Subgroups — C₆₇×#22300 #30701551

Order 17152.

618 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2402	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1940	C₂	1	2	yes	maximal
3	C₆₇×#1939	C₂	1	4	yes	maximal
4	C₆₇×#1916	C₂	1	4	yes	maximal
5	C₆₇×#1843	C₂	1	1	yes	maximal, characteristic
6	C₈×C₄⋊C₄×C₆₇	C₂	1	2	yes	maximal
7	C₆₇×#1746	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
9	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
10	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
11	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#400×C₆₇	C₄	1	4	yes	—
15	#398×C₆₇	C₄	1	4	yes	—
16	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
18	#376×C₆₇	C₂²	1	1	yes	characteristic
19	#336×C₆₇	C₂²	1	2	yes	—
20	#336×C₆₇	C₂²	1	2	yes	—
21	#336×C₆₇	C₂²	1	4	yes	—
22	#336×C₆₇	C₂²	1	4	yes	—
23	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
25	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
27	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
28	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
31	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
32	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₆₇	—	2	4	no	—
34	C₄⋊C₈×C₆₇	—	4	4	no	—
35	C₄⋊C₈×C₆₇	—	4	4	no	—
36	#97×C₆₇	—	2	2	no	—
37	C₄×C₈×C₆₇	—	4	4	no	—
38	C₄×C₈×C₆₇	Q₈	1	4	yes	—
39	#95×C₆₇	C₂×C₄	1	4	yes	—
40	#95×C₆₇	C₂×C₄	1	2	yes	—
41	#95×C₆₇	C₂×C₄	1	4	yes	—
42	#95×C₆₇	C₂×C₄	1	4	yes	—
43	#95×C₆₇	C₂×C₄	1	2	yes	—
44	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
45	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
48	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
49	C₂×C₈×C₆₇	—	2	2	no	—
50	C₂×C₈×C₆₇	—	2	2	no	—
51	C₂×C₈×C₆₇	—	2	4	no	—
52	C₂×C₈×C₆₇	—	2	1	no	—
53	C₂×C₈×C₆₇	—	2	1	no	—
54	C₂×C₈×C₆₇	—	2	4	no	—
55	C₂×C₈×C₆₇	—	2	2	no	—
56	C₂×C₈×C₆₇	—	2	4	no	—
57	C₂×C₈×C₆₇	—	2	2	no	—
58	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
59	C₄⋊C₄×C₆₇	—	2	8	no	—
60	C₄⋊C₄×C₆₇	—	2	8	no	—
61	C₄⋊C₄×C₆₇	—	2	8	no	—
62	C₄²×C₆₇	—	2	2	no	—
63	C₄²×C₆₇	—	2	4	no	—
64	C₄²×C₆₇	—	2	4	no	—
65	C₄²×C₆₇	—	2	2	no	—
66	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
67	C₄²×C₆₇	#41	1	2	yes	—
68	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
69	C₂×C₄×C₆₇	—	2	4	no	—
70	C₂×C₄×C₆₇	—	2	4	no	—
71	C₂×C₄×C₆₇	—	2	4	no	—
72	C₂×C₄×C₆₇	—	2	4	no	—
73	C₂×C₄×C₆₇	—	2	4	no	—
74	C₂×C₄×C₆₇	—	2	4	no	—
75	C₂×C₄×C₆₇	—	2	4	no	—
76	C₂×C₄×C₆₇	—	2	4	no	—
77	C₂×C₄×C₆₇	—	2	4	no	—
78	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
79	C₂×C₄×C₆₇	#131	1	8	yes	—
80	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
81	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
82	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
83	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
84	C₈×C₆₇	—	2	4	no	—
85	C₈×C₆₇	—	4	4	no	—
86	C₈×C₆₇	—	4	1	no	—
87	C₈×C₆₇	—	4	1	no	—
88	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
89	C₂²×C₆₇	#520	1	2	yes	—
90	C₂²×C₆₇	C₂×#131	1	4	yes	—
91	C₄×C₆₇	—	2	4	no	—
92	C₄×C₆₇	—	2	8	no	—
93	C₄×C₆₇	—	4	4	no	—
94	C₄×C₆₇	—	4	4	no	—
95	C₄×C₆₇	#540	1	1	yes	characteristic
96	C₄×C₆₇	#541	1	1	yes	characteristic
97	C₄×C₆₇	#544	1	2	yes	—
98	#22300	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂×C₆₇	#2330	1	1	yes	characteristic
100	C₂×C₆₇	#2960	1	2	yes	—
101	C₂×C₆₇	#2994	1	4	yes	—
102	#2402	C₂×C₆₇	1	1	yes	characteristic
103	#1940	C₂×C₆₇	1	2	yes	—
104	#1939	C₂×C₆₇	1	4	yes	—
105	#1916	C₂×C₆₇	1	4	yes	—
106	#1843	C₂×C₆₇	1	1	yes	characteristic
107	C₈×C₄⋊C₄	C₂×C₆₇	1	2	yes	—
108	#1746	C₂×C₆₇	1	1	yes	characteristic
109	C₆₇	#22300	1	1	yes	characteristic, direct product factor, Sylow
110	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
111	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
112	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
113	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	—
115	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
116	#400	C₄×C₆₇	1	4	yes	—
117	#398	C₄×C₆₇	1	4	yes	—
118	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
119	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
120	#376	C₂²×C₆₇	1	1	yes	characteristic
121	#336	C₂²×C₆₇	1	4	yes	—
122	#336	C₂²×C₆₇	1	2	yes	—
123	#336	C₂²×C₆₇	1	4	yes	—
124	#336	C₂²×C₆₇	1	2	yes	—
125	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
126	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
127	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
128	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
129	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
130	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
131	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
132	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
133	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
134	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
135	C₄⋊C₈	—	2	4	no	—
136	C₄⋊C₈	—	4	4	no	—
137	C₄⋊C₈	—	4	4	no	—
138	#97	—	2	2	no	—
139	C₄×C₈	—	4	4	no	—
140	C₄×C₈	Q₈×C₆₇	1	4	yes	—
141	#95	C₂×C₄×C₆₇	1	4	yes	—
142	#95	C₂×C₄×C₆₇	1	2	yes	—
143	#95	C₂×C₄×C₆₇	1	4	yes	—
144	#95	C₂×C₄×C₆₇	1	4	yes	—
145	#95	C₂×C₄×C₆₇	1	2	yes	—
146	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
147	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
148	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
149	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
150	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
161	C₄⋊C₄	—	2	8	no	—
162	C₄⋊C₄	—	2	8	no	—
163	C₄⋊C₄	—	2	8	no	—
164	C₄²	—	2	4	no	—
165	C₄²	—	2	2	no	—
166	C₄²	—	2	4	no	—
167	C₄²	—	2	2	no	—
168	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
169	C₄²	#41×C₆₇	1	2	yes	—
170	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	C₄×Q₈×C₆₇	1	4	yes	—
181	C₂×C₄	#131×C₆₇	1	8	yes	—
182	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
183	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
184	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
185	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
186	C₈	—	2	4	no	—
187	C₈	—	4	1	no	—
188	C₈	—	4	4	no	—
189	C₈	—	4	1	no	—
190	C₂²	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
191	C₂²	#520×C₆₇	1	2	yes	—
192	C₂²	C₂×#131×C₆₇	1	4	yes	—
193	C₄	—	2	4	no	—
194	C₄	—	2	8	no	—
195	C₄	—	4	4	no	—
196	C₄	—	4	4	no	—
197	C₄	#540×C₆₇	1	1	yes	characteristic
198	C₄	#541×C₆₇	1	1	yes	characteristic
199	C₄	#544×C₆₇	1	2	yes	—
200	C₂	C₆₇×#2330	1	1	yes	characteristic
201	C₂	C₆₇×#2960	1	2	yes	—
202	C₂	C₆₇×#2994	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
98. #22300	C₆₇	direct (N × H)
109. C₆₇	#22300	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×#2402	2
C₆₇×#1940	2
C₆₇×#1939	2
C₆₇×#1916	2
C₆₇×#1843	2
C₈×C₄⋊C₄×C₆₇	2
C₆₇×#1746	2
#22300	67

Subgroups — C67×#22300 #30701551

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22300 #30701551