Subgroups — C₆₇×#33067 #30712318

Order 17152.

750 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2859	C₂	1	1	yes	maximal, characteristic
2	C₄²×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1948	C₂	1	2	yes	maximal
4	C₆₇×#1919	C₂	1	4	yes	maximal
5	C₆₇×#1865	C₂	1	4	yes	maximal
6	C₆₇×#1831	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1803	C₂	1	2	yes	maximal
8	C₂×#125×C₆₇	—	2	1	no	—
9	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#106×C₆₇	C₂²	1	4	yes	—
12	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
13	C₂×#103×C₆₇	C₂²	1	4	yes	—
14	#389×C₆₇	—	2	2	no	—
15	#389×C₆₇	C₂²	1	4	yes	—
16	#384×C₆₇	C₂²	1	2	yes	—
17	#382×C₆₇	—	2	1	no	—
18	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
19	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
20	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₄³×C₆₇	—	2	1	no	—
23	C₄³×C₆₇	C₂²	1	1	yes	characteristic
24	#340×C₆₇	C₂²	1	4	yes	—
25	#340×C₆₇	C₂²	1	4	yes	—
26	#336×C₆₇	C₂²	1	4	yes	—
27	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
29	#125×C₆₇	—	4	4	no	—
30	C₄×Q₈×C₆₇	—	2	4	no	—
31	C₄×Q₈×C₆₇	—	2	4	no	—
32	C₄×Q₈×C₆₇	—	2	2	no	—
33	C₄×Q₈×C₆₇	Q₈	1	4	yes	—
34	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
42	C₂×C₄²×C₆₇	—	2	2	no	—
43	C₂×C₄²×C₆₇	—	2	1	no	—
44	C₂×C₄²×C₆₇	—	2	2	no	—
45	C₂×C₄²×C₆₇	—	2	1	no	—
46	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	D₄	1	2	yes	—
48	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
50	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	#106×C₆₇	—	2	8	no	—
52	C₄⋊C₈×C₆₇	—	2	4	no	—
53	#103×C₆₇	—	2	8	no	—
54	#95×C₆₇	—	2	2	no	—
55	#95×C₆₇	—	2	1	no	—
56	#95×C₆₇	—	2	4	no	—
57	C₂×Q₈×C₆₇	—	2	1	no	—
58	C₂×Q₈×C₆₇	—	2	2	no	—
59	C₂×Q₈×C₆₇	C₂×Q₈	1	2	yes	—
60	C₂×Q₈×C₆₇	#41	1	4	yes	—
61	C₂²×C₄×C₆₇	—	2	1	no	—
62	C₂²×C₄×C₆₇	—	2	1	no	—
63	C₂²×C₄×C₆₇	—	2	2	no	—
64	C₂²×C₄×C₆₇	—	2	4	no	—
65	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
70	C₂×C₈×C₆₇	—	2	4	no	—
71	C₂×C₈×C₆₇	—	2	4	no	—
72	C₂×C₈×C₆₇	—	2	4	no	—
73	C₄⋊C₄×C₆₇	—	2	8	no	—
74	C₄⋊C₄×C₆₇	—	2	4	no	—
75	C₄⋊C₄×C₆₇	—	2	1	no	—
76	C₄⋊C₄×C₆₇	—	2	2	no	—
77	C₄⋊C₄×C₆₇	—	2	4	no	—
78	C₄⋊C₄×C₆₇	—	2	4	no	—
79	C₄⋊C₄×C₆₇	—	4	4	no	—
80	C₄⋊C₄×C₆₇	—	4	8	no	—
81	C₄⋊C₄×C₆₇	—	4	4	no	—
82	C₄⋊C₄×C₆₇	C₂×Q₈	1	2	yes	—
83	C₄⋊C₄×C₆₇	#41	1	4	yes	—
84	C₄²×C₆₇	—	2	8	no	—
85	C₄²×C₆₇	—	2	4	no	—
86	C₄²×C₆₇	—	2	2	no	—
87	C₄²×C₆₇	—	2	2	no	—
88	C₄²×C₆₇	—	2	2	no	—
89	C₄²×C₆₇	—	4	2	no	—
90	C₄²×C₆₇	—	4	4	no	—
91	C₄²×C₆₇	—	4	2	no	—
92	C₄²×C₆₇	SD₁₆	1	4	yes	—
93	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
94	C₄²×C₆₇	#41	1	2	yes	—
95	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
96	Q₈×C₆₇	—	2	4	no	—
97	Q₈×C₆₇	—	2	2	no	—
98	Q₈×C₆₇	#125	1	4	yes	complemented
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	2	2	no	—
101	C₂×C₄×C₆₇	—	2	4	no	—
102	C₂×C₄×C₆₇	—	2	2	no	—
103	C₂×C₄×C₆₇	—	2	2	no	—
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	4	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	4	2	no	—
110	C₂×C₄×C₆₇	—	4	2	no	—
111	C₂×C₄×C₆₇	—	4	2	no	—
112	C₂×C₄×C₆₇	—	4	4	no	—
113	C₂×C₄×C₆₇	—	4	4	no	—
114	C₂×C₄×C₆₇	—	4	4	no	—
115	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
118	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
119	C₂×C₄×C₆₇	#137	1	2	yes	—
120	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
124	C₈×C₆₇	—	4	4	no	—
125	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
126	C₂²×C₆₇	#525	1	2	yes	—
127	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
128	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
129	C₂²×C₆₇	#575	1	2	yes	—
130	C₄×C₆₇	—	2	2	no	—
131	C₄×C₆₇	—	2	4	no	—
132	C₄×C₆₇	—	2	4	no	—
133	C₄×C₆₇	—	4	2	no	—
134	C₄×C₆₇	—	4	2	no	—
135	C₄×C₆₇	—	4	4	no	—
136	C₄×C₆₇	—	8	4	no	—
137	C₄×C₆₇	#475	1	4	yes	—
138	C₄×C₆₇	C₂×#125	1	1	yes	characteristic
139	C₄×C₆₇	#529	1	2	yes	—
140	C₄×C₆₇	#533	1	1	yes	characteristic
141	#33067	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂×C₆₇	#2851	1	1	yes	characteristic
143	C₂×C₆₇	C₂×#475	1	1	yes	characteristic
144	C₂×C₆₇	#3080	1	1	yes	characteristic
145	C₂×C₆₇	#3103	1	2	yes	—
146	C₂×C₆₇	#3116	1	2	yes	—
147	#2859	C₂×C₆₇	1	1	yes	characteristic
148	C₄²×Q₈	C₂×C₆₇	1	1	yes	characteristic
149	#1948	C₂×C₆₇	1	2	yes	—
150	#1919	C₂×C₆₇	1	4	yes	—
151	#1865	C₂×C₆₇	1	4	yes	—
152	#1831	C₂×C₆₇	1	1	yes	characteristic
153	#1803	C₂×C₆₇	1	2	yes	—
154	C₆₇	#33067	1	1	yes	characteristic, direct product factor, Sylow
155	C₂×#125	—	2	1	no	—
156	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
157	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
158	C₂×#106	C₂²×C₆₇	1	4	yes	—
159	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
160	C₂×#103	C₂²×C₆₇	1	4	yes	—
161	#389	—	2	2	no	—
162	#389	C₂²×C₆₇	1	4	yes	—
163	#384	C₂²×C₆₇	1	2	yes	—
164	#382	—	2	1	no	—
165	C₄×C₄⋊C₄	—	2	1	no	—
166	C₄×C₄⋊C₄	—	2	2	no	—
167	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
168	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
169	C₄³	—	2	1	no	—
170	C₄³	C₂²×C₆₇	1	1	yes	characteristic
171	#340	C₂²×C₆₇	1	4	yes	—
172	#340	C₂²×C₆₇	1	4	yes	—
173	#336	C₂²×C₆₇	1	4	yes	—
174	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
175	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
176	#125	—	4	4	no	—
177	C₄×Q₈	—	2	4	no	—
178	C₄×Q₈	—	2	4	no	—
179	C₄×Q₈	—	2	2	no	—
180	C₄×Q₈	Q₈×C₆₇	1	4	yes	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	4	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
187	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
188	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
194	C₂×C₄²	D₄×C₆₇	1	2	yes	—
195	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
196	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
197	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
198	#106	—	2	8	no	—
199	C₄⋊C₈	—	2	4	no	—
200	#103	—	2	8	no	—
201	#95	—	2	2	no	—
202	#95	—	2	1	no	—
203	#95	—	2	4	no	—
204	C₂×Q₈	—	2	1	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	C₂×Q₈×C₆₇	1	2	yes	—
207	C₂×Q₈	#41×C₆₇	1	4	yes	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
213	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
216	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₄⋊C₄	—	2	8	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	4	4	no	—
227	C₄⋊C₄	—	4	8	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	C₂×Q₈×C₆₇	1	2	yes	—
230	C₄⋊C₄	#41×C₆₇	1	4	yes	—
231	C₄²	—	2	8	no	—
232	C₄²	—	2	4	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	4	2	no	—
237	C₄²	—	4	4	no	—
238	C₄²	—	4	2	no	—
239	C₄²	SD₁₆×C₆₇	1	4	yes	—
240	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
241	C₄²	#41×C₆₇	1	2	yes	—
242	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
243	Q₈	—	2	4	no	—
244	Q₈	—	2	2	no	—
245	Q₈	#125×C₆₇	1	4	yes	complemented
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
263	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
264	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
265	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
266	C₂×C₄	#137×C₆₇	1	2	yes	—
267	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
268	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
269	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
270	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
271	C₈	—	4	4	no	—
272	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
273	C₂²	#525×C₆₇	1	2	yes	—
274	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
275	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
276	C₂²	#575×C₆₇	1	2	yes	—
277	C₄	—	2	2	no	—
278	C₄	—	2	4	no	—
279	C₄	—	2	4	no	—
280	C₄	—	4	2	no	—
281	C₄	—	4	2	no	—
282	C₄	—	4	4	no	—
283	C₄	—	8	4	no	—
284	C₄	#475×C₆₇	1	4	yes	—
285	C₄	C₂×#125×C₆₇	1	1	yes	characteristic
286	C₄	#529×C₆₇	1	2	yes	—
287	C₄	#533×C₆₇	1	1	yes	characteristic
288	C₂	C₆₇×#2851	1	1	yes	characteristic
289	C₂	C₂×#475×C₆₇	1	1	yes	characteristic
290	C₂	C₆₇×#3080	1	1	yes	characteristic
291	C₂	C₆₇×#3103	1	2	yes	—
292	C₂	C₆₇×#3116	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
98. Q₈×C₆₇	#125	semidirect
141. #33067	C₆₇	direct (N × H)
154. C₆₇	#33067	direct (N × H)
245. Q₈	#125×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2859	2
C₄²×Q₈×C₆₇	2
C₆₇×#1948	2
C₆₇×#1919	2
C₆₇×#1865	2
C₆₇×#1831	2
C₆₇×#1803	2
#33067	67

Subgroups — C67×#33067 #30712318

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33067 #30712318