂×C₆₇	semidirect
154. #2150	C₂×C₆₇	semidirect
155. #2141	C₂×C₆₇	semidirect
156. C₆₇	#33583	direct (N × H)
290. C₄	#510×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#510×C₆₇	2
C₂×#494×C₆₇	2
C₄×#135×C₆₇	2
C₆₇×#2209	2
C₆₇×#2208	2
C₆₇×#2150	2
C₆₇×#2141	2
#33583	67

Subgroups — C67×#33583 #30712834

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33583 #30712834