Subgroups — C₆₇×#33397 #30712648

Order 17152.

706 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₆₇	C₂	1	2	yes	maximal
2	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2151	C₂	1	8	yes	maximal, complemented
4	C₆₇×#2150	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₆₇	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
7	C₂×#117×C₆₇	C₄	1	1	yes	characteristic
8	C₂×M₃₂×C₆₇	—	2	8	no	—
9	C₂×M₃₂×C₆₇	C₂²	1	4	yes	—
10	C₂×M₃₂×C₆₇	C₂²	1	4	yes	—
11	C₂²×C₁₆×C₆₇	C₂²	1	4	yes	—
12	#405×C₆₇	C₄	1	4	yes	—
13	#405×C₆₇	C₂²	1	4	yes	—
14	C₂×#97×C₆₇	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
16	C₄⋊C₁₆×C₆₇	C₂²	1	16	yes	complemented
17	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
18	C₂×M₁₆×C₆₇	—	2	4	no	—
19	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₆₇	D₄	1	4	yes	—
24	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
25	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
26	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
27	#117×C₆₇	C₈	1	4	yes	—
28	#117×C₆₇	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₆₇	C₈	1	2	yes	—
30	C₂×C₂²⋊C₄×C₆₇	C₈	1	2	yes	—
31	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
32	M₃₂×C₆₇	—	2	8	no	—
33	M₃₂×C₆₇	—	2	8	no	—
34	C₂×C₁₆×C₆₇	—	2	8	no	—
35	C₂×C₁₆×C₆₇	C₂³	1	8	yes	—
36	#97×C₆₇	C₂×C₄	1	8	yes	—
37	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₆₇	C₂³	1	4	yes	—
39	C₂⁴×C₆₇	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
42	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
45	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
47	M₁₆×C₆₇	—	2	8	no	—
48	C₂×C₈×C₆₇	—	2	4	no	—
49	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
50	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	—
51	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
52	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
54	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
55	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
57	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
58	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₆₇	C₂×C₈	1	8	yes	—
60	C₂²⋊C₄×C₆₇	C₂×C₈	1	8	yes	—
61	C₄²×C₆₇	C₂×C₈	1	4	yes	—
62	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
63	C₁₆×C₆₇	—	2	8	no	—
64	C₂³×C₆₇	—	4	2	no	—
65	C₂³×C₆₇	C₄⋊C₈	1	4	yes	—
66	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
67	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₆₇	—	4	4	no	—
69	C₂×C₄×C₆₇	C₄⋊C₈	1	4	yes	—
70	C₂×C₄×C₆₇	C₄⋊C₈	1	8	yes	—
71	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
72	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
73	C₂×C₄×C₆₇	C₂²×C₈	1	8	yes	—
74	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
76	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
78	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
79	C₈×C₆₇	—	2	4	no	—
80	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
81	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
82	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
83	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
84	C₂²×C₆₇	—	2	2	no	—
85	C₂²×C₆₇	—	2	4	no	—
86	C₂²×C₆₇	—	4	4	no	—
87	C₂²×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
88	C₂²×C₆₇	C₂×C₄⋊C₈	1	4	yes	—
89	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
90	C₄×C₆₇	—	2	8	no	—
91	C₄×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
92	C₄×C₆₇	C₂×C₄⋊C₈	1	4	yes	—
93	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
94	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
95	#33397	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂×C₆₇	—	2	2	no	—
97	C₂×C₆₇	—	2	4	no	—
98	C₂×C₆₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
99	C₂×C₆₇	#3407	1	2	yes	—
100	C₂²×M₃₂	C₂×C₆₇	1	2	yes	—
101	C₂×#405	C₂×C₆₇	1	1	yes	characteristic
102	#2151	C₂×C₆₇	1	8	yes	complemented
103	#2150	C₂×C₆₇	1	4	yes	complemented
104	C₆₇	#33397	1	1	yes	characteristic, direct product factor, Sylow
105	C₂²×M₁₆	C₄×C₆₇	1	1	yes	characteristic
106	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
107	C₂×#117	C₄×C₆₇	1	1	yes	characteristic
108	C₂×M₃₂	—	2	8	no	—
109	C₂×M₃₂	C₂²×C₆₇	1	4	yes	—
110	C₂×M₃₂	C₂²×C₆₇	1	4	yes	—
111	C₂²×C₁₆	C₂²×C₆₇	1	4	yes	—
112	#405	C₄×C₆₇	1	4	yes	—
113	#405	C₂²×C₆₇	1	4	yes	—
114	C₂×#97	C₄×C₆₇	1	2	yes	—
115	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
116	C₄⋊C₁₆	C₂²×C₆₇	1	16	yes	complemented
117	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
118	C₂×M₁₆	—	2	4	no	—
119	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
120	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
121	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
122	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
123	C₂²×C₈	D₄×C₆₇	1	4	yes	—
124	C₂²×C₈	Q₈×C₆₇	1	4	yes	—
125	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
126	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
127	#117	C₈×C₆₇	1	4	yes	—
128	#117	C₂×C₄×C₆₇	1	4	yes	—
129	C₂×C₄⋊C₄	C₈×C₆₇	1	2	yes	—
130	C₂×C₂²⋊C₄	C₈×C₆₇	1	2	yes	—
131	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
132	M₃₂	—	2	8	no	—
133	M₃₂	—	2	8	no	—
134	C₂×C₁₆	—	2	8	no	—
135	C₂×C₁₆	C₂³×C₆₇	1	8	yes	—
136	#97	C₂×C₄×C₆₇	1	8	yes	—
137	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	—
138	C₄×C₈	C₂³×C₆₇	1	4	yes	—
139	C₂⁴	C₂×C₈×C₆₇	1	1	yes	characteristic
140	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
141	C₂²×C₄	C₂×C₈×C₆₇	1	4	yes	—
142	C₂²×C₄	C₂×C₈×C₆₇	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₈×C₆₇	1	2	yes	—
144	C₂²×C₄	M₁₆×C₆₇	1	4	yes	—
145	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
146	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
147	M₁₆	—	2	8	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	C₄⋊C₄×C₆₇	1	4	yes	—
150	C₂×C₈	C₄⋊C₄×C₆₇	1	8	yes	—
151	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
152	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
153	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
154	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
155	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
156	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
157	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
158	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
159	C₄⋊C₄	C₂×C₈×C₆₇	1	8	yes	—
160	C₂²⋊C₄	C₂×C₈×C₆₇	1	8	yes	—
161	C₄²	C₂×C₈×C₆₇	1	4	yes	—
162	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
163	C₁₆	—	2	8	no	—
164	C₂³	—	4	2	no	—
165	C₂³	C₄⋊C₈×C₆₇	1	4	yes	—
166	C₂³	C₂²×C₈×C₆₇	1	2	yes	—
167	C₂³	C₂²×C₈×C₆₇	1	1	yes	characteristic
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	C₄⋊C₈×C₆₇	1	4	yes	—
170	C₂×C₄	C₄⋊C₈×C₆₇	1	8	yes	—
171	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
172	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
173	C₂×C₄	C₂²×C₈×C₆₇	1	8	yes	—
174	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
175	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
176	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
177	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
178	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
179	C₈	—	2	4	no	—
180	C₈	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
181	C₈	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
182	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
183	C₈	C₂²×Q₈×C₆₇	1	1	yes	characteristic
184	C₂²	—	2	2	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	4	4	no	—
187	C₂²	C₂×C₄⋊C₈×C₆₇	1	2	yes	—
188	C₂²	C₂×C₄⋊C₈×C₆₇	1	4	yes	—
189	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
190	C₄	—	2	8	no	—
191	C₄	C₂×C₄⋊C₈×C₆₇	1	2	yes	—
192	C₄	C₂×C₄⋊C₈×C₆₇	1	4	yes	—
193	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
194	C₄	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	2	4	no	—
197	C₂	C₂²×C₄⋊C₈×C₆₇	1	1	yes	characteristic
198	C₂	C₆₇×#3407	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2151	C₂	semidirect
4. C₆₇×#2150	C₂	semidirect
16. C₄⋊C₁₆×C₆₇	C₂²	semidirect
95. #33397	C₆₇	direct (N × H)
102. #2151	C₂×C₆₇	semidirect
103. #2150	C₂×C₆₇	semidirect
104. C₆₇	#33397	direct (N × H)
116. C₄⋊C₁₆	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₃₂×C₆₇	2
C₂×#405×C₆₇	2
C₆₇×#2151	2
C₆₇×#2150	2
#33397	67

Subgroups — C67×#33397 #30712648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33397 #30712648