Subgroups — C₆₇×#59924 #30739175

Order 17152.

1772 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2811	C₂	1	4	yes	maximal
2	C₆₇×#2567	C₂	1	12	yes	maximal
3	C₆₇×#2551	C₂	1	6	yes	maximal
4	C₆₇×#2369	C₂	1	6	yes	maximal
5	C₄×C₄⋊Q₈×C₆₇	C₂	1	3	yes	maximal
6	C₂×C₄⋊Q₈×C₆₇	C₂²	1	12	yes	—
7	C₂×C₄⋊Q₈×C₆₇	C₂²	1	3	yes	—
8	C₂×C₄×Q₈×C₆₇	C₂²	1	6	yes	—
9	#395×C₆₇	C₂²	1	24	yes	—
10	#395×C₆₇	C₂²	1	8	yes	—
11	#391×C₆₇	C₂²	1	6	yes	—
12	#391×C₆₇	C₂²	1	24	yes	—
13	#389×C₆₇	C₂²	1	24	yes	—
14	#389×C₆₇	C₂²	1	12	yes	—
15	#384×C₆₇	C₂²	1	3	yes	—
16	#384×C₆₇	C₂²	1	8	yes	—
17	C₄×C₄⋊C₄×C₆₇	C₂²	1	12	yes	—
18	C₄×C₄⋊C₄×C₆₇	C₂²	1	12	yes	—
19	C₄³×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
21	C₂²×Q₈×C₆₇	C₂³	1	6	yes	—
22	C₄⋊Q₈×C₆₇	—	4	24	no	—
23	C₄⋊Q₈×C₆₇	Q₈	1	24	yes	complemented
24	C₄×Q₈×C₆₇	—	4	12	no	—
25	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
26	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
27	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
28	C₂×C₄⋊C₄×C₆₇	C₂³	1	24	yes	—
29	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
30	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
31	C₂×C₄²×C₆₇	C₂³	1	12	yes	—
32	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
34	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
35	#95×C₆₇	C₂³	1	24	yes	—
36	#95×C₆₇	C₂³	1	24	yes	—
37	C₂×Q₈×C₆₇	—	4	6	no	—
38	C₂×Q₈×C₆₇	—	4	6	no	—
39	C₂×Q₈×C₆₇	—	4	24	no	—
40	C₂×Q₈×C₆₇	C₂×Q₈	1	24	yes	—
41	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
43	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
44	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
45	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
46	C₄⋊C₄×C₆₇	—	2	24	no	—
47	C₄⋊C₄×C₆₇	—	4	12	no	—
48	C₄⋊C₄×C₆₇	—	4	6	no	—
49	C₄⋊C₄×C₆₇	—	4	24	no	—
50	C₄⋊C₄×C₆₇	C₂×Q₈	1	48	yes	—
51	C₄²×C₆₇	—	2	6	no	—
52	C₄²×C₆₇	—	2	24	no	—
53	C₄²×C₆₇	—	4	8	no	—
54	C₄²×C₆₇	—	4	1	no	—
55	C₄²×C₆₇	C₂×Q₈	1	12	yes	—
56	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
57	Q₈×C₆₇	—	4	24	no	—
58	Q₈×C₆₇	—	8	16	no	—
59	C₂×C₄×C₆₇	—	2	12	no	—
60	C₂×C₄×C₆₇	—	2	3	no	—
61	C₂×C₄×C₆₇	—	2	12	no	—
62	C₂×C₄×C₆₇	—	2	3	no	—
63	C₂×C₄×C₆₇	—	2	3	no	—
64	C₂×C₄×C₆₇	—	2	12	no	—
65	C₂×C₄×C₆₇	—	2	24	no	—
66	C₂×C₄×C₆₇	C₂²×Q₈	1	6	yes	—
67	C₂×C₄×C₆₇	C₂²×Q₈	1	24	yes	—
68	C₂×C₄×C₆₇	C₂²×Q₈	1	12	yes	—
69	C₂×C₄×C₆₇	H(5,𝔽₂)	1	6	yes	—
70	C₂²×C₆₇	C₂³×Q₈	1	3	yes	—
71	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	3	yes	—
72	C₂²×C₆₇	#585	1	1	yes	characteristic
73	C₄×C₆₇	—	2	24	no	—
74	C₄×C₆₇	—	2	6	no	—
75	C₄×C₆₇	—	4	1	no	—
76	C₄×C₆₇	—	4	12	no	—
77	C₄×C₆₇	Q₈²	1	12	yes	—
78	#59924	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂×C₆₇	C₂×Q₈²	1	3	yes	—
80	C₂×C₆₇	#3501	1	3	yes	—
81	C₂×C₆₇	#3549	1	1	yes	characteristic
82	#2811	C₂×C₆₇	1	4	yes	—
83	#2567	C₂×C₆₇	1	12	yes	—
84	#2551	C₂×C₆₇	1	6	yes	—
85	#2369	C₂×C₆₇	1	6	yes	—
86	C₄×C₄⋊Q₈	C₂×C₆₇	1	3	yes	—
87	C₆₇	#59924	1	1	yes	characteristic, direct product factor, Sylow
88	C₂×C₄⋊Q₈	C₂²×C₆₇	1	12	yes	—
89	C₂×C₄⋊Q₈	C₂²×C₆₇	1	3	yes	—
90	C₂×C₄×Q₈	C₂²×C₆₇	1	6	yes	—
91	#395	C₂²×C₆₇	1	8	yes	—
92	#395	C₂²×C₆₇	1	24	yes	—
93	#391	C₂²×C₆₇	1	6	yes	—
94	#391	C₂²×C₆₇	1	24	yes	—
95	#389	C₂²×C₆₇	1	12	yes	—
96	#389	C₂²×C₆₇	1	24	yes	—
97	#384	C₂²×C₆₇	1	3	yes	—
98	#384	C₂²×C₆₇	1	8	yes	—
99	C₄×C₄⋊C₄	C₂²×C₆₇	1	12	yes	—
100	C₄×C₄⋊C₄	C₂²×C₆₇	1	12	yes	—
101	C₄³	C₂²×C₆₇	1	1	yes	characteristic
102	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
103	C₂²×Q₈	C₂³×C₆₇	1	6	yes	—
104	C₄⋊Q₈	—	4	24	no	—
105	C₄⋊Q₈	Q₈×C₆₇	1	24	yes	complemented
106	C₄×Q₈	—	4	12	no	—
107	C₂×C₄⋊C₄	C₂³×C₆₇	1	12	yes	—
108	C₂×C₄⋊C₄	C₂³×C₆₇	1	12	yes	—
109	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
110	C₂×C₄⋊C₄	C₂³×C₆₇	1	24	yes	—
111	C₂×C₄⋊C₄	C₂³×C₆₇	1	12	yes	—
112	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
113	C₂×C₄²	C₂³×C₆₇	1	12	yes	—
114	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
115	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
116	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
117	#95	C₂³×C₆₇	1	24	yes	—
118	#95	C₂³×C₆₇	1	24	yes	—
119	C₂×Q₈	—	4	6	no	—
120	C₂×Q₈	—	4	6	no	—
121	C₂×Q₈	—	4	24	no	—
122	C₂×Q₈	C₂×Q₈×C₆₇	1	24	yes	—
123	C₂²×C₄	C₂⁴×C₆₇	1	12	yes	—
124	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
125	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
126	C₂²×C₄	C₂⁴×C₆₇	1	12	yes	—
127	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
128	C₄⋊C₄	—	2	24	no	—
129	C₄⋊C₄	—	4	12	no	—
130	C₄⋊C₄	—	4	6	no	—
131	C₄⋊C₄	—	4	24	no	—
132	C₄⋊C₄	C₂×Q₈×C₆₇	1	48	yes	—
133	C₄²	—	2	6	no	—
134	C₄²	—	2	24	no	—
135	C₄²	—	4	8	no	—
136	C₄²	—	4	1	no	—
137	C₄²	C₂×Q₈×C₆₇	1	12	yes	—
138	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
139	Q₈	—	4	24	no	—
140	Q₈	—	8	16	no	—
141	C₂×C₄	—	2	12	no	—
142	C₂×C₄	—	2	3	no	—
143	C₂×C₄	—	2	12	no	—
144	C₂×C₄	—	2	3	no	—
145	C₂×C₄	—	2	3	no	—
146	C₂×C₄	—	2	12	no	—
147	C₂×C₄	—	2	24	no	—
148	C₂×C₄	C₂²×Q₈×C₆₇	1	6	yes	—
149	C₂×C₄	C₂²×Q₈×C₆₇	1	24	yes	—
150	C₂×C₄	C₂²×Q₈×C₆₇	1	12	yes	—
151	C₂×C₄	H(5,𝔽₂)×C₆₇	1	6	yes	—
152	C₂²	C₂³×Q₈×C₆₇	1	3	yes	—
153	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	3	yes	—
154	C₂²	#585×C₆₇	1	1	yes	characteristic
155	C₄	—	2	6	no	—
156	C₄	—	2	24	no	—
157	C₄	—	4	12	no	—
158	C₄	—	4	1	no	—
159	C₄	Q₈²×C₆₇	1	12	yes	—
160	C₂	C₂×Q₈²×C₆₇	1	3	yes	—
161	C₂	C₆₇×#3501	1	3	yes	—
162	C₂	C₆₇×#3549	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
23. C₄⋊Q₈×C₆₇	Q₈	semidirect
78. #59924	C₆₇	direct (N × H)
87. C₆₇	#59924	direct (N × H)
105. C₄⋊Q₈	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×#2811	2
C₆₇×#2567	2
C₆₇×#2551	2
C₆₇×#2369	2
C₄×C₄⋊Q₈×C₆₇	2
#59924	67

Subgroups — C67×#59924 #30739175

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#59924 #30739175