Subgroups — C₂×C₆₇×#2241 #30712946

Order 17152.

622 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#494×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2241	C₂	1	8	yes	maximal, direct product factor
4	C₂×#358×C₆₇	C₂	1	4	yes	maximal
5	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₆₇	—	2	1	no	—
7	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
8	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
9	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
10	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
11	#500×C₆₇	—	2	2	no	—
12	#500×C₆₇	C₂²	1	4	yes	—
13	#494×C₆₇	—	2	2	no	—
14	#494×C₆₇	C₂²	1	4	yes	—
15	C₂×#107×C₆₇	—	2	1	no	—
16	C₂×#107×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#358×C₆₇	C₂²	1	16	yes	—
19	C₄×C₁₆×C₆₇	C₂²	1	4	yes	—
20	C₂²×Q₈×C₆₇	—	2	2	no	—
21	C₂²×Q₈×C₆₇	—	4	1	no	—
22	C₂×Q₁₆×C₆₇	—	2	2	no	—
23	C₂×Q₁₆×C₆₇	—	2	4	no	—
24	C₂×Q₁₆×C₆₇	—	2	8	no	—
25	C₂×Q₁₆×C₆₇	—	4	4	no	—
26	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
27	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈×C₆₇	—	2	4	no	—
30	C₄⋊Q₈×C₆₇	—	2	4	no	—
31	C₄⋊Q₈×C₆₇	—	4	2	no	—
32	C₄⋊Q₈×C₆₇	—	4	2	no	—
33	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
35	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
36	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
37	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
38	C₂×C₁₆×C₆₇	—	2	4	no	—
39	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
40	#107×C₆₇	—	2	2	no	—
41	#107×C₆₇	—	2	4	no	—
42	#107×C₆₇	C₂³	1	4	yes	—
43	C₄×C₈×C₆₇	D₄	1	2	yes	—
44	C₄×C₈×C₆₇	C₂³	1	2	yes	—
45	C₂×Q₈×C₆₇	—	2	4	no	—
46	C₂×Q₈×C₆₇	—	4	4	no	—
47	C₂×Q₈×C₆₇	—	4	2	no	—
48	C₂×Q₈×C₆₇	—	4	4	no	—
49	C₂×Q₈×C₆₇	—	4	4	no	—
50	C₂×Q₈×C₆₇	—	8	4	no	—
51	C₂²×C₄×C₆₇	—	4	2	no	—
52	C₂²×C₄×C₆₇	—	4	2	no	—
53	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	Q₁₆×C₆₇	—	2	8	no	—
57	Q₁₆×C₆₇	—	4	4	no	—
58	Q₁₆×C₆₇	—	4	8	no	—
59	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
60	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₆₇	#41	1	4	yes	—
63	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
64	C₄⋊C₄×C₆₇	—	2	4	no	—
65	C₄⋊C₄×C₆₇	—	4	2	no	—
66	C₄⋊C₄×C₆₇	—	4	4	no	—
67	C₄⋊C₄×C₆₇	—	4	4	no	—
68	C₄⋊C₄×C₆₇	—	4	4	no	—
69	C₄²×C₆₇	D₈	1	2	yes	—
70	C₄²×C₆₇	C₂×D₄	1	2	yes	—
71	C₁₆×C₆₇	—	2	4	no	—
72	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
73	Q₈×C₆₇	—	4	4	no	—
74	Q₈×C₆₇	—	4	4	no	—
75	Q₈×C₆₇	—	8	4	no	—
76	Q₈×C₆₇	—	8	8	no	—
77	C₂×C₄×C₆₇	—	4	4	no	—
78	C₂×C₄×C₆₇	—	4	4	no	—
79	C₂×C₄×C₆₇	—	8	4	no	—
80	C₂×C₄×C₆₇	—	8	4	no	—
81	C₂×C₄×C₆₇	SD₃₂	1	4	yes	—
82	C₂×C₄×C₆₇	Q₃₂	1	4	yes	—
83	C₂×C₄×C₆₇	#124	1	2	yes	—
84	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
86	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
87	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
88	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
89	C₈×C₆₇	#124	1	2	yes	—
90	C₈×C₆₇	C₂×SD₁₆	1	4	yes	—
91	C₈×C₆₇	C₂×#41	1	2	yes	—
92	C₂²×C₆₇	#486	1	2	yes	—
93	C₂²×C₆₇	C₂×SD₃₂	1	2	yes	—
94	C₂²×C₆₇	C₂×Q₃₂	1	2	yes	—
95	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
96	C₄×C₆₇	—	8	4	no	—
97	C₄×C₆₇	—	8	4	no	—
98	C₄×C₆₇	#486	1	2	yes	—
99	C₄×C₆₇	C₂×SD₃₂	1	4	yes	—
100	C₄×C₆₇	C₂×Q₃₂	1	4	yes	—
101	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
102	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
103	C₂×#2241	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂×C₆₇	#2241	1	4	yes	direct product factor
105	C₂×C₆₇	C₂×#486	1	1	yes	characteristic
106	C₂×C₆₇	C₂²×SD₃₂	1	1	yes	characteristic
107	C₂×C₆₇	C₂²×Q₃₂	1	1	yes	characteristic
108	C₂×#500	C₂×C₆₇	1	1	yes	characteristic
109	C₂×#494	C₂×C₆₇	1	1	yes	characteristic
110	#2241	C₂×C₆₇	1	8	yes	direct product factor
111	C₂×#358	C₂×C₆₇	1	4	yes	—
112	C₂×C₄×C₁₆	C₂×C₆₇	1	1	yes	characteristic
113	C₆₇	C₂×#2241	1	1	yes	characteristic, direct product factor, Sylow
114	C₂²×Q₁₆	—	2	1	no	—
115	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
116	C₂×C₄⋊Q₈	—	2	1	no	—
117	C₂×C₄⋊Q₈	—	2	1	no	—
118	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	—
119	#500	—	2	2	no	—
120	#500	C₂²×C₆₇	1	4	yes	—
121	#494	—	2	2	no	—
122	#494	C₂²×C₆₇	1	4	yes	—
123	C₂×#107	—	2	1	no	—
124	C₂×#107	C₂²×C₆₇	1	2	yes	—
125	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
126	#358	C₂²×C₆₇	1	16	yes	—
127	C₄×C₁₆	C₂²×C₆₇	1	4	yes	—
128	C₂²×Q₈	—	2	2	no	—
129	C₂²×Q₈	—	4	1	no	—
130	C₂×Q₁₆	—	2	4	no	—
131	C₂×Q₁₆	—	2	2	no	—
132	C₂×Q₁₆	—	2	8	no	—
133	C₂×Q₁₆	—	4	4	no	—
134	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
135	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
136	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
137	C₄⋊Q₈	—	2	4	no	—
138	C₄⋊Q₈	—	2	4	no	—
139	C₄⋊Q₈	—	4	2	no	—
140	C₄⋊Q₈	—	4	2	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	4	1	no	—
143	C₂×C₄⋊C₄	—	4	1	no	—
144	C₂×C₄⋊C₄	—	4	1	no	—
145	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
146	C₂×C₁₆	—	2	4	no	—
147	C₂×C₁₆	C₂³×C₆₇	1	4	yes	—
148	#107	—	2	4	no	—
149	#107	—	2	2	no	—
150	#107	C₂³×C₆₇	1	4	yes	—
151	C₄×C₈	D₄×C₆₇	1	2	yes	—
152	C₄×C₈	C₂³×C₆₇	1	2	yes	—
153	C₂×Q₈	—	2	4	no	—
154	C₂×Q₈	—	4	4	no	—
155	C₂×Q₈	—	4	2	no	—
156	C₂×Q₈	—	4	4	no	—
157	C₂×Q₈	—	4	4	no	—
158	C₂×Q₈	—	8	4	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	—	4	2	no	—
161	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
162	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
163	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
164	Q₁₆	—	2	8	no	—
165	Q₁₆	—	4	4	no	—
166	Q₁₆	—	4	8	no	—
167	C₂×C₈	SD₁₆×C₆₇	1	4	yes	—
168	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
169	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
170	C₂×C₈	#41×C₆₇	1	4	yes	—
171	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄⋊C₄	—	4	4	no	—
176	C₄⋊C₄	—	4	4	no	—
177	C₄²	D₈×C₆₇	1	2	yes	—
178	C₄²	C₂×D₄×C₆₇	1	2	yes	—
179	C₁₆	—	2	4	no	—
180	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
181	Q₈	—	4	4	no	—
182	Q₈	—	4	4	no	—
183	Q₈	—	8	4	no	—
184	Q₈	—	8	8	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	—	8	4	no	—
188	C₂×C₄	—	8	4	no	—
189	C₂×C₄	SD₃₂×C₆₇	1	4	yes	—
190	C₂×C₄	Q₃₂×C₆₇	1	4	yes	—
191	C₂×C₄	#124×C₆₇	1	2	yes	—
192	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
193	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
194	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
195	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
196	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
197	C₈	#124×C₆₇	1	2	yes	—
198	C₈	C₂×SD₁₆×C₆₇	1	4	yes	—
199	C₈	C₂×#41×C₆₇	1	2	yes	—
200	C₂²	#486×C₆₇	1	2	yes	—
201	C₂²	C₂×SD₃₂×C₆₇	1	2	yes	—
202	C₂²	C₂×Q₃₂×C₆₇	1	2	yes	—
203	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
204	C₄	—	8	4	no	—
205	C₄	—	8	4	no	—
206	C₄	#486×C₆₇	1	2	yes	—
207	C₄	C₂×SD₃₂×C₆₇	1	4	yes	—
208	C₄	C₂×Q₃₂×C₆₇	1	4	yes	—
209	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
210	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
211	C₂	C₆₇×#2241	1	4	yes	direct product factor
212	C₂	C₂×#486×C₆₇	1	1	yes	characteristic
213	C₂	C₂²×SD₃₂×C₆₇	1	1	yes	characteristic
214	C₂	C₂²×Q₃₂×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2241	C₂	direct (N × H)
103. C₂×#2241	C₆₇	direct (N × H)
104. C₂×C₆₇	#2241	direct (N × H)
110. #2241	C₂×C₆₇	direct (N × H)
113. C₆₇	C₂×#2241	direct (N × H)
211. C₂	C₆₇×#2241	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#500×C₆₇	2
C₂×#494×C₆₇	2
C₆₇×#2241	2
C₂×#358×C₆₇	2
C₂×C₄×C₁₆×C₆₇	2
C₂×#2241	67

Subgroups — C2×C67×#2241 #30712946

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2241 #30712946