Subgroups — C₆₇×#33333 #30712584

Order 17152.

1162 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#131×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₃₂×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2113	C₂	1	8	yes	maximal, complemented
4	C₆₇×#2112	C₂	1	4	yes	maximal, complemented
5	C₂²×#41×C₆₇	C₄	1	1	yes	characteristic
6	C₂×#131×C₆₇	—	2	8	no	—
7	C₂×#131×C₆₇	C₄	1	4	yes	—
8	C₂×#131×C₆₇	C₂²	1	4	yes	—
9	C₂²×M₁₆×C₆₇	C₄	1	1	yes	characteristic
10	C₂²×M₁₆×C₆₇	C₄	1	2	yes	—
11	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
12	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×M₃₂×C₆₇	—	2	8	no	—
14	C₂×M₃₂×C₆₇	C₂²	1	4	yes	—
15	C₂×M₃₂×C₆₇	C₂²	1	4	yes	complemented
16	C₂²×C₁₆×C₆₇	C₂²	1	4	yes	complemented
17	C₂²⋊C₁₆×C₆₇	C₂²	1	16	yes	complemented
18	C₂×#41×C₆₇	—	2	4	no	—
19	C₂×#41×C₆₇	—	2	4	no	—
20	C₂×#41×C₆₇	C₈	1	4	yes	—
21	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
22	C₂²×Q₈×C₆₇	C₈	1	1	yes	characteristic
23	C₂²×D₄×C₆₇	C₈	1	2	yes	—
24	C₂²×D₄×C₆₇	C₈	1	1	yes	characteristic
25	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
27	#131×C₆₇	—	2	16	no	—
28	#131×C₆₇	—	2	8	no	—
29	#131×C₆₇	—	2	8	no	—
30	C₂×M₁₆×C₆₇	—	2	4	no	—
31	C₂×M₁₆×C₆₇	—	2	8	no	—
32	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
33	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	—
34	C₂²×C₈×C₆₇	—	2	8	no	—
35	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₆₇	D₄	1	8	yes	—
40	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
41	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
42	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	M₃₂×C₆₇	—	2	8	no	—
44	M₃₂×C₆₇	—	2	8	no	—
45	C₂×C₁₆×C₆₇	—	2	8	no	—
46	C₂×C₁₆×C₆₇	C₂³	1	8	yes	—
47	C₂⁴×C₆₇	C₂×C₈	1	2	yes	—
48	C₂⁴×C₆₇	C₂×C₈	1	1	yes	characteristic
49	#41×C₆₇	—	2	16	no	—
50	#41×C₆₇	—	2	8	no	—
51	#41×C₆₇	—	2	8	no	—
52	C₂×Q₈×C₆₇	—	2	4	no	—
53	C₂×Q₈×C₆₇	C₂×C₈	1	4	yes	—
54	C₂×D₄×C₆₇	—	2	8	no	—
55	C₂×D₄×C₆₇	—	2	4	no	—
56	C₂×D₄×C₆₇	C₂×C₈	1	8	yes	—
57	C₂×D₄×C₆₇	C₂×C₈	1	4	yes	—
58	C₂²×C₄×C₆₇	—	2	4	no	—
59	C₂²×C₄×C₆₇	—	2	4	no	—
60	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
61	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
63	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
66	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
69	M₁₆×C₆₇	—	2	8	no	—
70	M₁₆×C₆₇	—	2	16	no	—
71	C₂×C₈×C₆₇	—	2	4	no	—
72	C₂×C₈×C₆₇	—	2	8	no	—
73	C₂×C₈×C₆₇	—	2	8	no	—
74	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
75	C₂×C₈×C₆₇	C₂²⋊C₄	1	8	yes	—
76	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
77	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
80	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
81	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
82	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
83	C₁₆×C₆₇	—	2	8	no	—
84	C₂³×C₆₇	—	2	4	no	—
85	C₂³×C₆₇	—	4	2	no	—
86	C₂³×C₆₇	—	4	4	no	—
87	C₂³×C₆₇	C₂²⋊C₈	1	4	yes	—
88	C₂³×C₆₇	C₂²×C₈	1	4	yes	—
89	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
90	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
91	Q₈×C₆₇	—	2	8	no	—
92	D₄×C₆₇	—	2	16	no	—
93	D₄×C₆₇	—	2	8	no	—
94	C₂×C₄×C₆₇	—	2	4	no	—
95	C₂×C₄×C₆₇	—	2	4	no	—
96	C₂×C₄×C₆₇	—	2	4	no	—
97	C₂×C₄×C₆₇	—	2	8	no	—
98	C₂×C₄×C₆₇	C₂²⋊C₈	1	4	yes	—
99	C₂×C₄×C₆₇	C₂²⋊C₈	1	8	yes	—
100	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
101	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
102	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
103	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
106	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
107	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
108	C₈×C₆₇	—	2	4	no	—
109	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
110	C₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
111	C₈×C₆₇	C₂²×D₄	1	2	yes	—
112	C₂²×C₆₇	—	2	2	no	—
113	C₂²×C₆₇	—	2	4	no	—
114	C₂²×C₆₇	—	2	4	no	—
115	C₂²×C₆₇	—	2	8	no	—
116	C₂²×C₆₇	—	4	4	no	—
117	C₂²×C₆₇	—	4	8	no	—
118	C₂²×C₆₇	C₂×C₂²⋊C₈	1	2	yes	—
119	C₂²×C₆₇	C₂×C₂²⋊C₈	1	4	yes	—
120	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
121	C₄×C₆₇	—	2	4	no	—
122	C₄×C₆₇	C₂×C₂²⋊C₈	1	2	yes	—
123	C₄×C₆₇	C₂×C₂²⋊C₈	1	4	yes	—
124	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
125	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
126	#33333	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂×C₆₇	—	2	2	no	—
128	C₂×C₆₇	—	2	4	no	—
129	C₂×C₆₇	—	4	4	no	—
130	C₂×C₆₇	C₂²×C₂²⋊C₈	1	1	yes	characteristic
131	C₂×C₆₇	#3407	1	2	yes	—
132	C₂²×#131	C₂×C₆₇	1	1	yes	characteristic
133	C₂²×M₃₂	C₂×C₆₇	1	2	yes	complemented
134	#2113	C₂×C₆₇	1	8	yes	complemented
135	#2112	C₂×C₆₇	1	4	yes	complemented
136	C₆₇	#33333	1	1	yes	characteristic, direct product factor, Sylow
137	C₂²×#41	C₄×C₆₇	1	1	yes	characteristic
138	C₂×#131	—	2	8	no	—
139	C₂×#131	C₄×C₆₇	1	4	yes	—
140	C₂×#131	C₂²×C₆₇	1	4	yes	—
141	C₂²×M₁₆	C₄×C₆₇	1	1	yes	characteristic
142	C₂²×M₁₆	C₄×C₆₇	1	2	yes	—
143	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
144	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
145	C₂×M₃₂	—	2	8	no	—
146	C₂×M₃₂	C₂²×C₆₇	1	4	yes	—
147	C₂×M₃₂	C₂²×C₆₇	1	4	yes	complemented
148	C₂²×C₁₆	C₂²×C₆₇	1	4	yes	complemented
149	C₂²⋊C₁₆	C₂²×C₆₇	1	16	yes	complemented
150	C₂×#41	—	2	4	no	—
151	C₂×#41	—	2	4	no	—
152	C₂×#41	C₈×C₆₇	1	4	yes	—
153	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
154	C₂²×Q₈	C₈×C₆₇	1	1	yes	characteristic
155	C₂²×D₄	C₈×C₆₇	1	2	yes	—
156	C₂²×D₄	C₈×C₆₇	1	1	yes	characteristic
157	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
158	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
159	#131	—	2	16	no	—
160	#131	—	2	8	no	—
161	#131	—	2	8	no	—
162	C₂×M₁₆	—	2	4	no	—
163	C₂×M₁₆	—	2	8	no	—
164	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
165	C₂×M₁₆	C₂×C₄×C₆₇	1	8	yes	—
166	C₂²×C₈	—	2	8	no	—
167	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
168	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
169	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
170	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
171	C₂²×C₈	D₄×C₆₇	1	8	yes	—
172	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
173	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
175	M₃₂	—	2	8	no	—
176	M₃₂	—	2	8	no	—
177	C₂×C₁₆	—	2	8	no	—
178	C₂×C₁₆	C₂³×C₆₇	1	8	yes	—
179	C₂⁴	C₂×C₈×C₆₇	1	2	yes	—
180	C₂⁴	C₂×C₈×C₆₇	1	1	yes	characteristic
181	#41	—	2	8	no	—
182	#41	—	2	8	no	—
183	#41	—	2	16	no	—
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	C₂×C₈×C₆₇	1	4	yes	—
186	C₂×D₄	—	2	8	no	—
187	C₂×D₄	—	2	4	no	—
188	C₂×D₄	C₂×C₈×C₆₇	1	8	yes	—
189	C₂×D₄	C₂×C₈×C₆₇	1	4	yes	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
193	C₂²×C₄	C₂×C₈×C₆₇	1	2	yes	—
194	C₂²×C₄	C₂×C₈×C₆₇	1	4	yes	—
195	C₂²×C₄	C₂×C₈×C₆₇	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₈×C₆₇	1	2	yes	—
197	C₂²×C₄	M₁₆×C₆₇	1	4	yes	—
198	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
199	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
201	M₁₆	—	2	8	no	—
202	M₁₆	—	2	16	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	8	no	—
205	C₂×C₈	—	2	8	no	—
206	C₂×C₈	C₂²⋊C₄×C₆₇	1	4	yes	—
207	C₂×C₈	C₂²⋊C₄×C₆₇	1	8	yes	—
208	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
209	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
210	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
211	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
212	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
213	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
214	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
215	C₁₆	—	2	8	no	—
216	C₂³	—	2	4	no	—
217	C₂³	—	4	2	no	—
218	C₂³	—	4	4	no	—
219	C₂³	C₂²⋊C₈×C₆₇	1	4	yes	—
220	C₂³	C₂²×C₈×C₆₇	1	4	yes	—
221	C₂³	C₂²×C₈×C₆₇	1	1	yes	characteristic
222	C₂³	C₂²×C₈×C₆₇	1	2	yes	—
223	Q₈	—	2	8	no	—
224	D₄	—	2	16	no	—
225	D₄	—	2	8	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	C₂²⋊C₈×C₆₇	1	4	yes	—
231	C₂×C₄	C₂²⋊C₈×C₆₇	1	8	yes	—
232	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
233	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
234	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
235	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
236	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
237	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
238	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
239	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
240	C₈	—	2	4	no	—
241	C₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
242	C₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
243	C₈	C₂²×D₄×C₆₇	1	2	yes	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	2	8	no	—
248	C₂²	—	4	4	no	—
249	C₂²	—	4	8	no	—
250	C₂²	C₂×C₂²⋊C₈×C₆₇	1	2	yes	—
251	C₂²	C₂×C₂²⋊C₈×C₆₇	1	4	yes	—
252	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
253	C₄	—	2	4	no	—
254	C₄	C₂×C₂²⋊C₈×C₆₇	1	2	yes	—
255	C₄	C₂×C₂²⋊C₈×C₆₇	1	4	yes	—
256	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
257	C₄	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	2	4	no	—
260	C₂	—	4	4	no	—
261	C₂	C₂²×C₂²⋊C₈×C₆₇	1	1	yes	characteristic
262	C₂	C₆₇×#3407	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₆₇	C₂⁴
commutator	C₂²	C₂³×C₈×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂³×C₈
prime radical	C₂²×#41×C₆₇	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×M₃₂×C₆₇	C₂	semidirect
3. C₆₇×#2113	C₂	semidirect
4. C₆₇×#2112	C₂	semidirect
15. C₂×M₃₂×C₆₇	C₂²	semidirect
16. C₂²×C₁₆×C₆₇	C₂²	semidirect
17. C₂²⋊C₁₆×C₆₇	C₂²	semidirect
126. #33333	C₆₇	direct (N × H)
133. C₂²×M₃₂	C₂×C₆₇	semidirect
134. #2113	C₂×C₆₇	semidirect
135. #2112	C₂×C₆₇	semidirect
136. C₆₇	#33333	direct (N × H)
147. C₂×M₃₂	C₂²×C₆₇	semidirect
148. C₂²×C₁₆	C₂²×C₆₇	semidirect
149. C₂²⋊C₁₆	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#131×C₆₇	2
C₂²×M₃₂×C₆₇	2
C₆₇×#2113	2
C₆₇×#2112	2
#33333	67

Subgroups — C67×#33333 #30712584

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33333 #30712584