Subgroups — C₆₇×#10842 #30690093

Order 17152.

950 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#382×C₆₇	C₂	1	2	yes	maximal
3	C₂×#340×C₆₇	C₂	1	4	yes	maximal
4	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
5	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
6	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈×C₆₇	—	2	4	no	—
8	C₂×C₄×C₈×C₆₇	C₄	1	4	yes	complemented
9	#382×C₆₇	—	2	4	no	—
10	#382×C₆₇	C₄	1	8	yes	complemented
11	C₂×#95×C₆₇	—	2	4	no	—
12	#340×C₆₇	—	2	8	no	—
13	C₂³×C₄×C₆₇	—	2	4	no	—
14	C₂³×C₄×C₆₇	D₄	1	2	yes	—
15	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
16	C₂²×C₈×C₆₇	—	2	4	no	—
17	C₂²×C₈×C₆₇	—	2	4	no	—
18	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
19	C₂²×C₈×C₆₇	D₄	1	4	yes	—
20	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
21	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
22	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
23	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
24	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
25	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₆₇	D₄	1	4	yes	—
28	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
29	C₄×C₈×C₆₇	—	2	8	no	—
30	#95×C₆₇	—	2	16	no	—
31	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₄×C₆₇	—	2	8	no	—
33	C₂²×C₄×C₆₇	—	2	8	no	—
34	C₂²×C₄×C₆₇	—	4	4	no	—
35	C₂²×C₄×C₆₇	—	4	4	no	—
36	C₂²×C₄×C₆₇	—	4	16	no	—
37	C₂²×C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
38	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
39	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
42	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
45	C₂²×C₄×C₆₇	#41	1	4	yes	—
46	C₂×C₈×C₆₇	—	2	4	no	—
47	C₂×C₈×C₆₇	—	2	4	no	—
48	C₂×C₈×C₆₇	—	2	8	no	—
49	C₂×C₈×C₆₇	—	2	8	no	—
50	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	—
51	C₄⋊C₄×C₆₇	—	2	16	no	—
52	C₄⋊C₄×C₆₇	—	4	8	no	—
53	C₄²×C₆₇	C₄²	1	4	yes	—
54	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
55	C₄²×C₆₇	C₄⋊C₄	1	8	yes	—
56	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
57	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂³×C₆₇	#124	1	2	yes	—
59	C₂³×C₆₇	#127	1	1	yes	characteristic
60	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
61	C₂³×C₆₇	#135	1	8	yes	—
62	C₂×C₄×C₆₇	—	2	16	no	—
63	C₂×C₄×C₆₇	—	2	8	no	—
64	C₂×C₄×C₆₇	—	4	8	no	—
65	C₂×C₄×C₆₇	—	4	16	no	—
66	C₂×C₄×C₆₇	—	8	8	no	—
67	C₂×C₄×C₆₇	—	8	8	no	—
68	C₂×C₄×C₆₇	#95	1	8	yes	—
69	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
73	C₂×C₄×C₆₇	#117	1	2	yes	—
74	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
75	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
76	C₂×C₄×C₆₇	#121	1	4	yes	—
77	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
78	C₂×C₄×C₆₇	#125	1	2	yes	—
79	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
80	C₈×C₆₇	—	2	8	no	—
81	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
82	C₂²×C₆₇	#382	1	1	yes	characteristic
83	C₂²×C₆₇	#384	1	1	yes	characteristic
84	C₂²×C₆₇	#390	1	2	yes	—
85	C₂²×C₆₇	#391	1	2	yes	—
86	C₂²×C₆₇	#416	1	8	yes	—
87	C₂²×C₆₇	#427	1	4	yes	—
88	C₂²×C₆₇	#488	1	8	yes	—
89	C₂²×C₆₇	#495	1	4	yes	—
90	C₂²×C₆₇	#499	1	4	yes	—
91	C₄×C₆₇	—	2	8	no	—
92	C₄×C₆₇	—	8	8	no	—
93	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
94	C₄×C₆₇	#380	1	2	yes	—
95	C₄×C₆₇	#389	1	4	yes	—
96	#10842	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
98	C₂×C₆₇	#1735	1	2	yes	—
99	C₂×C₆₇	#1835	1	2	yes	—
100	C₂×C₆₇	#1846	1	2	yes	—
101	C₂×C₆₇	#1958	1	4	yes	—
102	C₂×C₆₇	#1991	1	4	yes	—
103	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic
104	C₂×#382	C₂×C₆₇	1	2	yes	—
105	C₂×#340	C₂×C₆₇	1	4	yes	—
106	C₆₇	#10842	1	1	yes	characteristic, direct product factor, Sylow
107	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
108	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
109	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
110	C₂×C₄×C₈	—	2	4	no	—
111	C₂×C₄×C₈	C₄×C₆₇	1	4	yes	complemented
112	#382	—	2	4	no	—
113	#382	C₄×C₆₇	1	8	yes	complemented
114	C₂×#95	—	2	4	no	—
115	#340	—	2	8	no	—
116	C₂³×C₄	—	2	4	no	—
117	C₂³×C₄	D₄×C₆₇	1	2	yes	—
118	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
119	C₂²×C₈	—	2	4	no	—
120	C₂²×C₈	—	2	4	no	—
121	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
122	C₂²×C₈	D₄×C₆₇	1	4	yes	—
123	C₂²×C₈	Q₈×C₆₇	1	4	yes	—
124	C₂×C₄⋊C₄	—	2	8	no	—
125	C₂×C₄⋊C₄	—	2	8	no	—
126	C₂×C₄⋊C₄	—	2	4	no	—
127	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	8	yes	—
128	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
129	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
130	C₂×C₄²	D₄×C₆₇	1	4	yes	—
131	C₂×C₄²	Q₈×C₆₇	1	2	yes	—
132	C₄×C₈	—	2	8	no	—
133	#95	—	2	16	no	—
134	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
135	C₂²×C₄	—	2	8	no	—
136	C₂²×C₄	—	2	8	no	—
137	C₂²×C₄	—	4	4	no	—
138	C₂²×C₄	—	4	4	no	—
139	C₂²×C₄	—	4	16	no	—
140	C₂²×C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
141	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
142	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
143	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
144	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
145	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
146	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
147	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
148	C₂²×C₄	#41×C₆₇	1	4	yes	—
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	2	8	no	—
152	C₂×C₈	—	2	8	no	—
153	C₂×C₈	C₄⋊C₄×C₆₇	1	8	yes	—
154	C₄⋊C₄	—	2	16	no	—
155	C₄⋊C₄	—	4	8	no	—
156	C₄²	C₄²×C₆₇	1	4	yes	—
157	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
158	C₄²	C₄⋊C₄×C₆₇	1	8	yes	—
159	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
160	C₂³	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
161	C₂³	#124×C₆₇	1	2	yes	—
162	C₂³	#127×C₆₇	1	1	yes	characteristic
163	C₂³	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
164	C₂³	#135×C₆₇	1	8	yes	—
165	C₂×C₄	—	2	16	no	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	4	8	no	—
168	C₂×C₄	—	4	16	no	—
169	C₂×C₄	—	8	8	no	—
170	C₂×C₄	—	8	8	no	—
171	C₂×C₄	#95×C₆₇	1	8	yes	—
172	C₂×C₄	C₂×C₄²×C₆₇	1	1	yes	characteristic
173	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
174	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
175	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
176	C₂×C₄	#117×C₆₇	1	2	yes	—
177	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
178	C₂×C₄	C₄×Q₈×C₆₇	1	4	yes	—
179	C₂×C₄	#121×C₆₇	1	4	yes	—
180	C₂×C₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
181	C₂×C₄	#125×C₆₇	1	2	yes	—
182	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
183	C₈	—	2	8	no	—
184	C₂²	C₂×#95×C₆₇	1	1	yes	characteristic
185	C₂²	#382×C₆₇	1	1	yes	characteristic
186	C₂²	#384×C₆₇	1	1	yes	characteristic
187	C₂²	#390×C₆₇	1	2	yes	—
188	C₂²	#391×C₆₇	1	2	yes	—
189	C₂²	#416×C₆₇	1	8	yes	—
190	C₂²	#427×C₆₇	1	4	yes	—
191	C₂²	#488×C₆₇	1	8	yes	—
192	C₂²	#495×C₆₇	1	4	yes	—
193	C₂²	#499×C₆₇	1	4	yes	—
194	C₄	—	2	8	no	—
195	C₄	—	8	8	no	—
196	C₄	C₄×C₄⋊C₄×C₆₇	1	2	yes	—
197	C₄	#380×C₆₇	1	2	yes	—
198	C₄	#389×C₆₇	1	4	yes	—
199	C₂	C₆₇×C₄⋊#95	1	1	yes	characteristic
200	C₂	C₆₇×#1735	1	2	yes	—
201	C₂	C₆₇×#1835	1	2	yes	—
202	C₂	C₆₇×#1846	1	2	yes	—
203	C₂	C₆₇×#1958	1	4	yes	—
204	C₂	C₆₇×#1991	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₆₇	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₂×C₄×C₈×C₆₇	C₄	semidirect
10. #382×C₆₇	C₄	semidirect
96. #10842	C₆₇	direct (N × H)
106. C₆₇	#10842	direct (N × H)
111. C₂×C₄×C₈	C₄×C₆₇	semidirect
113. #382	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₈×C₆₇	2
C₂×#382×C₆₇	2
C₂×#340×C₆₇	2
#10842	67

Subgroups — C67×#10842 #30690093

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10842 #30690093