Subgroups — C₆₇×#7020 #30686271

Order 17152.

430 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#1797	C₂	1	2	yes	maximal
2	C₂×#336×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₈×C₆₇	C₄	1	2	yes	—
4	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
5	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
6	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
7	#336×C₆₇	—	2	2	no	—
8	#336×C₆₇	C₄	1	4	yes	—
9	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
10	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
11	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
12	C₂²×C₈×C₆₇	—	2	2	no	—
13	C₂²×C₈×C₆₇	—	2	2	no	—
14	C₂²×C₈×C₆₇	—	2	4	no	—
15	C₂²×C₈×C₆₇	—	4	2	no	—
16	C₂²×C₈×C₆₇	—	4	2	no	—
17	C₂²×C₈×C₆₇	C₈	1	4	yes	complemented
18	C₂×C₄²×C₆₇	—	4	2	no	—
19	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₆₇	D₄	1	2	yes	—
21	C₄⋊C₈×C₆₇	—	4	4	no	—
22	C₂²⋊C₈×C₆₇	—	4	2	no	—
23	C₂²⋊C₈×C₆₇	—	4	2	no	—
24	C₂⁴×C₆₇	C₄⋊C₄	1	1	yes	characteristic
25	C₂²×C₄×C₆₇	—	2	2	no	—
26	C₂²×C₄×C₆₇	—	2	1	no	—
27	C₂²×C₄×C₆₇	—	2	1	no	—
28	C₂²×C₄×C₆₇	—	2	2	no	—
29	C₂²×C₄×C₆₇	—	4	1	no	—
30	C₂²×C₄×C₆₇	—	4	1	no	—
31	C₂²×C₄×C₆₇	—	4	1	no	—
32	C₂²×C₄×C₆₇	—	4	1	no	—
33	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
34	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
37	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
38	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
39	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
41	C₂²×C₄×C₆₇	Q₁₆	1	1	yes	characteristic
42	C₂×C₈×C₆₇	—	2	4	no	—
43	C₂×C₈×C₆₇	—	4	4	no	—
44	C₂×C₈×C₆₇	—	4	2	no	—
45	C₂×C₈×C₆₇	—	4	2	no	—
46	C₂×C₈×C₆₇	—	4	4	no	—
47	C₂×C₈×C₆₇	—	4	4	no	—
48	C₂×C₈×C₆₇	—	4	4	no	—
49	C₂×C₈×C₆₇	—	4	4	no	—
50	C₄²×C₆₇	—	2	4	no	—
51	C₄²×C₆₇	—	4	2	no	—
52	C₂³×C₆₇	—	2	1	no	—
53	C₂³×C₆₇	—	2	1	no	—
54	C₂³×C₆₇	—	2	1	no	—
55	C₂³×C₆₇	—	2	1	no	—
56	C₂³×C₆₇	#95	1	1	yes	characteristic
57	C₂³×C₆₇	C₄⋊C₈	1	2	yes	—
58	C₂³×C₆₇	#106	1	1	yes	characteristic
59	C₂³×C₆₇	#107	1	1	yes	characteristic
60	C₂³×C₆₇	#108	1	2	yes	—
61	C₂×C₄×C₆₇	—	2	2	no	—
62	C₂×C₄×C₆₇	—	2	2	no	—
63	C₂×C₄×C₆₇	—	2	1	no	—
64	C₂×C₄×C₆₇	—	2	1	no	—
65	C₂×C₄×C₆₇	—	2	2	no	—
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	—	2	2	no	—
68	C₂×C₄×C₆₇	—	2	2	no	—
69	C₂×C₄×C₆₇	—	2	2	no	—
70	C₂×C₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	—	2	2	no	—
72	C₂×C₄×C₆₇	—	4	1	no	—
73	C₂×C₄×C₆₇	—	4	2	no	—
74	C₂×C₄×C₆₇	—	4	1	no	—
75	C₂×C₄×C₆₇	—	4	4	no	—
76	C₂×C₄×C₆₇	C₄×C₈	1	1	yes	characteristic
77	C₂×C₄×C₆₇	#97	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
79	C₂×C₄×C₆₇	#99	1	2	yes	—
80	C₂×C₄×C₆₇	#100	1	1	yes	characteristic
81	C₂×C₄×C₆₇	#101	1	1	yes	characteristic
82	C₂×C₄×C₆₇	#102	1	2	yes	—
83	C₂×C₄×C₆₇	#103	1	2	yes	—
84	C₈×C₆₇	—	4	4	no	—
85	C₈×C₆₇	—	8	4	no	—
86	C₂²×C₆₇	—	2	2	no	—
87	C₂²×C₆₇	—	2	1	no	—
88	C₂²×C₆₇	—	2	1	no	—
89	C₂²×C₆₇	—	2	2	no	—
90	C₂²×C₆₇	—	2	1	no	—
91	C₂²×C₆₇	—	2	1	no	—
92	C₂²×C₆₇	—	2	2	no	—
93	C₂²×C₆₇	—	2	2	no	—
94	C₂²×C₆₇	#334	1	2	yes	—
95	C₂²×C₆₇	#335	1	2	yes	—
96	C₂²×C₆₇	#336	1	1	yes	characteristic
97	C₂²×C₆₇	H(3,ℤ/4)	1	1	yes	characteristic
98	C₂²×C₆₇	#338	1	1	yes	characteristic
99	C₂²×C₆₇	#340	1	1	yes	characteristic
100	C₂²×C₆₇	#341	1	1	yes	characteristic
101	C₂²×C₆₇	#342	1	1	yes	characteristic
102	C₂²×C₆₇	#343	1	1	yes	characteristic
103	C₄×C₆₇	—	2	1	no	—
104	C₄×C₆₇	—	2	1	no	—
105	C₄×C₆₇	—	2	2	no	—
106	C₄×C₆₇	—	2	2	no	—
107	C₄×C₆₇	—	4	2	no	—
108	C₄×C₆₇	#323	1	2	yes	—
109	C₄×C₆₇	#324	1	2	yes	—
110	#7020	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂×C₆₇	—	2	2	no	—
112	C₂×C₆₇	—	2	2	no	—
113	C₂×C₆₇	#1275	1	1	yes	characteristic
114	C₂×C₆₇	#1277	1	1	yes	characteristic
115	C₂×C₆₇	#1280	1	1	yes	characteristic
116	C₂×C₆₇	#1289	1	1	yes	characteristic
117	C₂×C₆₇	#1290	1	1	yes	characteristic
118	C₂×C₆₇	#1292	1	1	yes	characteristic
119	C₂×C₆₇	#1293	1	1	yes	characteristic
120	#1797	C₂×C₆₇	1	2	yes	—
121	C₂×#336	C₂×C₆₇	1	1	yes	characteristic
122	C₆₇	#7020	1	1	yes	characteristic, direct product factor, Sylow
123	C₂³×C₈	C₄×C₆₇	1	2	yes	—
124	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
125	C₂×C₄⋊C₈	—	2	2	no	—
126	C₂×C₂²⋊C₈	—	2	2	no	—
127	#336	—	2	2	no	—
128	#336	C₄×C₆₇	1	4	yes	—
129	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
130	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
131	C₂³×C₄	Q₈×C₆₇	1	1	yes	characteristic
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	2	4	no	—
135	C₂²×C₈	—	4	2	no	—
136	C₂²×C₈	—	4	2	no	—
137	C₂²×C₈	C₈×C₆₇	1	4	yes	complemented
138	C₂×C₄²	—	4	2	no	—
139	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
140	C₂×C₄²	D₄×C₆₇	1	2	yes	—
141	C₄⋊C₈	—	4	4	no	—
142	C₂²⋊C₈	—	4	2	no	—
143	C₂²⋊C₈	—	4	2	no	—
144	C₂⁴	C₄⋊C₄×C₆₇	1	1	yes	characteristic
145	C₂²×C₄	—	2	1	no	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	4	1	no	—
150	C₂²×C₄	—	4	1	no	—
151	C₂²×C₄	—	4	1	no	—
152	C₂²×C₄	—	4	1	no	—
153	C₂²×C₄	C₄²×C₆₇	1	1	yes	characteristic
154	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
155	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
156	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
157	C₂²×C₄	C₂×C₈×C₆₇	1	2	yes	—
158	C₂²×C₄	M₁₆×C₆₇	1	2	yes	—
159	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
160	C₂²×C₄	SD₁₆×C₆₇	1	2	yes	—
161	C₂²×C₄	Q₁₆×C₆₇	1	1	yes	characteristic
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	—	4	4	no	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	—	4	4	no	—
167	C₂×C₈	—	4	4	no	—
168	C₂×C₈	—	4	4	no	—
169	C₂×C₈	—	4	4	no	—
170	C₄²	—	2	4	no	—
171	C₄²	—	4	2	no	—
172	C₂³	—	2	1	no	—
173	C₂³	—	2	1	no	—
174	C₂³	—	2	1	no	—
175	C₂³	—	2	1	no	—
176	C₂³	#95×C₆₇	1	1	yes	characteristic
177	C₂³	C₄⋊C₈×C₆₇	1	2	yes	—
178	C₂³	#106×C₆₇	1	1	yes	characteristic
179	C₂³	#107×C₆₇	1	1	yes	characteristic
180	C₂³	#108×C₆₇	1	2	yes	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	C₄×C₈×C₆₇	1	1	yes	characteristic
197	C₂×C₄	#97×C₆₇	1	1	yes	characteristic
198	C₂×C₄	C₂²⋊C₈×C₆₇	1	2	yes	—
199	C₂×C₄	#99×C₆₇	1	2	yes	—
200	C₂×C₄	#100×C₆₇	1	1	yes	characteristic
201	C₂×C₄	#101×C₆₇	1	1	yes	characteristic
202	C₂×C₄	#102×C₆₇	1	2	yes	—
203	C₂×C₄	#103×C₆₇	1	2	yes	—
204	C₈	—	4	4	no	—
205	C₈	—	8	4	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	—	2	2	no	—
214	C₂²	#334×C₆₇	1	2	yes	—
215	C₂²	#335×C₆₇	1	2	yes	—
216	C₂²	#336×C₆₇	1	1	yes	characteristic
217	C₂²	H(3,ℤ/4)×C₆₇	1	1	yes	characteristic
218	C₂²	#338×C₆₇	1	1	yes	characteristic
219	C₂²	#340×C₆₇	1	1	yes	characteristic
220	C₂²	#341×C₆₇	1	1	yes	characteristic
221	C₂²	#342×C₆₇	1	1	yes	characteristic
222	C₂²	#343×C₆₇	1	1	yes	characteristic
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	—	4	2	no	—
228	C₄	#323×C₆₇	1	2	yes	—
229	C₄	#324×C₆₇	1	2	yes	—
230	C₂	—	2	2	no	—
231	C₂	—	2	2	no	—
232	C₂	C₆₇×#1275	1	1	yes	characteristic
233	C₂	C₆₇×#1277	1	1	yes	characteristic
234	C₂	C₆₇×#1280	1	1	yes	characteristic
235	C₂	C₆₇×#1289	1	1	yes	characteristic
236	C₂	C₆₇×#1290	1	1	yes	characteristic
237	C₂	C₆₇×#1292	1	1	yes	characteristic
238	C₂	C₆₇×#1293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	#95
commutator	C₂×C₄	C₄×C₈×C₆₇
Frattini, cosocle	C₂²×C₄²	C₂²×C₆₇
prime radical	C₂⁴×C₆₇	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
17. C₂²×C₈×C₆₇	C₈	semidirect
110. #7020	C₆₇	direct (N × H)
122. C₆₇	#7020	direct (N × H)
137. C₂²×C₈	C₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#1797	2
C₂×#336×C₆₇	2
#7020	67

Subgroups — C67×#7020 #30686271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#7020 #30686271