Subgroups — C₆₇×#22257 #30701508

Order 17152.

586 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3162	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2960	C₂	1	2	yes	maximal
3	C₆₇×#1995	C₂	1	2	yes	maximal
4	C₆₇×#1922	C₂	1	8	yes	maximal
5	C₆₇×#1838	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1746	C₂	1	1	yes	maximal, characteristic
7	#533×C₆₇	C₂²	1	2	yes	—
8	#501×C₆₇	C₄	1	4	yes	complemented
9	#501×C₆₇	C₄	1	4	yes	complemented
10	#446×C₆₇	—	2	2	no	—
11	#446×C₆₇	—	2	2	no	—
12	C₈×Q₈×C₆₇	—	2	2	no	—
13	C₈×Q₈×C₆₇	—	2	2	no	—
14	#427×C₆₇	C₂²	1	4	yes	—
15	#424×C₆₇	C₂²	1	4	yes	—
16	#423×C₆₇	C₂²	1	2	yes	—
17	#405×C₆₇	C₂²	1	4	yes	—
18	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	#376×C₆₇	C₂²	1	1	yes	characteristic
21	#341×C₆₇	C₂²	1	4	yes	—
22	#339×C₆₇	C₂²	1	8	yes	—
23	#336×C₆₇	C₂²	1	4	yes	—
24	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
25	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
26	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
27	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
28	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
29	#125×C₆₇	C₂×C₄	1	4	yes	—
30	C₂²⋊Q₈×C₆₇	—	2	2	no	—
31	C₂²⋊Q₈×C₆₇	—	2	2	no	—
32	C₄×Q₈×C₆₇	—	2	2	no	—
33	C₄×Q₈×C₆₇	—	2	2	no	—
34	#117×C₆₇	C₂³	1	4	yes	—
35	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
36	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
37	#107×C₆₇	C₂×C₄	1	8	yes	—
38	#106×C₆₇	C₂×C₄	1	4	yes	—
39	#106×C₆₇	C₂×C₄	1	4	yes	—
40	C₄⋊C₈×C₆₇	—	2	4	no	—
41	C₄⋊C₈×C₆₇	—	2	4	no	—
42	C₄⋊C₈×C₆₇	—	2	2	no	—
43	C₄⋊C₈×C₆₇	—	2	2	no	—
44	#97×C₆₇	—	2	4	no	—
45	#97×C₆₇	C₂×C₄	1	4	yes	—
46	C₄×C₈×C₆₇	—	2	4	no	—
47	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
48	C₄×C₈×C₆₇	D₄	1	2	yes	—
49	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
50	#95×C₆₇	—	2	2	no	—
51	C₂×Q₈×C₆₇	—	2	2	no	—
52	C₂×Q₈×C₆₇	—	2	2	no	—
53	C₂²×C₄×C₆₇	—	2	2	no	—
54	C₂²×C₄×C₆₇	#41	1	2	yes	—
55	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
56	M₁₆×C₆₇	—	2	8	no	—
57	C₂×C₈×C₆₇	—	2	4	no	—
58	C₂×C₈×C₆₇	—	2	4	no	—
59	C₂×C₈×C₆₇	—	2	4	no	—
60	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
61	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
63	C₄⋊C₄×C₆₇	—	2	4	no	—
64	C₄⋊C₄×C₆₇	—	2	4	no	—
65	C₄⋊C₄×C₆₇	—	2	2	no	—
66	C₄⋊C₄×C₆₇	—	2	2	no	—
67	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
68	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
69	C₂²⋊C₄×C₆₇	—	2	4	no	—
70	C₄²×C₆₇	—	2	4	no	—
71	C₄²×C₆₇	—	2	4	no	—
72	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
73	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
76	Q₈×C₆₇	—	4	4	no	—
77	Q₈×C₆₇	—	4	4	no	—
78	C₂×C₄×C₆₇	—	2	4	no	—
79	C₂×C₄×C₆₇	—	2	4	no	—
80	C₂×C₄×C₆₇	—	4	2	no	—
81	C₂×C₄×C₆₇	—	4	4	no	—
82	C₂×C₄×C₆₇	—	4	2	no	—
83	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
87	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
88	C₈×C₆₇	—	2	4	no	—
89	C₈×C₆₇	—	2	2	no	—
90	C₈×C₆₇	—	2	2	no	—
91	C₈×C₆₇	—	4	4	no	—
92	C₂²×C₆₇	—	2	2	no	—
93	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
94	C₂²×C₆₇	#541	1	2	yes	—
95	C₄×C₆₇	—	2	2	no	—
96	C₄×C₆₇	—	2	2	no	—
97	C₄×C₆₇	—	4	4	no	—
98	C₄×C₆₇	—	4	4	no	—
99	C₄×C₆₇	—	8	2	no	—
100	C₄×C₆₇	#519	1	2	yes	—
101	C₄×C₆₇	#536	1	2	yes	—
102	#22257	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂×C₆₇	—	2	2	no	—
104	C₂×C₆₇	#2327	1	1	yes	characteristic
105	C₂×C₆₇	#2955	1	2	yes	—
106	#3162	C₂×C₆₇	1	1	yes	characteristic
107	#2960	C₂×C₆₇	1	2	yes	—
108	#1995	C₂×C₆₇	1	2	yes	—
109	#1922	C₂×C₆₇	1	8	yes	—
110	#1838	C₂×C₆₇	1	1	yes	characteristic
111	#1746	C₂×C₆₇	1	1	yes	characteristic
112	C₆₇	#22257	1	1	yes	characteristic, direct product factor, Sylow
113	#533	C₂²×C₆₇	1	2	yes	—
114	#501	C₄×C₆₇	1	4	yes	complemented
115	#501	C₄×C₆₇	1	4	yes	complemented
116	#446	—	2	2	no	—
117	#446	—	2	2	no	—
118	C₈×Q₈	—	2	2	no	—
119	C₈×Q₈	—	2	2	no	—
120	#427	C₂²×C₆₇	1	4	yes	—
121	#424	C₂²×C₆₇	1	4	yes	—
122	#423	C₂²×C₆₇	1	2	yes	—
123	#405	C₂²×C₆₇	1	4	yes	—
124	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
125	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
126	#376	C₂²×C₆₇	1	1	yes	characteristic
127	#341	C₂²×C₆₇	1	4	yes	—
128	#339	C₂²×C₆₇	1	8	yes	—
129	#336	C₂²×C₆₇	1	4	yes	—
130	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
131	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
132	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
133	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
134	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
135	#125	C₂×C₄×C₆₇	1	4	yes	—
136	C₂²⋊Q₈	—	2	2	no	—
137	C₂²⋊Q₈	—	2	2	no	—
138	C₄×Q₈	—	2	2	no	—
139	C₄×Q₈	—	2	2	no	—
140	#117	C₂³×C₆₇	1	4	yes	—
141	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
142	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
143	#107	C₂×C₄×C₆₇	1	8	yes	—
144	#106	C₂×C₄×C₆₇	1	4	yes	—
145	#106	C₂×C₄×C₆₇	1	4	yes	—
146	C₄⋊C₈	—	2	4	no	—
147	C₄⋊C₈	—	2	4	no	—
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	#97	—	2	4	no	—
151	#97	C₂×C₄×C₆₇	1	4	yes	—
152	C₄×C₈	—	2	4	no	—
153	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
154	C₄×C₈	D₄×C₆₇	1	2	yes	—
155	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
156	#95	—	2	2	no	—
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	—	2	2	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	#41×C₆₇	1	2	yes	—
161	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
162	M₁₆	—	2	8	no	—
163	C₂×C₈	—	2	4	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
167	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
168	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
174	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
175	C₂²⋊C₄	—	2	4	no	—
176	C₄²	—	2	4	no	—
177	C₄²	—	2	4	no	—
178	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
179	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
180	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
181	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
182	Q₈	—	4	4	no	—
183	Q₈	—	4	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
190	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
191	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
192	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
193	C₂×C₄	2₋¹⁺⁴×C₆₇	1	4	yes	—
194	C₈	—	2	4	no	—
195	C₈	—	2	2	no	—
196	C₈	—	2	2	no	—
197	C₈	—	4	4	no	—
198	C₂²	—	2	2	no	—
199	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
200	C₂²	#541×C₆₇	1	2	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	4	4	no	—
204	C₄	—	4	4	no	—
205	C₄	—	8	2	no	—
206	C₄	#519×C₆₇	1	2	yes	—
207	C₄	#536×C₆₇	1	2	yes	—
208	C₂	—	2	2	no	—
209	C₂	C₆₇×#2327	1	1	yes	characteristic
210	C₂	C₆₇×#2955	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
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
8. #501×C₆₇	C₄	semidirect
9. #501×C₆₇	C₄	semidirect
102. #22257	C₆₇	direct (N × H)
112. C₆₇	#22257	direct (N × H)
114. #501	C₄×C₆₇	semidirect
115. #501	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3162	2
C₆₇×#2960	2
C₆₇×#1995	2
C₆₇×#1922	2
C₆₇×#1838	2
C₆₇×#1746	2
#22257	67

Subgroups — C67×#22257 #30701508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22257 #30701508