Subgroups — C₆₇×#33289 #30712540

Order 17152.

862 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2868	C₂	1	2	yes	maximal
2	C₆₇×#1990	C₂	1	8	yes	maximal
3	C₆₇×#1834	C₂	1	4	yes	maximal
4	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
6	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
7	C₂×C₄⋊Q₈×C₆₇	C₂²	1	4	yes	—
8	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
9	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
10	#384×C₆₇	—	2	2	no	—
11	#384×C₆₇	—	2	2	no	—
12	#384×C₆₇	C₂²	1	4	yes	—
13	#384×C₆₇	C₂²	1	4	yes	—
14	C₄³×C₆₇	C₂²	1	1	yes	characteristic
15	#340×C₆₇	C₂²	1	16	yes	—
16	C₂²×Q₈×C₆₇	—	2	4	no	—
17	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
18	C₄⋊Q₈×C₆₇	—	2	16	no	—
19	C₄⋊Q₈×C₆₇	—	4	8	no	—
20	C₄⋊Q₈×C₆₇	—	4	8	no	—
21	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
22	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
23	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
24	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
25	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
26	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
27	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₆₇	D₄	1	2	yes	—
29	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
32	C₄×C₈×C₆₇	—	2	8	no	—
33	C₄×C₈×C₆₇	Q₈	1	8	yes	complemented
34	C₂×Q₈×C₆₇	—	4	8	no	—
35	C₂×Q₈×C₆₇	—	4	8	no	—
36	C₂×Q₈×C₆₇	—	4	8	no	—
37	C₂²×C₄×C₆₇	—	2	8	no	—
38	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
39	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
40	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
43	C₂×C₈×C₆₇	—	2	8	no	—
44	C₂×C₈×C₆₇	C₂×Q₈	1	8	yes	—
45	C₄⋊C₄×C₆₇	—	2	16	no	—
46	C₄⋊C₄×C₆₇	—	4	8	no	—
47	C₄⋊C₄×C₆₇	—	4	8	no	—
48	C₄⋊C₄×C₆₇	—	4	8	no	—
49	C₄⋊C₄×C₆₇	—	4	8	no	—
50	C₄⋊C₄×C₆₇	—	4	8	no	—
51	C₄²×C₆₇	D₈	1	4	yes	—
52	C₄²×C₆₇	SD₁₆	1	8	yes	—
53	C₄²×C₆₇	Q₁₆	1	4	yes	—
54	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
55	C₄²×C₆₇	#41	1	8	yes	—
56	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
57	Q₈×C₆₇	—	8	16	no	—
58	C₂×C₄×C₆₇	—	4	8	no	—
59	C₂×C₄×C₆₇	—	4	8	no	—
60	C₂×C₄×C₆₇	—	4	8	no	—
61	C₂×C₄×C₆₇	#124	1	4	yes	—
62	C₂×C₄×C₆₇	#124	1	4	yes	—
63	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
64	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
65	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
66	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
67	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
68	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
69	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
70	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
71	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
72	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
73	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
74	C₈×C₆₇	—	2	8	no	—
75	C₂²×C₆₇	C₂×#124	1	2	yes	—
76	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
77	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
78	C₂²×C₆₇	C₂²×SD₁₆	1	2	yes	—
79	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
80	C₄×C₆₇	—	8	8	no	—
81	C₄×C₆₇	#486	1	8	yes	—
82	C₄×C₆₇	#487	1	8	yes	—
83	C₄×C₆₇	#498	1	4	yes	—
84	C₄×C₆₇	#500	1	4	yes	—
85	C₄×C₆₇	#533	1	4	yes	—
86	#33289	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂×C₆₇	#2862	1	1	yes	characteristic
88	C₂×C₆₇	C₂×#486	1	2	yes	—
89	C₂×C₆₇	C₂×#487	1	2	yes	—
90	C₂×C₆₇	C₂×#498	1	1	yes	characteristic
91	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
92	#2868	C₂×C₆₇	1	2	yes	—
93	#1990	C₂×C₆₇	1	8	yes	—
94	#1834	C₂×C₆₇	1	4	yes	—
95	C₄²×C₈	C₂×C₆₇	1	1	yes	characteristic
96	C₆₇	#33289	1	1	yes	characteristic, direct product factor, Sylow
97	C₂×C₄⋊Q₈	—	2	2	no	—
98	C₂×C₄⋊Q₈	—	2	2	no	—
99	C₂×C₄⋊Q₈	C₂²×C₆₇	1	4	yes	—
100	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	—
101	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
102	#384	—	2	2	no	—
103	#384	—	2	2	no	—
104	#384	C₂²×C₆₇	1	4	yes	—
105	#384	C₂²×C₆₇	1	4	yes	—
106	C₄³	C₂²×C₆₇	1	1	yes	characteristic
107	#340	C₂²×C₆₇	1	16	yes	—
108	C₂²×Q₈	—	2	4	no	—
109	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
110	C₄⋊Q₈	—	2	16	no	—
111	C₄⋊Q₈	—	4	8	no	—
112	C₄⋊Q₈	—	4	8	no	—
113	C₂×C₄⋊C₄	—	2	4	no	—
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₄⋊C₄	—	2	4	no	—
116	C₂×C₄⋊C₄	—	2	4	no	—
117	C₂×C₄⋊C₄	—	2	4	no	—
118	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
119	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
120	C₂×C₄²	D₄×C₆₇	1	2	yes	—
121	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
122	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
123	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
124	C₄×C₈	—	2	8	no	—
125	C₄×C₈	Q₈×C₆₇	1	8	yes	complemented
126	C₂×Q₈	—	4	8	no	—
127	C₂×Q₈	—	4	8	no	—
128	C₂×Q₈	—	4	8	no	—
129	C₂²×C₄	—	2	8	no	—
130	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
131	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
132	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
133	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
134	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
135	C₂×C₈	—	2	8	no	—
136	C₂×C₈	C₂×Q₈×C₆₇	1	8	yes	—
137	C₄⋊C₄	—	2	16	no	—
138	C₄⋊C₄	—	4	8	no	—
139	C₄⋊C₄	—	4	8	no	—
140	C₄⋊C₄	—	4	8	no	—
141	C₄⋊C₄	—	4	8	no	—
142	C₄⋊C₄	—	4	8	no	—
143	C₄²	D₈×C₆₇	1	4	yes	—
144	C₄²	SD₁₆×C₆₇	1	8	yes	—
145	C₄²	Q₁₆×C₆₇	1	4	yes	—
146	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
147	C₄²	#41×C₆₇	1	8	yes	—
148	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
149	Q₈	—	8	16	no	—
150	C₂×C₄	—	4	8	no	—
151	C₂×C₄	—	4	8	no	—
152	C₂×C₄	—	4	8	no	—
153	C₂×C₄	#124×C₆₇	1	4	yes	—
154	C₂×C₄	#124×C₆₇	1	4	yes	—
155	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
156	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
157	C₂×C₄	C₂×D₈×C₆₇	1	4	yes	—
158	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
159	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
160	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
161	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
162	C₂×C₄	C₂×Q₁₆×C₆₇	1	4	yes	—
163	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
164	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
165	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
166	C₈	—	2	8	no	—
167	C₂²	C₂×#124×C₆₇	1	2	yes	—
168	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
169	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
170	C₂²	C₂²×SD₁₆×C₆₇	1	2	yes	—
171	C₂²	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
172	C₄	—	8	8	no	—
173	C₄	#486×C₆₇	1	8	yes	—
174	C₄	#487×C₆₇	1	8	yes	—
175	C₄	#498×C₆₇	1	4	yes	—
176	C₄	#500×C₆₇	1	4	yes	—
177	C₄	#533×C₆₇	1	4	yes	—
178	C₂	C₆₇×#2862	1	1	yes	characteristic
179	C₂	C₂×#486×C₆₇	1	2	yes	—
180	C₂	C₂×#487×C₆₇	1	2	yes	—
181	C₂	C₂×#498×C₆₇	1	1	yes	characteristic
182	C₂	C₂×#500×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
33. C₄×C₈×C₆₇	Q₈	semidirect
86. #33289	C₆₇	direct (N × H)
96. C₆₇	#33289	direct (N × H)
125. 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×#2868	2
C₆₇×#1990	2
C₆₇×#1834	2
C₄²×C₈×C₆₇	2
#33289	67

Subgroups — C67×#33289 #30712540

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33289 #30712540