Subgroups — C₆₇×#33258 #30712509

Order 17152.

718 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×#2859	C₂	1	2	yes	maximal
2	C₆₇×#1942	C₂	1	8	yes	maximal
3	C₆₇×#1845	C₂	1	2	yes	maximal
4	C₆₇×#1834	C₂	1	2	yes	maximal
5	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#125×C₆₇	—	2	2	no	—
7	C₂×#107×C₆₇	C₂²	1	8	yes	—
8	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
9	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	#389×C₆₇	—	2	4	no	—
12	#389×C₆₇	C₂²	1	8	yes	—
13	#384×C₆₇	C₂²	1	4	yes	—
14	#382×C₆₇	—	2	2	no	—
15	C₄³×C₆₇	C₂²	1	1	yes	characteristic
16	#340×C₆₇	C₂²	1	8	yes	—
17	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
18	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
19	#125×C₆₇	—	4	8	no	—
20	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
21	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
22	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
23	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
24	C₂×C₄²×C₆₇	D₄	1	2	yes	—
25	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
28	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
29	#107×C₆₇	—	2	16	no	—
30	C₄×C₈×C₆₇	—	2	8	no	—
31	C₄×C₈×C₆₇	Q₈	1	4	yes	—
32	C₄×C₈×C₆₇	Q₈	1	4	yes	—
33	#95×C₆₇	—	2	8	no	—
34	C₂²×C₄×C₆₇	—	2	8	no	—
35	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
36	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
37	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
40	C₂×C₈×C₆₇	—	2	4	no	—
41	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
42	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
43	C₂×C₈×C₆₇	#41	1	8	yes	—
44	C₄⋊C₄×C₆₇	—	2	16	no	—
45	C₄⋊C₄×C₆₇	—	4	16	no	—
46	C₄⋊C₄×C₆₇	—	4	8	no	—
47	C₄⋊C₄×C₆₇	—	4	8	no	—
48	C₄²×C₆₇	—	2	4	no	—
49	C₄²×C₆₇	—	2	4	no	—
50	C₄²×C₆₇	D₈	1	4	yes	—
51	C₄²×C₆₇	Q₁₆	1	4	yes	—
52	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
53	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
54	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
55	C₂×C₄×C₆₇	—	2	2	no	—
56	C₂×C₄×C₆₇	—	2	2	no	—
57	C₂×C₄×C₆₇	—	2	2	no	—
58	C₂×C₄×C₆₇	—	2	2	no	—
59	C₂×C₄×C₆₇	—	4	8	no	—
60	C₂×C₄×C₆₇	—	4	8	no	—
61	C₂×C₄×C₆₇	—	4	8	no	—
62	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
63	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
64	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
65	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
66	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
67	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
68	C₂×C₄×C₆₇	#135	1	4	yes	—
69	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
72	C₈×C₆₇	—	2	4	no	—
73	C₈×C₆₇	#125	1	8	yes	complemented
74	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
75	C₂²×C₆₇	#532	1	2	yes	—
76	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
77	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
78	C₂²×C₆₇	C₂×#135	1	2	yes	—
79	C₄×C₆₇	—	2	4	no	—
80	C₄×C₆₇	—	2	4	no	—
81	C₄×C₆₇	—	8	8	no	—
82	C₄×C₆₇	#499	1	4	yes	—
83	C₄×C₆₇	#500	1	4	yes	—
84	C₄×C₆₇	C₂×#125	1	2	yes	—
85	C₄×C₆₇	#533	1	2	yes	—
86	#33258	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂×C₆₇	#2859	1	1	yes	characteristic
88	C₂×C₆₇	C₂×#499	1	1	yes	characteristic
89	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
90	C₂×C₆₇	#3169	1	2	yes	—
91	C₂×C₆₇	#3170	1	2	yes	—
92	#2859	C₂×C₆₇	1	2	yes	—
93	#1942	C₂×C₆₇	1	8	yes	—
94	#1845	C₂×C₆₇	1	2	yes	—
95	#1834	C₂×C₆₇	1	2	yes	—
96	C₄²×C₈	C₂×C₆₇	1	1	yes	characteristic
97	C₆₇	#33258	1	1	yes	characteristic, direct product factor, Sylow
98	C₂×#125	—	2	2	no	—
99	C₂×#107	C₂²×C₆₇	1	8	yes	—
100	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	—
101	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
102	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
103	#389	—	2	4	no	—
104	#389	C₂²×C₆₇	1	8	yes	—
105	#384	C₂²×C₆₇	1	4	yes	—
106	#382	—	2	2	no	—
107	C₄³	C₂²×C₆₇	1	1	yes	characteristic
108	#340	C₂²×C₆₇	1	8	yes	—
109	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
110	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
111	#125	—	4	8	no	—
112	C₂×C₄⋊C₄	—	2	4	no	—
113	C₂×C₄⋊C₄	—	2	8	no	—
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
116	C₂×C₄²	D₄×C₆₇	1	2	yes	—
117	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
118	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
119	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
120	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
121	#107	—	2	16	no	—
122	C₄×C₈	—	2	8	no	—
123	C₄×C₈	Q₈×C₆₇	1	4	yes	—
124	C₄×C₈	Q₈×C₆₇	1	4	yes	—
125	#95	—	2	8	no	—
126	C₂²×C₄	—	2	8	no	—
127	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
128	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
129	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
130	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
131	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
134	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
135	C₂×C₈	#41×C₆₇	1	8	yes	—
136	C₄⋊C₄	—	2	16	no	—
137	C₄⋊C₄	—	4	16	no	—
138	C₄⋊C₄	—	4	8	no	—
139	C₄⋊C₄	—	4	8	no	—
140	C₄²	—	2	4	no	—
141	C₄²	—	2	4	no	—
142	C₄²	D₈×C₆₇	1	4	yes	—
143	C₄²	Q₁₆×C₆₇	1	4	yes	—
144	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
145	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
146	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	8	no	—
152	C₂×C₄	—	4	8	no	—
153	C₂×C₄	—	4	8	no	—
154	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
155	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
156	C₂×C₄	C₂×D₈×C₆₇	1	4	yes	—
157	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
158	C₂×C₄	C₂×Q₁₆×C₆₇	1	4	yes	—
159	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
160	C₂×C₄	#135×C₆₇	1	4	yes	—
161	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
162	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
163	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
164	C₈	—	2	4	no	—
165	C₈	#125×C₆₇	1	8	yes	complemented
166	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
167	C₂²	#532×C₆₇	1	2	yes	—
168	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
169	C₂²	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
170	C₂²	C₂×#135×C₆₇	1	2	yes	—
171	C₄	—	2	4	no	—
172	C₄	—	2	4	no	—
173	C₄	—	8	8	no	—
174	C₄	#499×C₆₇	1	4	yes	—
175	C₄	#500×C₆₇	1	4	yes	—
176	C₄	C₂×#125×C₆₇	1	2	yes	—
177	C₄	#533×C₆₇	1	2	yes	—
178	C₂	C₆₇×#2859	1	1	yes	characteristic
179	C₂	C₂×#499×C₆₇	1	1	yes	characteristic
180	C₂	C₂×#500×C₆₇	1	1	yes	characteristic
181	C₂	C₆₇×#3169	1	2	yes	—
182	C₂	C₆₇×#3170	1	2	yes	—

	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
73. C₈×C₆₇	#125	semidirect
86. #33258	C₆₇	direct (N × H)
97. C₆₇	#33258	direct (N × H)
165. C₈	#125×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2859	2
C₆₇×#1942	2
C₆₇×#1845	2
C₆₇×#1834	2
C₄²×C₈×C₆₇	2
#33258	67

Subgroups — C67×#33258 #30712509

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33258 #30712509