Subgroups — C₆₇×#33657 #30712908

Order 17152.

566 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3058	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2916	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2238	C₂	1	8	yes	maximal, complemented
4	C₆₇×#2155	C₂	1	2	yes	maximal, complemented
5	C₂×#361×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2115	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
8	#568×C₆₇	—	2	1	no	—
9	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
10	#535×C₆₇	—	2	1	no	—
11	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
12	#469×C₆₇	C₂²	1	4	yes	—
13	#466×C₆₇	—	2	2	no	—
14	#430×C₆₇	—	2	1	no	—
15	#430×C₆₇	C₂²	1	4	yes	complemented
16	C₂×#108×C₆₇	—	2	2	no	—
17	C₂×#108×C₆₇	C₂²	1	2	yes	—
18	#427×C₆₇	C₂²	1	1	yes	characteristic
19	#418×C₆₇	—	2	1	no	—
20	#365×C₆₇	C₂²	1	8	yes	complemented
21	#361×C₆₇	C₂²	1	8	yes	—
22	#349×C₆₇	C₂²	1	4	yes	complemented
23	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
24	C₂²×D₄×C₆₇	—	2	1	no	—
25	C₂×D₈×C₆₇	—	2	1	no	—
26	C₂×D₈×C₆₇	—	2	2	no	—
27	C₂×D₈×C₆₇	—	2	2	no	—
28	C₂×D₈×C₆₇	C₂³	1	2	yes	—
29	#131×C₆₇	—	2	2	no	—
30	#131×C₆₇	—	2	2	no	—
31	#131×C₆₇	—	4	4	no	—
32	C₂×M₁₆×C₆₇	—	2	4	no	—
33	C₂×M₁₆×C₆₇	—	2	2	no	—
34	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
35	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
36	C₂²×C₈×C₆₇	D₄	1	2	yes	—
37	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
38	#127×C₆₇	—	4	1	no	—
39	#124×C₆₇	—	4	1	no	—
40	#121×C₆₇	—	2	4	no	—
41	#120×C₆₇	—	4	2	no	—
42	#117×C₆₇	—	2	1	no	—
43	M₃₂×C₆₇	—	2	2	no	—
44	M₃₂×C₆₇	C₂³	1	4	yes	—
45	C₂×C₁₆×C₆₇	—	2	2	no	—
46	#108×C₆₇	—	2	2	no	—
47	#108×C₆₇	—	2	4	no	—
48	#108×C₆₇	C₂³	1	4	yes	—
49	#107×C₆₇	—	2	1	no	—
50	#106×C₆₇	C₂³	1	2	yes	—
51	#102×C₆₇	—	2	4	no	—
52	C₂⁴×C₆₇	—	4	1	no	—
53	#41×C₆₇	—	2	2	no	—
54	#41×C₆₇	—	2	2	no	—
55	C₂×Q₈×C₆₇	SD₁₆	1	1	yes	characteristic
56	C₂×D₄×C₆₇	—	2	2	no	—
57	C₂×D₄×C₆₇	—	4	4	no	—
58	C₂×D₄×C₆₇	—	4	2	no	—
59	C₂×D₄×C₆₇	—	4	1	no	—
60	C₂×D₄×C₆₇	—	4	2	no	—
61	C₂×D₄×C₆₇	SD₁₆	1	2	yes	—
62	C₂×D₄×C₆₇	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	D₈×C₆₇	—	2	4	no	—
66	D₈×C₆₇	—	4	1	no	—
67	D₈×C₆₇	—	4	1	no	—
68	M₁₆×C₆₇	—	2	2	no	—
69	M₁₆×C₆₇	—	2	4	no	—
70	M₁₆×C₆₇	—	2	4	no	—
71	M₁₆×C₆₇	—	4	4	no	—
72	M₁₆×C₆₇	—	4	2	no	—
73	C₂×C₈×C₆₇	—	2	2	no	—
74	C₂×C₈×C₆₇	—	2	2	no	—
75	C₂×C₈×C₆₇	—	4	4	no	—
76	C₂×C₈×C₆₇	—	4	2	no	—
77	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₆₇	#41	1	2	yes	—
80	C₂×C₈×C₆₇	#41	1	2	yes	—
81	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
82	C₄⋊C₄×C₆₇	—	2	2	no	—
83	C₂²⋊C₄×C₆₇	—	4	1	no	—
84	C₂²⋊C₄×C₆₇	—	4	4	no	—
85	C₄²×C₆₇	—	4	1	no	—
86	C₁₆×C₆₇	—	2	4	no	—
87	C₂³×C₆₇	—	4	2	no	—
88	C₂³×C₆₇	—	8	2	no	—
89	C₂³×C₆₇	—	16	2	no	—
90	C₂³×C₆₇	C₂×SD₁₆	1	2	yes	—
91	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
92	Q₈×C₆₇	—	2	2	no	—
93	D₄×C₆₇	—	2	4	no	—
94	D₄×C₆₇	—	2	2	no	—
95	D₄×C₆₇	—	4	2	no	—
96	D₄×C₆₇	—	4	2	no	—
97	D₄×C₆₇	—	8	2	no	—
98	D₄×C₆₇	—	16	4	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	2	2	no	—
101	C₂×C₄×C₆₇	—	4	2	no	—
102	C₂×C₄×C₆₇	#123	1	2	yes	—
103	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
104	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
106	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
107	C₈×C₆₇	—	2	2	no	—
108	C₈×C₆₇	—	4	4	no	—
109	C₈×C₆₇	#123	1	1	yes	characteristic
110	C₈×C₆₇	#123	1	1	yes	characteristic
111	C₈×C₆₇	C₂×#41	1	2	yes	—
112	C₂²×C₆₇	—	2	2	no	—
113	C₂²×C₆₇	—	4	1	no	—
114	C₂²×C₆₇	—	4	2	no	—
115	C₂²×C₆₇	—	8	4	no	—
116	C₂²×C₆₇	—	8	2	no	—
117	C₂²×C₆₇	—	16	4	no	—
118	C₂²×C₆₇	#481	1	2	yes	—
119	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
120	C₄×C₆₇	—	2	2	no	—
121	C₄×C₆₇	—	8	2	no	—
122	C₄×C₆₇	#481	1	2	yes	—
123	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
124	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
125	#33657	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂×C₆₇	—	2	1	no	—
127	C₂×C₆₇	—	2	2	no	—
128	C₂×C₆₇	—	4	2	no	—
129	C₂×C₆₇	—	16	2	no	—
130	C₂×C₆₇	C₂×#481	1	1	yes	characteristic
131	#3058	C₂×C₆₇	1	1	yes	characteristic
132	#2916	C₂×C₆₇	1	1	yes	characteristic, complemented
133	#2238	C₂×C₆₇	1	8	yes	complemented
134	#2155	C₂×C₆₇	1	2	yes	complemented
135	C₂×#361	C₂×C₆₇	1	2	yes	complemented
136	#2115	C₂×C₆₇	1	1	yes	characteristic, complemented
137	C₆₇	#33657	1	1	yes	characteristic, direct product factor, Sylow
138	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
139	#568	—	2	1	no	—
140	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
141	#535	—	2	1	no	—
142	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
143	#469	C₂²×C₆₇	1	4	yes	—
144	#466	—	2	2	no	—
145	#430	—	2	1	no	—
146	#430	C₂²×C₆₇	1	4	yes	complemented
147	C₂×#108	—	2	2	no	—
148	C₂×#108	C₂²×C₆₇	1	2	yes	—
149	#427	C₂²×C₆₇	1	1	yes	characteristic
150	#418	—	2	1	no	—
151	#365	C₂²×C₆₇	1	8	yes	complemented
152	#361	C₂²×C₆₇	1	8	yes	—
153	#349	C₂²×C₆₇	1	4	yes	complemented
154	C₂×#41	D₄×C₆₇	1	1	yes	characteristic
155	C₂²×D₄	—	2	1	no	—
156	C₂×D₈	—	2	1	no	—
157	C₂×D₈	—	2	2	no	—
158	C₂×D₈	—	2	2	no	—
159	C₂×D₈	C₂³×C₆₇	1	2	yes	—
160	#131	—	2	2	no	—
161	#131	—	2	2	no	—
162	#131	—	4	4	no	—
163	C₂×M₁₆	—	2	4	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
166	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
167	C₂²×C₈	D₄×C₆₇	1	2	yes	—
168	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
169	#127	—	4	1	no	—
170	#124	—	4	1	no	—
171	#121	—	2	4	no	—
172	#120	—	4	2	no	—
173	#117	—	2	1	no	—
174	M₃₂	—	2	2	no	—
175	M₃₂	C₂³×C₆₇	1	4	yes	—
176	C₂×C₁₆	—	2	2	no	—
177	#108	—	2	2	no	—
178	#108	—	2	4	no	—
179	#108	C₂³×C₆₇	1	4	yes	—
180	#107	—	2	1	no	—
181	#106	C₂³×C₆₇	1	2	yes	—
182	#102	—	2	4	no	—
183	C₂⁴	—	4	1	no	—
184	#41	—	2	2	no	—
185	#41	—	2	2	no	—
186	C₂×Q₈	SD₁₆×C₆₇	1	1	yes	characteristic
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	4	4	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂×D₄	—	4	1	no	—
191	C₂×D₄	—	4	2	no	—
192	C₂×D₄	SD₁₆×C₆₇	1	2	yes	—
193	C₂×D₄	SD₁₆×C₆₇	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
195	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
196	D₈	—	2	4	no	—
197	D₈	—	4	1	no	—
198	D₈	—	4	1	no	—
199	M₁₆	—	2	2	no	—
200	M₁₆	—	2	4	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	—	4	4	no	—
203	M₁₆	—	4	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
209	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
210	C₂×C₈	#41×C₆₇	1	2	yes	—
211	C₂×C₈	#41×C₆₇	1	2	yes	—
212	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
213	C₄⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	4	1	no	—
215	C₂²⋊C₄	—	4	4	no	—
216	C₄²	—	4	1	no	—
217	C₁₆	—	2	4	no	—
218	C₂³	—	4	2	no	—
219	C₂³	—	8	2	no	—
220	C₂³	—	16	2	no	—
221	C₂³	C₂×SD₁₆×C₆₇	1	2	yes	—
222	C₂³	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
223	Q₈	—	2	2	no	—
224	D₄	—	2	4	no	—
225	D₄	—	2	2	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	2	no	—
228	D₄	—	8	2	no	—
229	D₄	—	16	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	#123×C₆₇	1	2	yes	—
234	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
235	C₂×C₄	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
236	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
237	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	4	4	no	—
240	C₈	#123×C₆₇	1	1	yes	characteristic
241	C₈	#123×C₆₇	1	1	yes	characteristic
242	C₈	C₂×#41×C₆₇	1	2	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	8	4	no	—
247	C₂²	—	8	2	no	—
248	C₂²	—	16	4	no	—
249	C₂²	#481×C₆₇	1	2	yes	—
250	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	8	2	no	—
253	C₄	#481×C₆₇	1	2	yes	—
254	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
255	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
256	C₂	—	2	1	no	—
257	C₂	—	2	2	no	—
258	C₂	—	4	2	no	—
259	C₂	—	16	2	no	—
260	C₂	C₂×#481×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	C₂×#481
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₆₇×#3058	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂×#131×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2916	C₂	semidirect
3. C₆₇×#2238	C₂	semidirect
4. C₆₇×#2155	C₂	semidirect
5. C₂×#361×C₆₇	C₂	semidirect
6. C₆₇×#2115	C₂	semidirect
15. #430×C₆₇	C₂²	semidirect
20. #365×C₆₇	C₂²	semidirect
22. #349×C₆₇	C₂²	semidirect
125. #33657	C₆₇	direct (N × H)
132. #2916	C₂×C₆₇	semidirect
133. #2238	C₂×C₆₇	semidirect
134. #2155	C₂×C₆₇	semidirect
135. C₂×#361	C₂×C₆₇	semidirect
136. #2115	C₂×C₆₇	semidirect
137. C₆₇	#33657	direct (N × H)
146. #430	C₂²×C₆₇	semidirect
151. #365	C₂²×C₆₇	semidirect
153. #349	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3058	2
C₆₇×#2916	2
C₆₇×#2238	2
C₆₇×#2155	2
C₂×#361×C₆₇	2
C₆₇×#2115	2
#33657	67

Subgroups — C67×#33657 #30712908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33657 #30712908