Subgroups — C₆₇×#33475 #30712726

Order 17152.

490 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2183	C₂	1	8	yes	maximal, complemented
3	C₆₇×#2151	C₂	1	2	yes	maximal
4	C₆₇×#2150	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2108	C₂	1	2	yes	maximal
6	C₂×#346×C₆₇	C₂	1	1	yes	maximal, characteristic
7	#533×C₆₇	C₄	1	1	yes	characteristic
8	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
9	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	#446×C₆₇	C₄	1	4	yes	—
12	C₈×Q₈×C₆₇	C₂²	1	4	yes	—
13	#424×C₆₇	C₄	1	2	yes	—
14	#423×C₆₇	C₄	1	1	yes	characteristic
15	#405×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₁₆×C₆₇	C₂²	1	4	yes	—
18	C₄⋊C₁₆×C₆₇	C₂²	1	8	yes	—
19	#346×C₆₇	C₂²	1	4	yes	—
20	#346×C₆₇	C₂²	1	4	yes	—
21	C₄×C₁₆×C₆₇	C₂²	1	4	yes	—
22	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
25	C₄⋊Q₈×C₆₇	C₈	1	2	yes	—
26	#125×C₆₇	C₈	1	2	yes	—
27	C₂²⋊Q₈×C₆₇	C₈	1	4	yes	—
28	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
29	#117×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
31	M₃₂×C₆₇	—	2	4	no	—
32	C₂×C₁₆×C₆₇	—	2	2	no	—
33	C₂×C₁₆×C₆₇	Q₈	1	4	yes	complemented
34	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
35	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
36	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
37	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
38	C₄⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
39	#97×C₆₇	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₆₇	C₂³	1	4	yes	—
42	C₄×C₈×C₆₇	C₂³	1	2	yes	—
43	C₂×Q₈×C₆₇	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	M₁₆×C₆₇	—	2	4	no	—
47	C₂×C₈×C₆₇	—	2	2	no	—
48	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
50	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₆₇	#41	1	2	yes	—
53	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
54	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
55	C₄⋊C₄×C₆₇	C₂×C₈	1	8	yes	—
56	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
57	C₂²⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
58	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
59	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
60	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
61	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
62	C₁₆×C₆₇	—	2	2	no	—
63	C₁₆×C₆₇	—	2	4	no	—
64	C₁₆×C₆₇	C₂×Q₈	1	4	yes	—
65	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
66	Q₈×C₆₇	—	4	4	no	—
67	C₂×C₄×C₆₇	—	4	2	no	—
68	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
69	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
70	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
71	C₂×C₄×C₆₇	C₂²×C₈	1	8	yes	—
72	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₆₇	#131	1	2	yes	—
74	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
75	C₈×C₆₇	—	2	2	no	—
76	C₈×C₆₇	—	2	4	no	—
77	C₈×C₆₇	C₄×Q₈	1	2	yes	—
78	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
79	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
80	C₂²×C₆₇	—	2	2	no	—
81	C₂²×C₆₇	C₈×Q₈	1	2	yes	—
82	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
83	C₄×C₆₇	—	2	2	no	—
84	C₄×C₆₇	—	2	2	no	—
85	C₄×C₆₇	—	2	8	no	—
86	C₄×C₆₇	C₈×Q₈	1	2	yes	—
87	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
88	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
89	#33475	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂×C₆₇	—	2	2	no	—
91	C₂×C₆₇	C₂×C₈×Q₈	1	1	yes	characteristic
92	C₂×C₆₇	#3407	1	2	yes	—
93	#2960	C₂×C₆₇	1	1	yes	characteristic
94	#2183	C₂×C₆₇	1	8	yes	complemented
95	#2151	C₂×C₆₇	1	2	yes	—
96	#2150	C₂×C₆₇	1	1	yes	characteristic
97	#2108	C₂×C₆₇	1	2	yes	—
98	C₂×#346	C₂×C₆₇	1	1	yes	characteristic
99	C₆₇	#33475	1	1	yes	characteristic, direct product factor, Sylow
100	#533	C₄×C₆₇	1	1	yes	characteristic
101	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
102	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
103	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
104	#446	C₄×C₆₇	1	4	yes	—
105	C₈×Q₈	C₂²×C₆₇	1	4	yes	—
106	#424	C₄×C₆₇	1	2	yes	—
107	#423	C₄×C₆₇	1	1	yes	characteristic
108	#405	C₂²×C₆₇	1	2	yes	—
109	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
110	C₄⋊C₁₆	C₂²×C₆₇	1	4	yes	—
111	C₄⋊C₁₆	C₂²×C₆₇	1	8	yes	—
112	#346	C₂²×C₆₇	1	4	yes	—
113	#346	C₂²×C₆₇	1	4	yes	—
114	C₄×C₁₆	C₂²×C₆₇	1	4	yes	—
115	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
116	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
117	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
118	C₄⋊Q₈	C₈×C₆₇	1	2	yes	—
119	#125	C₈×C₆₇	1	2	yes	—
120	C₂²⋊Q₈	C₈×C₆₇	1	4	yes	—
121	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
122	#117	C₂×C₄×C₆₇	1	2	yes	—
123	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
124	M₃₂	—	2	4	no	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	Q₈×C₆₇	1	4	yes	complemented
127	C₂×C₁₆	C₂³×C₆₇	1	4	yes	—
128	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
129	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
130	C₄⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
131	C₄⋊C₈	C₂×C₄×C₆₇	1	8	yes	—
132	#97	C₂×C₄×C₆₇	1	4	yes	—
133	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
134	C₄×C₈	C₂³×C₆₇	1	4	yes	—
135	C₄×C₈	C₂³×C₆₇	1	2	yes	—
136	C₂×Q₈	C₂×C₈×C₆₇	1	4	yes	—
137	C₂²×C₄	C₂×C₈×C₆₇	1	2	yes	—
138	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
139	M₁₆	—	2	4	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
142	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
143	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
144	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
145	C₂×C₈	#41×C₆₇	1	2	yes	—
146	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
147	C₄⋊C₄	C₂×C₈×C₆₇	1	4	yes	—
148	C₄⋊C₄	C₂×C₈×C₆₇	1	8	yes	—
149	C₄⋊C₄	C₂×C₈×C₆₇	1	4	yes	—
150	C₂²⋊C₄	C₂×C₈×C₆₇	1	4	yes	—
151	C₄²	C₂×C₈×C₆₇	1	1	yes	characteristic
152	C₄²	C₂×C₈×C₆₇	1	1	yes	characteristic
153	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
154	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
155	C₁₆	—	2	2	no	—
156	C₁₆	—	2	4	no	—
157	C₁₆	C₂×Q₈×C₆₇	1	4	yes	—
158	C₂³	C₂²×C₈×C₆₇	1	1	yes	characteristic
159	Q₈	—	4	4	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
162	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
163	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
164	C₂×C₄	C₂²×C₈×C₆₇	1	8	yes	—
165	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
166	C₂×C₄	#131×C₆₇	1	2	yes	—
167	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	2	4	no	—
170	C₈	C₄×Q₈×C₆₇	1	2	yes	—
171	C₈	C₂²×Q₈×C₆₇	1	1	yes	characteristic
172	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
173	C₂²	—	2	2	no	—
174	C₂²	C₈×Q₈×C₆₇	1	2	yes	—
175	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	2	8	no	—
179	C₄	C₈×Q₈×C₆₇	1	2	yes	—
180	C₄	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
181	C₄	C₂×#131×C₆₇	1	1	yes	characteristic
182	C₂	—	2	2	no	—
183	C₂	C₂×C₈×Q₈×C₆₇	1	1	yes	characteristic
184	C₂	C₆₇×#3407	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₆₇	C₂⁴
commutator	C₂²	C₂³×C₈×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂³×C₈
prime radical	C₂³×C₆₇	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2183	C₂	semidirect
33. C₂×C₁₆×C₆₇	Q₈	semidirect
89. #33475	C₆₇	direct (N × H)
94. #2183	C₂×C₆₇	semidirect
99. C₆₇	#33475	direct (N × H)
126. C₂×C₁₆	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×#2960	2
C₆₇×#2183	2
C₆₇×#2151	2
C₆₇×#2150	2
C₆₇×#2108	2
C₂×#346×C₆₇	2
#33475	67

Subgroups — C67×#33475 #30712726

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33475 #30712726