Subgroups — C₆₇×#21737 #30700988

Order 17152.

662 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3162	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2960	C₂	1	2	yes	maximal
3	C₆₇×#1995	C₂	1	2	yes	maximal
4	C₆₇×#1945	C₂	1	4	yes	maximal
5	C₆₇×#1849	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1773	C₂	1	4	yes	maximal
7	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic
8	#533×C₆₇	C₂²	1	2	yes	—
9	#501×C₆₇	C₄	1	8	yes	complemented
10	#446×C₆₇	—	2	4	no	—
11	C₈×Q₈×C₆₇	—	2	4	no	—
12	C₂×#108×C₆₇	C₂²	1	4	yes	—
13	#427×C₆₇	C₂²	1	4	yes	—
14	#424×C₆₇	C₂²	1	4	yes	—
15	#423×C₆₇	C₂²	1	2	yes	—
16	#405×C₆₇	C₂²	1	4	yes	—
17	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#97×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	#376×C₆₇	C₂²	1	1	yes	characteristic
22	#339×C₆₇	C₂²	1	8	yes	—
23	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
24	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
26	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
27	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
28	#125×C₆₇	C₂×C₄	1	4	yes	—
29	C₂²⋊Q₈×C₆₇	—	2	4	no	—
30	C₄×Q₈×C₆₇	—	2	4	no	—
31	#117×C₆₇	C₂³	1	4	yes	—
32	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
33	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
34	#108×C₆₇	—	2	8	no	—
35	#107×C₆₇	C₂×C₄	1	8	yes	—
36	#106×C₆₇	C₂×C₄	1	8	yes	—
37	C₄⋊C₈×C₆₇	—	2	8	no	—
38	C₄⋊C₈×C₆₇	—	2	4	no	—
39	#97×C₆₇	—	2	4	no	—
40	#97×C₆₇	C₂×C₄	1	4	yes	—
41	#97×C₆₇	Q₈	1	8	yes	complemented
42	C₄×C₈×C₆₇	—	2	4	no	—
43	C₄×C₈×C₆₇	—	2	4	no	—
44	C₄×C₈×C₆₇	D₄	1	4	yes	—
45	#95×C₆₇	—	2	2	no	—
46	C₂×Q₈×C₆₇	—	2	4	no	—
47	C₂²×C₄×C₆₇	—	2	2	no	—
48	C₂²×C₄×C₆₇	#41	1	2	yes	—
49	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
50	M₁₆×C₆₇	—	2	8	no	—
51	C₂×C₈×C₆₇	—	2	2	no	—
52	C₂×C₈×C₆₇	—	2	4	no	—
53	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
54	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
57	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
58	C₂×C₈×C₆₇	#41	1	4	yes	—
59	C₄⋊C₄×C₆₇	—	2	8	no	—
60	C₄⋊C₄×C₆₇	—	2	4	no	—
61	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
62	C₂²⋊C₄×C₆₇	—	2	4	no	—
63	C₄²×C₆₇	—	2	4	no	—
64	C₄²×C₆₇	—	2	2	no	—
65	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
66	C₄²×C₆₇	C₂×D₄	1	2	yes	—
67	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
68	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
69	Q₈×C₆₇	—	4	8	no	—
70	C₂×C₄×C₆₇	—	2	8	no	—
71	C₂×C₄×C₆₇	—	2	4	no	—
72	C₂×C₄×C₆₇	—	4	2	no	—
73	C₂×C₄×C₆₇	—	4	4	no	—
74	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
75	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
78	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
79	C₈×C₆₇	—	2	2	no	—
80	C₈×C₆₇	—	4	4	no	—
81	C₈×C₆₇	C₄×Q₈	1	8	yes	—
82	C₈×C₆₇	C₄⋊Q₈	1	4	yes	—
83	C₂²×C₆₇	—	2	2	no	—
84	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
85	C₂²×C₆₇	#533	1	2	yes	—
86	C₄×C₆₇	—	2	4	no	—
87	C₄×C₆₇	—	4	4	no	—
88	C₄×C₆₇	—	4	8	no	—
89	C₄×C₆₇	C₂×C₄×Q₈	1	2	yes	—
90	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
91	C₄×C₆₇	#532	1	1	yes	characteristic
92	#21737	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂×C₆₇	—	2	2	no	—
94	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
95	C₂×C₆₇	#2955	1	2	yes	—
96	#3162	C₂×C₆₇	1	1	yes	characteristic
97	#2960	C₂×C₆₇	1	2	yes	—
98	#1995	C₂×C₆₇	1	2	yes	—
99	#1945	C₂×C₆₇	1	4	yes	—
100	#1849	C₂×C₆₇	1	1	yes	characteristic
101	#1773	C₂×C₆₇	1	4	yes	—
102	#1726	C₂×C₆₇	1	1	yes	characteristic
103	C₆₇	#21737	1	1	yes	characteristic, direct product factor, Sylow
104	#533	C₂²×C₆₇	1	2	yes	—
105	#501	C₄×C₆₇	1	8	yes	complemented
106	#446	—	2	4	no	—
107	C₈×Q₈	—	2	4	no	—
108	C₂×#108	C₂²×C₆₇	1	4	yes	—
109	#427	C₂²×C₆₇	1	4	yes	—
110	#424	C₂²×C₆₇	1	4	yes	—
111	#423	C₂²×C₆₇	1	2	yes	—
112	#405	C₂²×C₆₇	1	4	yes	—
113	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
114	C₂×#97	C₂²×C₆₇	1	2	yes	—
115	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
116	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
117	#376	C₂²×C₆₇	1	1	yes	characteristic
118	#339	C₂²×C₆₇	1	8	yes	—
119	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
120	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
121	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
122	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
123	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
124	#125	C₂×C₄×C₆₇	1	4	yes	—
125	C₂²⋊Q₈	—	2	4	no	—
126	C₄×Q₈	—	2	4	no	—
127	#117	C₂³×C₆₇	1	4	yes	—
128	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
129	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
130	#108	—	2	8	no	—
131	#107	C₂×C₄×C₆₇	1	8	yes	—
132	#106	C₂×C₄×C₆₇	1	8	yes	—
133	C₄⋊C₈	—	2	8	no	—
134	C₄⋊C₈	—	2	4	no	—
135	#97	—	2	4	no	—
136	#97	C₂×C₄×C₆₇	1	4	yes	—
137	#97	Q₈×C₆₇	1	8	yes	complemented
138	C₄×C₈	—	2	4	no	—
139	C₄×C₈	—	2	4	no	—
140	C₄×C₈	D₄×C₆₇	1	4	yes	—
141	#95	—	2	2	no	—
142	C₂×Q₈	—	2	4	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	#41×C₆₇	1	2	yes	—
145	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
146	M₁₆	—	2	8	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
150	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
151	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
152	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
153	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
154	C₂×C₈	#41×C₆₇	1	4	yes	—
155	C₄⋊C₄	—	2	8	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
158	C₂²⋊C₄	—	2	4	no	—
159	C₄²	—	2	4	no	—
160	C₄²	—	2	2	no	—
161	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
162	C₄²	C₂×D₄×C₆₇	1	2	yes	—
163	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
164	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
165	Q₈	—	4	8	no	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
171	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
172	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
173	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
174	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
175	C₈	—	2	2	no	—
176	C₈	—	4	4	no	—
177	C₈	C₄×Q₈×C₆₇	1	8	yes	—
178	C₈	C₄⋊Q₈×C₆₇	1	4	yes	—
179	C₂²	—	2	2	no	—
180	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
181	C₂²	#533×C₆₇	1	2	yes	—
182	C₄	—	2	4	no	—
183	C₄	—	4	4	no	—
184	C₄	—	4	8	no	—
185	C₄	C₂×C₄×Q₈×C₆₇	1	2	yes	—
186	C₄	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
187	C₄	#532×C₆₇	1	1	yes	characteristic
188	C₂	—	2	2	no	—
189	C₂	C₄×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
190	C₂	C₆₇×#2955	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
9. #501×C₆₇	C₄	semidirect
41. #97×C₆₇	Q₈	semidirect
92. #21737	C₆₇	direct (N × H)
103. C₆₇	#21737	direct (N × H)
105. #501	C₄×C₆₇	semidirect
137. #97	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×#3162	2
C₆₇×#2960	2
C₆₇×#1995	2
C₆₇×#1945	2
C₆₇×#1849	2
C₆₇×#1773	2
C₆₇×#1726	2
#21737	67

Subgroups — C67×#21737 #30700988

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21737 #30700988