Subgroups — C₆₇×#33658 #30712909

Order 17152.

534 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3059	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2916	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2239	C₂	1	8	yes	maximal, complemented
4	C₆₇×#2155	C₂	1	2	yes	maximal, complemented
5	C₂×#362×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2115	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
8	#568×C₆₇	—	2	1	no	—
9	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
10	#536×C₆₇	—	2	1	no	—
11	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
12	#470×C₆₇	C₂²	1	4	yes	—
13	#467×C₆₇	—	2	2	no	—
14	#430×C₆₇	—	2	1	no	—
15	#430×C₆₇	C₂²	1	4	yes	—
16	C₂×#108×C₆₇	—	2	2	no	—
17	C₂×#108×C₆₇	C₂²	1	2	yes	—
18	#427×C₆₇	C₂²	1	1	yes	characteristic
19	#419×C₆₇	—	2	1	no	—
20	#365×C₆₇	C₂²	1	8	yes	—
21	#362×C₆₇	C₂²	1	8	yes	—
22	#349×C₆₇	C₂²	1	4	yes	—
23	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
24	C₂²×Q₈×C₆₇	—	2	1	no	—
25	C₂×Q₁₆×C₆₇	—	2	1	no	—
26	C₂×Q₁₆×C₆₇	—	2	2	no	—
27	C₂×Q₁₆×C₆₇	—	2	2	no	—
28	C₂×Q₁₆×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	C₄⋊Q₈×C₆₇	—	4	1	no	—
39	#124×C₆₇	—	4	1	no	—
40	#123×C₆₇	—	4	2	no	—
41	C₂²⋊Q₈×C₆₇	—	2	4	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	#103×C₆₇	—	2	4	no	—
52	#41×C₆₇	—	2	2	no	—
53	#41×C₆₇	—	2	2	no	—
54	C₂×Q₈×C₆₇	—	2	2	no	—
55	C₂×Q₈×C₆₇	—	4	1	no	—
56	C₂×Q₈×C₆₇	—	4	2	no	—
57	C₂×Q₈×C₆₇	—	4	2	no	—
58	C₂×Q₈×C₆₇	SD₁₆	1	1	yes	characteristic
59	C₂×D₄×C₆₇	SD₁₆	1	2	yes	—
60	C₂×D₄×C₆₇	SD₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	—	4	1	no	—
62	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
64	Q₁₆×C₆₇	—	2	4	no	—
65	Q₁₆×C₆₇	—	4	1	no	—
66	Q₁₆×C₆₇	—	4	1	no	—
67	M₁₆×C₆₇	—	2	2	no	—
68	M₁₆×C₆₇	—	2	4	no	—
69	M₁₆×C₆₇	—	2	4	no	—
70	M₁₆×C₆₇	—	4	4	no	—
71	M₁₆×C₆₇	—	4	2	no	—
72	C₂×C₈×C₆₇	—	2	2	no	—
73	C₂×C₈×C₆₇	—	2	2	no	—
74	C₂×C₈×C₆₇	—	4	4	no	—
75	C₂×C₈×C₆₇	—	4	2	no	—
76	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	#41	1	2	yes	—
79	C₂×C₈×C₆₇	#41	1	2	yes	—
80	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
81	C₄⋊C₄×C₆₇	—	2	2	no	—
82	C₄⋊C₄×C₆₇	—	4	4	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	C₂×SD₁₆	1	2	yes	—
88	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
89	Q₈×C₆₇	—	2	2	no	—
90	Q₈×C₆₇	—	4	2	no	—
91	Q₈×C₆₇	—	4	2	no	—
92	Q₈×C₆₇	—	8	2	no	—
93	D₄×C₆₇	—	2	4	no	—
94	D₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	—	2	2	no	—
96	C₂×C₄×C₆₇	—	2	2	no	—
97	C₂×C₄×C₆₇	—	4	2	no	—
98	C₂×C₄×C₆₇	—	4	2	no	—
99	C₂×C₄×C₆₇	—	8	2	no	—
100	C₂×C₄×C₆₇	#123	1	2	yes	—
101	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
102	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
104	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
105	C₈×C₆₇	—	2	2	no	—
106	C₈×C₆₇	—	4	4	no	—
107	C₈×C₆₇	#123	1	1	yes	characteristic
108	C₈×C₆₇	#123	1	1	yes	characteristic
109	C₈×C₆₇	C₂×#41	1	2	yes	—
110	C₂²×C₆₇	—	2	2	no	—
111	C₂²×C₆₇	—	4	1	no	—
112	C₂²×C₆₇	—	4	2	no	—
113	C₂²×C₆₇	#481	1	2	yes	—
114	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
115	C₄×C₆₇	—	2	2	no	—
116	C₄×C₆₇	—	8	2	no	—
117	C₄×C₆₇	—	8	2	no	—
118	C₄×C₆₇	#481	1	2	yes	—
119	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
120	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
121	#33658	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂×C₆₇	—	2	1	no	—
123	C₂×C₆₇	—	2	2	no	—
124	C₂×C₆₇	—	4	2	no	—
125	C₂×C₆₇	C₂×#481	1	1	yes	characteristic
126	#3059	C₂×C₆₇	1	1	yes	characteristic
127	#2916	C₂×C₆₇	1	1	yes	characteristic
128	#2239	C₂×C₆₇	1	8	yes	complemented
129	#2155	C₂×C₆₇	1	2	yes	complemented
130	C₂×#362	C₂×C₆₇	1	2	yes	complemented
131	#2115	C₂×C₆₇	1	1	yes	characteristic
132	C₆₇	#33658	1	1	yes	characteristic, direct product factor, Sylow
133	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
134	#568	—	2	1	no	—
135	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
136	#536	—	2	1	no	—
137	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
138	#470	C₂²×C₆₇	1	4	yes	—
139	#467	—	2	2	no	—
140	#430	—	2	1	no	—
141	#430	C₂²×C₆₇	1	4	yes	—
142	C₂×#108	—	2	2	no	—
143	C₂×#108	C₂²×C₆₇	1	2	yes	—
144	#427	C₂²×C₆₇	1	1	yes	characteristic
145	#419	—	2	1	no	—
146	#365	C₂²×C₆₇	1	8	yes	—
147	#362	C₂²×C₆₇	1	8	yes	—
148	#349	C₂²×C₆₇	1	4	yes	—
149	C₂×#41	D₄×C₆₇	1	1	yes	characteristic
150	C₂²×Q₈	—	2	1	no	—
151	C₂×Q₁₆	—	2	1	no	—
152	C₂×Q₁₆	—	2	2	no	—
153	C₂×Q₁₆	—	2	2	no	—
154	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
155	#131	—	2	2	no	—
156	#131	—	2	2	no	—
157	#131	—	4	4	no	—
158	C₂×M₁₆	—	2	4	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
161	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
162	C₂²×C₈	D₄×C₆₇	1	2	yes	—
163	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
164	C₄⋊Q₈	—	4	1	no	—
165	#124	—	4	1	no	—
166	#123	—	4	2	no	—
167	C₂²⋊Q₈	—	2	4	no	—
168	#117	—	2	1	no	—
169	M₃₂	—	2	2	no	—
170	M₃₂	C₂³×C₆₇	1	4	yes	—
171	C₂×C₁₆	—	2	2	no	—
172	#108	—	2	2	no	—
173	#108	—	2	4	no	—
174	#108	C₂³×C₆₇	1	4	yes	—
175	#107	—	2	1	no	—
176	#106	C₂³×C₆₇	1	2	yes	—
177	#103	—	2	4	no	—
178	#41	—	2	2	no	—
179	#41	—	2	2	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	—	4	1	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	SD₁₆×C₆₇	1	1	yes	characteristic
185	C₂×D₄	SD₁₆×C₆₇	1	2	yes	—
186	C₂×D₄	SD₁₆×C₆₇	1	1	yes	characteristic
187	C₂²×C₄	—	4	1	no	—
188	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
189	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
190	Q₁₆	—	2	4	no	—
191	Q₁₆	—	4	1	no	—
192	Q₁₆	—	4	1	no	—
193	M₁₆	—	2	2	no	—
194	M₁₆	—	2	4	no	—
195	M₁₆	—	2	4	no	—
196	M₁₆	—	4	4	no	—
197	M₁₆	—	4	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	4	4	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
203	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
204	C₂×C₈	#41×C₆₇	1	2	yes	—
205	C₂×C₈	#41×C₆₇	1	2	yes	—
206	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	4	4	no	—
209	C₂²⋊C₄	—	4	1	no	—
210	C₂²⋊C₄	—	4	4	no	—
211	C₄²	—	4	1	no	—
212	C₁₆	—	2	4	no	—
213	C₂³	C₂×SD₁₆×C₆₇	1	2	yes	—
214	C₂³	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
215	Q₈	—	2	2	no	—
216	Q₈	—	4	2	no	—
217	Q₈	—	4	2	no	—
218	Q₈	—	8	2	no	—
219	D₄	—	2	4	no	—
220	D₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	#123×C₆₇	1	2	yes	—
227	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
228	C₂×C₄	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
229	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
230	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	—	4	4	no	—
233	C₈	#123×C₆₇	1	1	yes	characteristic
234	C₈	#123×C₆₇	1	1	yes	characteristic
235	C₈	C₂×#41×C₆₇	1	2	yes	—
236	C₂²	—	2	2	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	2	no	—
239	C₂²	#481×C₆₇	1	2	yes	—
240	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	8	2	no	—
243	C₄	—	8	2	no	—
244	C₄	#481×C₆₇	1	2	yes	—
245	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
246	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
247	C₂	—	2	1	no	—
248	C₂	—	2	2	no	—
249	C₂	—	4	2	no	—
250	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	C₂×#41×C₆₇	D₄
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
3. C₆₇×#2239	C₂	semidirect
4. C₆₇×#2155	C₂	semidirect
5. C₂×#362×C₆₇	C₂	semidirect
121. #33658	C₆₇	direct (N × H)
128. #2239	C₂×C₆₇	semidirect
129. #2155	C₂×C₆₇	semidirect
130. C₂×#362	C₂×C₆₇	semidirect
132. C₆₇	#33658	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3059	2
C₆₇×#2916	2
C₆₇×#2239	2
C₆₇×#2155	2
C₂×#362×C₆₇	2
C₆₇×#2115	2
#33658	67

Subgroups — C67×#33658 #30712909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33658 #30712909