Subgroups — C₆₇×#17962 #30697213

Order 17152.

734 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3169	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2931	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1594	C₂	1	4	yes	maximal, complemented
4	C₆₇×#1580	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1566	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1492	C₂	1	2	yes	maximal, complemented
7	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₆₇	—	2	2	no	—
9	#532×C₆₇	C₂²	1	2	yes	—
10	#500×C₆₇	C₄	1	1	yes	characteristic
11	#493×C₆₇	C₄	1	1	yes	characteristic
12	#486×C₆₇	C₄	1	2	yes	—
13	#466×C₆₇	C₄	1	4	yes	—
14	C₄×D₈×C₆₇	C₂²	1	4	yes	—
15	#436×C₆₇	—	2	2	no	—
16	#436×C₆₇	C₂²	1	4	yes	—
17	C₈×D₄×C₆₇	—	2	2	no	—
18	C₈×D₄×C₆₇	C₂²	1	4	yes	—
19	#423×C₆₇	C₂²	1	2	yes	—
20	#408×C₆₇	—	2	2	no	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	#334×C₆₇	C₂²	1	4	yes	—
25	D₄⋊C₈×C₆₇	C₂²	1	8	yes	—
26	#322×C₆₇	C₂²	1	2	yes	—
27	#322×C₆₇	C₂²	1	2	yes	—
28	C₂×#41×C₆₇	—	2	2	no	—
29	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	—
30	#131×C₆₇	—	4	8	no	—
31	C₂×M₁₆×C₆₇	—	2	2	no	—
32	C₂×M₁₆×C₆₇	—	2	4	no	—
33	C₂²×C₈×C₆₇	—	2	1	no	—
34	C₂²×C₈×C₆₇	—	2	4	no	—
35	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₆₇	D₄	1	2	yes	—
37	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
38	#127×C₆₇	C₂×C₄	1	2	yes	—
39	#124×C₆₇	—	2	2	no	—
40	#121×C₆₇	—	2	2	no	—
41	#121×C₆₇	C₂×C₄	1	4	yes	—
42	C₄×D₄×C₆₇	—	2	2	no	—
43	C₄×D₄×C₆₇	C₂³	1	4	yes	—
44	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
45	#107×C₆₇	C₂×C₄	1	4	yes	—
46	C₄⋊C₈×C₆₇	—	2	2	no	—
47	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
48	#102×C₆₇	C₂×C₄	1	8	yes	—
49	C₂²⋊C₈×C₆₇	—	2	4	no	—
50	C₂²⋊C₈×C₆₇	—	2	4	no	—
51	C₄×C₈×C₆₇	—	2	1	no	—
52	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₆₇	D₄	1	2	yes	—
55	C₄×C₈×C₆₇	C₂³	1	2	yes	—
56	C₄×C₈×C₆₇	C₂³	1	2	yes	—
57	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
59	#41×C₆₇	—	4	4	no	—
60	#41×C₆₇	—	4	4	no	—
61	C₂×Q₈×C₆₇	—	2	2	no	—
62	C₂×D₄×C₆₇	—	2	4	no	—
63	C₂×D₄×C₆₇	—	2	2	no	—
64	C₂×D₄×C₆₇	—	2	2	no	—
65	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₆₇	—	2	4	no	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
70	D₈×C₆₇	—	2	8	no	—
71	M₁₆×C₆₇	—	4	8	no	—
72	M₁₆×C₆₇	—	4	4	no	—
73	C₂×C₈×C₆₇	—	2	4	no	—
74	C₂×C₈×C₆₇	—	2	4	no	—
75	C₂×C₈×C₆₇	—	2	4	no	—
76	C₂×C₈×C₆₇	—	2	2	no	—
77	C₂×C₈×C₆₇	—	4	8	no	—
78	C₂×C₈×C₆₇	—	4	2	no	—
79	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
83	C₄⋊C₄×C₆₇	—	2	2	no	—
84	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
85	C₂²⋊C₄×C₆₇	—	2	4	no	—
86	C₂²⋊C₄×C₆₇	—	2	4	no	—
87	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₆₇	#41	1	1	yes	characteristic
90	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
91	C₂³×C₆₇	—	2	4	no	—
92	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
93	Q₈×C₆₇	—	4	4	no	—
94	D₄×C₆₇	—	2	8	no	—
95	D₄×C₆₇	—	4	8	no	—
96	D₄×C₆₇	—	4	4	no	—
97	D₄×C₆₇	—	4	4	no	—
98	C₂×C₄×C₆₇	—	2	2	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	2	4	no	—
101	C₂×C₄×C₆₇	—	4	4	no	—
102	C₂×C₄×C₆₇	—	4	4	no	—
103	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
107	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
110	C₈×C₆₇	—	2	2	no	—
111	C₈×C₆₇	—	2	2	no	—
112	C₈×C₆₇	—	2	2	no	—
113	C₈×C₆₇	—	4	2	no	—
114	C₈×C₆₇	—	4	4	no	—
115	C₈×C₆₇	#131	1	4	yes	complemented
116	C₂²×C₆₇	—	2	1	no	—
117	C₂²×C₆₇	—	2	1	no	—
118	C₂²×C₆₇	—	4	4	no	—
119	C₂²×C₆₇	—	4	4	no	—
120	C₂²×C₆₇	—	4	4	no	—
121	C₂²×C₆₇	#442	1	2	yes	—
122	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
123	C₄×C₆₇	—	2	1	no	—
124	C₄×C₆₇	—	2	1	no	—
125	C₄×C₆₇	—	4	4	no	—
126	C₄×C₆₇	#442	1	2	yes	—
127	C₄×C₆₇	#444	1	2	yes	—
128	C₄×C₆₇	C₂×#131	1	2	yes	—
129	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
130	#17962	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂×C₆₇	—	2	2	no	—
132	C₂×C₆₇	—	8	4	no	—
133	C₂×C₆₇	#2931	1	1	yes	characteristic
134	C₂×C₆₇	C₂×#442	1	1	yes	characteristic
135	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
136	#3169	C₂×C₆₇	1	1	yes	characteristic
137	#2931	C₂×C₆₇	1	2	yes	complemented
138	#1594	C₂×C₆₇	1	4	yes	complemented
139	#1580	C₂×C₆₇	1	4	yes	complemented
140	#1566	C₂×C₆₇	1	1	yes	characteristic, complemented
141	#1492	C₂×C₆₇	1	2	yes	complemented
142	C₂×#322	C₂×C₆₇	1	1	yes	characteristic, complemented
143	C₆₇	#17962	1	1	yes	characteristic, direct product factor, Sylow
144	C₂×#131	—	2	2	no	—
145	#532	C₂²×C₆₇	1	2	yes	—
146	#500	C₄×C₆₇	1	1	yes	characteristic
147	#493	C₄×C₆₇	1	1	yes	characteristic
148	#486	C₄×C₆₇	1	2	yes	—
149	#466	C₄×C₆₇	1	4	yes	—
150	C₄×D₈	C₂²×C₆₇	1	4	yes	—
151	#436	—	2	2	no	—
152	#436	C₂²×C₆₇	1	4	yes	—
153	C₈×D₄	—	2	2	no	—
154	C₈×D₄	C₂²×C₆₇	1	4	yes	—
155	#423	C₂²×C₆₇	1	2	yes	—
156	#408	—	2	2	no	—
157	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
160	#334	C₂²×C₆₇	1	4	yes	—
161	D₄⋊C₈	C₂²×C₆₇	1	8	yes	—
162	#322	C₂²×C₆₇	1	2	yes	—
163	#322	C₂²×C₆₇	1	2	yes	—
164	C₂×#41	—	2	2	no	—
165	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	—
166	#131	—	4	8	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	4	no	—
169	C₂²×C₈	—	2	1	no	—
170	C₂²×C₈	—	2	4	no	—
171	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
172	C₂²×C₈	D₄×C₆₇	1	2	yes	—
173	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
174	#127	C₂×C₄×C₆₇	1	2	yes	—
175	#124	—	2	2	no	—
176	#121	—	2	2	no	—
177	#121	C₂×C₄×C₆₇	1	4	yes	—
178	C₄×D₄	—	2	2	no	—
179	C₄×D₄	C₂³×C₆₇	1	4	yes	—
180	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
181	#107	C₂×C₄×C₆₇	1	4	yes	—
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
184	#102	C₂×C₄×C₆₇	1	8	yes	—
185	C₂²⋊C₈	—	2	4	no	—
186	C₂²⋊C₈	—	2	4	no	—
187	C₄×C₈	—	2	1	no	—
188	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
189	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
190	C₄×C₈	D₄×C₆₇	1	2	yes	—
191	C₄×C₈	C₂³×C₆₇	1	2	yes	—
192	C₄×C₈	C₂³×C₆₇	1	2	yes	—
193	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
194	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
195	#41	—	4	4	no	—
196	#41	—	4	4	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
205	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
206	D₈	—	2	8	no	—
207	M₁₆	—	4	8	no	—
208	M₁₆	—	4	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	4	8	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
216	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
217	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
218	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
221	C₂²⋊C₄	—	2	4	no	—
222	C₂²⋊C₄	—	2	4	no	—
223	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
224	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
225	C₄²	#41×C₆₇	1	1	yes	characteristic
226	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
227	C₂³	—	2	4	no	—
228	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
229	Q₈	—	4	4	no	—
230	D₄	—	2	8	no	—
231	D₄	—	4	8	no	—
232	D₄	—	4	4	no	—
233	D₄	—	4	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
240	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
241	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
242	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
243	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
244	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
245	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	—	4	2	no	—
250	C₈	—	4	4	no	—
251	C₈	#131×C₆₇	1	4	yes	complemented
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	4	4	no	—
257	C₂²	#442×C₆₇	1	2	yes	—
258	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	4	4	no	—
262	C₄	#442×C₆₇	1	2	yes	—
263	C₄	#444×C₆₇	1	2	yes	—
264	C₄	C₂×#131×C₆₇	1	2	yes	—
265	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
266	C₂	—	2	2	no	—
267	C₂	—	8	4	no	—
268	C₂	C₆₇×#2931	1	1	yes	characteristic
269	C₂	C₂×#442×C₆₇	1	1	yes	characteristic
270	C₂	C₂×#444×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₆₇×#3169	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2931	C₂	semidirect
3. C₆₇×#1594	C₂	semidirect
4. C₆₇×#1580	C₂	semidirect
5. C₆₇×#1566	C₂	semidirect
6. C₆₇×#1492	C₂	semidirect
7. C₂×#322×C₆₇	C₂	semidirect
115. C₈×C₆₇	#131	semidirect
130. #17962	C₆₇	direct (N × H)
137. #2931	C₂×C₆₇	semidirect
138. #1594	C₂×C₆₇	semidirect
139. #1580	C₂×C₆₇	semidirect
140. #1566	C₂×C₆₇	semidirect
141. #1492	C₂×C₆₇	semidirect
142. C₂×#322	C₂×C₆₇	semidirect
143. C₆₇	#17962	direct (N × H)
251. C₈	#131×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3169	2
C₆₇×#2931	2
C₆₇×#1594	2
C₆₇×#1580	2
C₆₇×#1566	2
C₆₇×#1492	2
C₂×#322×C₆₇	2
#17962	67

Subgroups — C67×#17962 #30697213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17962 #30697213