Subgroups — C₆₇×#22308 #30701559

Order 17152.

650 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2399	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1984	C₂	1	4	yes	maximal
3	C₆₇×#1940	C₂	1	2	yes	maximal
4	C₆₇×#1916	C₂	1	4	yes	maximal
5	C₆₇×#1832	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×Q₈×C₆₇	C₂²	1	2	yes	—
9	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	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	#398×C₆₇	C₄	1	4	yes	—
15	#395×C₆₇	C₄	1	4	yes	—
16	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	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	2	yes	—
23	#336×C₆₇	C₂²	1	2	yes	—
24	#336×C₆₇	C₂²	1	4	yes	—
25	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
27	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
28	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
29	C₄×Q₈×C₆₇	—	4	4	no	—
30	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
36	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₆₇	—	2	4	no	—
39	C₄⋊C₈×C₆₇	—	4	4	no	—
40	#97×C₆₇	—	2	2	no	—
41	C₄×C₈×C₆₇	—	4	4	no	—
42	C₄×C₈×C₆₇	Q₈	1	4	yes	complemented
43	#95×C₆₇	C₂×C₄	1	4	yes	—
44	#95×C₆₇	C₂×C₄	1	2	yes	—
45	#95×C₆₇	C₂×C₄	1	2	yes	—
46	#95×C₆₇	C₂×C₄	1	2	yes	—
47	#95×C₆₇	C₂×C₄	1	2	yes	—
48	C₂×Q₈×C₆₇	—	2	8	no	—
49	C₂×Q₈×C₆₇	—	4	2	no	—
50	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
56	C₂×C₈×C₆₇	—	2	2	no	—
57	C₂×C₈×C₆₇	—	2	2	no	—
58	C₂×C₈×C₆₇	—	2	4	no	—
59	C₂×C₈×C₆₇	—	2	2	no	—
60	C₂×C₈×C₆₇	—	2	2	no	—
61	C₂×C₈×C₆₇	—	2	4	no	—
62	C₂×C₈×C₆₇	—	2	4	no	—
63	C₂×C₈×C₆₇	—	2	2	no	—
64	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
65	C₄⋊C₄×C₆₇	—	2	8	no	—
66	C₄⋊C₄×C₆₇	—	2	8	no	—
67	C₄⋊C₄×C₆₇	—	4	4	no	—
68	C₄⋊C₄×C₆₇	—	4	2	no	—
69	C₄²×C₆₇	—	2	4	no	—
70	C₄²×C₆₇	—	2	2	no	—
71	C₄²×C₆₇	—	2	4	no	—
72	C₄²×C₆₇	—	2	2	no	—
73	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
74	C₄²×C₆₇	#41	1	2	yes	—
75	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
76	Q₈×C₆₇	—	4	8	no	—
77	C₂×C₄×C₆₇	—	2	4	no	—
78	C₂×C₄×C₆₇	—	2	4	no	—
79	C₂×C₄×C₆₇	—	2	4	no	—
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	2	4	no	—
82	C₂×C₄×C₆₇	—	2	2	no	—
83	C₂×C₄×C₆₇	—	2	4	no	—
84	C₂×C₄×C₆₇	—	2	4	no	—
85	C₂×C₄×C₆₇	—	2	4	no	—
86	C₂×C₄×C₆₇	—	2	4	no	—
87	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
88	C₂×C₄×C₆₇	#131	1	8	yes	—
89	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
91	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
92	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
93	C₈×C₆₇	—	2	4	no	—
94	C₈×C₆₇	—	4	4	no	—
95	C₈×C₆₇	—	4	2	no	—
96	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
97	C₂²×C₆₇	#518	1	1	yes	characteristic
98	C₂²×C₆₇	#519	1	1	yes	characteristic
99	C₂²×C₆₇	C₂×#131	1	4	yes	—
100	C₄×C₆₇	—	2	4	no	—
101	C₄×C₆₇	—	2	8	no	—
102	C₄×C₆₇	—	4	4	no	—
103	C₄×C₆₇	—	4	2	no	—
104	C₄×C₆₇	—	4	2	no	—
105	C₄×C₆₇	#542	1	2	yes	—
106	C₄×C₆₇	#544	1	2	yes	—
107	#22308	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂×C₆₇	#2331	1	1	yes	characteristic
109	C₂×C₆₇	#2960	1	2	yes	—
110	C₂×C₆₇	#2977	1	2	yes	—
111	C₂×C₆₇	#2986	1	2	yes	—
112	#2399	C₂×C₆₇	1	1	yes	characteristic
113	#1984	C₂×C₆₇	1	4	yes	—
114	#1940	C₂×C₆₇	1	2	yes	—
115	#1916	C₂×C₆₇	1	4	yes	—
116	#1832	C₂×C₆₇	1	1	yes	characteristic
117	C₈×C₄⋊C₄	C₂×C₆₇	1	2	yes	—
118	#1746	C₂×C₆₇	1	1	yes	characteristic
119	C₆₇	#22308	1	1	yes	characteristic, direct product factor, Sylow
120	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
121	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
122	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
123	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
124	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	—
125	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
126	#398	C₄×C₆₇	1	4	yes	—
127	#395	C₄×C₆₇	1	4	yes	—
128	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
129	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
130	#376	C₂²×C₆₇	1	1	yes	characteristic
131	#336	C₂²×C₆₇	1	2	yes	—
132	#336	C₂²×C₆₇	1	2	yes	—
133	#336	C₂²×C₆₇	1	2	yes	—
134	#336	C₂²×C₆₇	1	4	yes	—
135	#336	C₂²×C₆₇	1	4	yes	—
136	#336	C₂²×C₆₇	1	2	yes	—
137	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
138	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
139	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
140	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
141	C₄×Q₈	—	4	4	no	—
142	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
143	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
144	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
145	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
146	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
147	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
148	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
149	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
150	C₄⋊C₈	—	2	4	no	—
151	C₄⋊C₈	—	4	4	no	—
152	#97	—	2	2	no	—
153	C₄×C₈	—	4	4	no	—
154	C₄×C₈	Q₈×C₆₇	1	4	yes	complemented
155	#95	C₂×C₄×C₆₇	1	4	yes	—
156	#95	C₂×C₄×C₆₇	1	2	yes	—
157	#95	C₂×C₄×C₆₇	1	2	yes	—
158	#95	C₂×C₄×C₆₇	1	2	yes	—
159	#95	C₂×C₄×C₆₇	1	2	yes	—
160	C₂×Q₈	—	2	8	no	—
161	C₂×Q₈	—	4	2	no	—
162	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
163	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
164	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
165	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
166	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
167	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
177	C₄⋊C₄	—	2	8	no	—
178	C₄⋊C₄	—	2	8	no	—
179	C₄⋊C₄	—	4	4	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄²	—	2	4	no	—
182	C₄²	—	2	4	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
186	C₄²	#41×C₆₇	1	2	yes	—
187	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
188	Q₈	—	4	8	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	C₄×Q₈×C₆₇	1	4	yes	—
200	C₂×C₄	#131×C₆₇	1	8	yes	—
201	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
202	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
203	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
204	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
205	C₈	—	2	4	no	—
206	C₈	—	4	2	no	—
207	C₈	—	4	4	no	—
208	C₂²	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
209	C₂²	#518×C₆₇	1	1	yes	characteristic
210	C₂²	#519×C₆₇	1	1	yes	characteristic
211	C₂²	C₂×#131×C₆₇	1	4	yes	—
212	C₄	—	2	4	no	—
213	C₄	—	2	8	no	—
214	C₄	—	4	2	no	—
215	C₄	—	4	4	no	—
216	C₄	—	4	2	no	—
217	C₄	#542×C₆₇	1	2	yes	—
218	C₄	#544×C₆₇	1	2	yes	—
219	C₂	C₆₇×#2331	1	1	yes	characteristic
220	C₂	C₆₇×#2960	1	2	yes	—
221	C₂	C₆₇×#2977	1	2	yes	—
222	C₂	C₆₇×#2986	1	2	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
42. C₄×C₈×C₆₇	Q₈	semidirect
107. #22308	C₆₇	direct (N × H)
119. C₆₇	#22308	direct (N × H)
154. C₄×C₈	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2399	2
C₆₇×#1984	2
C₆₇×#1940	2
C₆₇×#1916	2
C₆₇×#1832	2
C₈×C₄⋊C₄×C₆₇	2
C₆₇×#1746	2
#22308	67

Subgroups — C67×#22308 #30701559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22308 #30701559