Subgroups — C₆₇×#33999 #30713250

Order 17152.

3598 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₆₇	C₂	1	3	yes	maximal
2	C₆₇×#2276	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2275	C₂	1	12	yes	maximal, complemented
4	C₄×#117×C₆₇	C₂	1	9	yes	maximal, complemented
5	C₂×#376×C₆₇	C₂	1	3	yes	maximal, complemented
6	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
7	C₄×#41×C₆₇	C₄	1	48	yes	complemented
8	C₂×C₄×Q₈×C₆₇	C₂²	1	6	yes	—
9	C₂×C₄×D₄×C₆₇	C₂²	1	18	yes	—
10	C₂×#117×C₆₇	C₂²	1	9	yes	—
11	C₂²×C₄²×C₆₇	C₂²	1	9	yes	—
12	C₄×C₄⋊C₄×C₆₇	C₂²	1	36	yes	—
13	C₄×C₂²⋊C₄×C₆₇	C₂²	1	36	yes	—
14	#376×C₆₇	C₂²	1	1	yes	characteristic
15	#376×C₆₇	C₂²	1	9	yes	—
16	#376×C₆₇	C₂²	1	12	yes	—
17	C₂×#95×C₆₇	C₂²	1	12	yes	—
18	C₄³×C₆₇	C₂²	1	6	yes	—
19	C₂×#41×C₆₇	C₂×C₄	1	24	yes	—
20	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
21	C₂²×D₄×C₆₇	C₂³	1	3	yes	—
22	C₂³×C₄×C₆₇	C₂³	1	18	yes	—
23	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
24	C₄×Q₈×C₆₇	C₂×C₄	1	48	yes	complemented
25	C₄×D₄×C₆₇	C₂×C₄	1	144	yes	complemented
26	#117×C₆₇	C₂×C₄	1	72	yes	complemented
27	C₂×C₄⋊C₄×C₆₇	C₂³	1	18	yes	—
28	C₂×C₂²⋊C₄×C₆₇	C₂³	1	18	yes	—
29	C₂×C₄²×C₆₇	C₂×C₄	1	72	yes	complemented
30	C₂×C₄²×C₆₇	C₂³	1	24	yes	—
31	C₂×C₄²×C₆₇	C₂³	1	18	yes	—
32	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
33	C₂×C₄²×C₆₇	C₂³	1	9	yes	—
34	#95×C₆₇	C₂³	1	4	yes	—
35	#95×C₆₇	C₂³	1	36	yes	—
36	C₂⁴×C₆₇	C₂⁴	1	3	yes	—
37	#41×C₆₇	C₄²	1	64	yes	complemented
38	C₂×Q₈×C₆₇	C₂²×C₄	1	12	yes	—
39	C₂×D₄×C₆₇	C₂²×C₄	1	36	yes	—
40	C₂²×C₄×C₆₇	—	4	36	no	—
41	C₂²×C₄×C₆₇	C₂²×C₄	1	72	yes	—
42	C₂²×C₄×C₆₇	C₂²×C₄	1	36	yes	—
43	C₂²×C₄×C₆₇	C₂⁴	1	18	yes	—
44	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
45	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
46	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
47	C₄⋊C₄×C₆₇	C₂²×C₄	1	72	yes	—
48	C₂²⋊C₄×C₆₇	C₂²×C₄	1	72	yes	—
49	C₄²×C₆₇	—	4	24	no	—
50	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
51	C₄²×C₆₇	C₂²×C₄	1	36	yes	—
52	C₄²×C₆₇	C₂²×C₄	1	72	yes	—
53	C₂³×C₆₇	—	2	12	no	—
54	C₂³×C₆₇	C₂³×C₄	1	18	yes	—
55	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
56	Q₈×C₆₇	C₂×C₄²	1	16	yes	—
57	D₄×C₆₇	C₂×C₄²	1	48	yes	—
58	C₂×C₄×C₆₇	—	2	36	no	—
59	C₂×C₄×C₆₇	—	2	12	no	—
60	C₂×C₄×C₆₇	—	4	72	no	—
61	C₂×C₄×C₆₇	C₂×C₄²	1	48	yes	—
62	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
63	C₂×C₄×C₆₇	C₂³×C₄	1	36	yes	—
64	C₂×C₄×C₆₇	C₂³×C₄	1	6	yes	—
65	C₂×C₄×C₆₇	C₂³×C₄	1	18	yes	—
66	C₂²×C₆₇	—	2	36	no	—
67	C₂²×C₆₇	C₂²×C₄²	1	12	yes	—
68	C₂²×C₆₇	C₂⁴×C₄	1	3	yes	—
69	C₂²×C₆₇	#585	1	4	yes	—
70	C₄×C₆₇	—	2	12	no	—
71	C₄×C₆₇	—	2	36	no	—
72	C₄×C₆₇	C₂²×C₄²	1	4	yes	—
73	C₄×C₆₇	C₂²×C₄²	1	12	yes	—
74	#33999	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂×C₆₇	—	2	12	no	—
76	C₂×C₆₇	C₂³×C₄²	1	1	yes	characteristic
77	C₂×C₆₇	#3428	1	6	yes	—
78	C₂×C₄×#41	C₂×C₆₇	1	3	yes	—
79	#2276	C₂×C₆₇	1	4	yes	complemented
80	#2275	C₂×C₆₇	1	12	yes	complemented
81	C₄×#117	C₂×C₆₇	1	9	yes	complemented
82	C₂×#376	C₂×C₆₇	1	3	yes	complemented
83	C₆₇	#33999	1	1	yes	characteristic, direct product factor, Sylow
84	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
85	C₄×#41	C₄×C₆₇	1	48	yes	complemented
86	C₂×C₄×Q₈	C₂²×C₆₇	1	6	yes	—
87	C₂×C₄×D₄	C₂²×C₆₇	1	18	yes	—
88	C₂×#117	C₂²×C₆₇	1	9	yes	—
89	C₂²×C₄²	C₂²×C₆₇	1	9	yes	—
90	C₄×C₄⋊C₄	C₂²×C₆₇	1	36	yes	—
91	C₄×C₂²⋊C₄	C₂²×C₆₇	1	36	yes	—
92	#376	C₂²×C₆₇	1	1	yes	characteristic
93	#376	C₂²×C₆₇	1	9	yes	—
94	#376	C₂²×C₆₇	1	12	yes	—
95	C₂×#95	C₂²×C₆₇	1	12	yes	—
96	C₄³	C₂²×C₆₇	1	6	yes	—
97	C₂×#41	C₂×C₄×C₆₇	1	24	yes	—
98	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
99	C₂²×D₄	C₂³×C₆₇	1	3	yes	—
100	C₂³×C₄	C₂³×C₆₇	1	18	yes	—
101	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
102	C₄×Q₈	C₂×C₄×C₆₇	1	48	yes	complemented
103	C₄×D₄	C₂×C₄×C₆₇	1	144	yes	complemented
104	#117	C₂×C₄×C₆₇	1	72	yes	complemented
105	C₂×C₄⋊C₄	C₂³×C₆₇	1	18	yes	—
106	C₂×C₂²⋊C₄	C₂³×C₆₇	1	18	yes	—
107	C₂×C₄²	C₂×C₄×C₆₇	1	72	yes	complemented
108	C₂×C₄²	C₂³×C₆₇	1	24	yes	—
109	C₂×C₄²	C₂³×C₆₇	1	9	yes	—
110	C₂×C₄²	C₂³×C₆₇	1	18	yes	—
111	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
112	#95	C₂³×C₆₇	1	4	yes	—
113	#95	C₂³×C₆₇	1	36	yes	—
114	C₂⁴	C₂⁴×C₆₇	1	3	yes	—
115	#41	C₄²×C₆₇	1	64	yes	complemented
116	C₂×Q₈	C₂²×C₄×C₆₇	1	12	yes	—
117	C₂×D₄	C₂²×C₄×C₆₇	1	36	yes	—
118	C₂²×C₄	—	4	36	no	—
119	C₂²×C₄	C₂²×C₄×C₆₇	1	72	yes	—
120	C₂²×C₄	C₂²×C₄×C₆₇	1	36	yes	—
121	C₂²×C₄	C₂⁴×C₆₇	1	6	yes	—
122	C₂²×C₄	C₂⁴×C₆₇	1	18	yes	—
123	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
124	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
125	C₄⋊C₄	C₂²×C₄×C₆₇	1	72	yes	—
126	C₂²⋊C₄	C₂²×C₄×C₆₇	1	72	yes	—
127	C₄²	—	4	24	no	—
128	C₄²	C₂²×C₄×C₆₇	1	12	yes	—
129	C₄²	C₂²×C₄×C₆₇	1	36	yes	—
130	C₄²	C₂²×C₄×C₆₇	1	72	yes	—
131	C₂³	—	2	12	no	—
132	C₂³	C₂³×C₄×C₆₇	1	18	yes	—
133	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
134	Q₈	C₂×C₄²×C₆₇	1	16	yes	—
135	D₄	C₂×C₄²×C₆₇	1	48	yes	—
136	C₂×C₄	—	2	36	no	—
137	C₂×C₄	—	2	12	no	—
138	C₂×C₄	—	4	72	no	—
139	C₂×C₄	C₂×C₄²×C₆₇	1	48	yes	—
140	C₂×C₄	C₂³×C₄×C₆₇	1	12	yes	—
141	C₂×C₄	C₂³×C₄×C₆₇	1	36	yes	—
142	C₂×C₄	C₂³×C₄×C₆₇	1	6	yes	—
143	C₂×C₄	C₂³×C₄×C₆₇	1	18	yes	—
144	C₂²	—	2	36	no	—
145	C₂²	C₂²×C₄²×C₆₇	1	12	yes	—
146	C₂²	C₂⁴×C₄×C₆₇	1	3	yes	—
147	C₂²	#585×C₆₇	1	4	yes	—
148	C₄	—	2	12	no	—
149	C₄	—	2	36	no	—
150	C₄	C₂²×C₄²×C₆₇	1	4	yes	—
151	C₄	C₂²×C₄²×C₆₇	1	12	yes	—
152	C₂	—	2	12	no	—
153	C₂	C₂³×C₄²×C₆₇	1	1	yes	characteristic
154	C₂	C₆₇×#3428	1	6	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂	C₂³×C₄²×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂⁵
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2276	C₂	semidirect
3. C₆₇×#2275	C₂	semidirect
4. C₄×#117×C₆₇	C₂	semidirect
5. C₂×#376×C₆₇	C₂	semidirect
7. C₄×#41×C₆₇	C₄	semidirect
24. C₄×Q₈×C₆₇	C₂×C₄	semidirect
25. C₄×D₄×C₆₇	C₂×C₄	semidirect
26. #117×C₆₇	C₂×C₄	semidirect
29. C₂×C₄²×C₆₇	C₂×C₄	semidirect
37. #41×C₆₇	C₄²	semidirect
74. #33999	C₆₇	direct (N × H)
79. #2276	C₂×C₆₇	semidirect
80. #2275	C₂×C₆₇	semidirect
81. C₄×#117	C₂×C₆₇	semidirect
82. C₂×#376	C₂×C₆₇	semidirect
83. C₆₇	#33999	direct (N × H)
85. C₄×#41	C₄×C₆₇	semidirect
102. C₄×Q₈	C₂×C₄×C₆₇	semidirect
103. C₄×D₄	C₂×C₄×C₆₇	semidirect
104. #117	C₂×C₄×C₆₇	semidirect
107. C₂×C₄²	C₂×C₄×C₆₇	semidirect
115. #41	C₄²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₆₇	2
C₆₇×#2276	2
C₆₇×#2275	2
C₄×#117×C₆₇	2
C₂×#376×C₆₇	2
#33999	67

Subgroups — C67×#33999 #30713250

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33999 #30713250