Subgroups — C₂×Q₈×Q₁₆×C₆₇ #30740606

Order 17152.

1830 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈²×C₆₇	C₂	1	2	yes	maximal
2	Q₈×Q₁₆×C₆₇	C₂	1	16	yes	maximal, direct product factor
3	C₂×#500×C₆₇	C₂	1	3	yes	maximal
4	C₂×#477×C₆₇	C₂	1	6	yes	maximal
5	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₁₆×C₆₇	C₂	1	3	yes	maximal
7	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
8	Q₈²×C₆₇	—	2	8	no	—
9	Q₈²×C₆₇	C₂²	1	16	yes	—
10	C₂×C₄⋊Q₈×C₆₇	—	2	6	no	—
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	6	yes	—
12	C₂×C₄×Q₈×C₆₇	—	2	2	no	—
13	C₂×C₄×Q₈×C₆₇	C₂²	1	6	yes	—
14	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	#500×C₆₇	C₂²	1	24	yes	—
16	#477×C₆₇	C₂²	1	48	yes	—
17	C₈×Q₈×C₆₇	C₂²	1	8	yes	—
18	C₄×Q₁₆×C₆₇	C₂²	1	24	yes	—
19	C₂×#107×C₆₇	C₂²	1	6	yes	—
20	C₂×#107×C₆₇	C₂²	1	3	yes	—
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	3	yes	—
22	C₂×#103×C₆₇	C₂²	1	6	yes	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
24	C₂²×Q₈×C₆₇	—	2	6	no	—
25	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
26	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
27	C₂×Q₁₆×C₆₇	Q₈	1	8	yes	direct product factor
28	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
29	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
30	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
31	C₄⋊Q₈×C₆₇	—	2	48	no	—
32	C₄⋊Q₈×C₆₇	—	2	12	no	—
33	C₄⋊Q₈×C₆₇	C₂³	1	24	yes	—
34	C₄×Q₈×C₆₇	—	2	16	no	—
35	C₄×Q₈×C₆₇	—	2	12	no	—
36	C₄×Q₈×C₆₇	D₄	1	4	yes	—
37	C₄×Q₈×C₆₇	C₂³	1	24	yes	—
38	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
39	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
41	C₂×C₄⋊C₄×C₆₇	D₄	1	3	yes	—
42	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
43	C₂×C₄²×C₆₇	—	2	6	no	—
44	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
45	#107×C₆₇	C₂³	1	24	yes	—
46	#107×C₆₇	C₂³	1	12	yes	—
47	C₄⋊C₈×C₆₇	C₂³	1	12	yes	—
48	#103×C₆₇	C₂³	1	24	yes	—
49	C₄×C₈×C₆₇	C₂³	1	12	yes	—
50	C₂×Q₈×C₆₇	—	2	2	no	—
51	C₂×Q₈×C₆₇	—	2	24	no	—
52	C₂×Q₈×C₆₇	—	2	4	no	—
53	C₂×Q₈×C₆₇	—	8	12	no	—
54	C₂×Q₈×C₆₇	Q₁₆	1	8	yes	direct product factor
55	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
56	C₂×Q₈×C₆₇	C₂×Q₈	1	8	yes	—
57	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
58	C₂²×C₄×C₆₇	—	2	2	no	—
59	C₂²×C₄×C₆₇	—	2	6	no	—
60	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
61	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
62	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
63	Q₁₆×C₆₇	C₂×Q₈	1	16	yes	direct product factor
64	C₂×C₈×C₆₇	—	2	6	no	—
65	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
66	C₂×C₈×C₆₇	C₂⁴	1	6	yes	—
67	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
68	C₄⋊C₄×C₆₇	—	2	24	no	—
69	C₄⋊C₄×C₆₇	—	2	24	no	—
70	C₄⋊C₄×C₆₇	—	2	6	no	—
71	C₄⋊C₄×C₆₇	C₂×D₄	1	12	yes	—
72	C₄⋊C₄×C₆₇	C₂⁴	1	12	yes	—
73	C₄²×C₆₇	—	2	24	no	—
74	C₄²×C₆₇	C₂×D₄	1	6	yes	—
75	C₄²×C₆₇	C₂⁴	1	6	yes	—
76	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
77	Q₈×C₆₇	—	2	8	no	—
78	Q₈×C₆₇	—	2	4	no	—
79	Q₈×C₆₇	—	8	24	no	—
80	Q₈×C₆₇	C₂×Q₁₆	1	16	yes	direct product factor
81	Q₈×C₆₇	C₂²×Q₈	1	8	yes	—
82	C₂×C₄×C₆₇	—	2	8	no	—
83	C₂×C₄×C₆₇	—	2	6	no	—
84	C₂×C₄×C₆₇	—	2	4	no	—
85	C₂×C₄×C₆₇	—	2	12	no	—
86	C₂×C₄×C₆₇	—	4	12	no	—
87	C₂×C₄×C₆₇	C₂×Q₁₆	1	12	yes	—
88	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
89	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
91	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
92	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
93	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
94	C₈×C₆₇	—	2	6	no	—
95	C₈×C₆₇	C₂²×Q₈	1	4	yes	—
96	C₂²×C₆₇	D₄×Q₈	1	2	yes	—
97	C₂²×C₆₇	C₂²×Q₁₆	1	2	yes	—
98	C₂²×C₆₇	#576	1	2	yes	—
99	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
100	C₄×C₆₇	—	2	8	no	—
101	C₄×C₆₇	—	2	6	no	—
102	C₄×C₆₇	—	4	12	no	—
103	C₄×C₆₇	D₄×Q₈	1	2	yes	—
104	C₄×C₆₇	C₂²×Q₁₆	1	12	yes	—
105	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
106	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
107	C₂×Q₈×Q₁₆	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂×C₆₇	Q₈×Q₁₆	1	4	yes	direct product factor
109	C₂×C₆₇	C₂×D₄×Q₈	1	1	yes	characteristic
110	C₂×C₆₇	C₂³×Q₁₆	1	1	yes	characteristic
111	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
112	C₂×Q₈²	C₂×C₆₇	1	2	yes	—
113	Q₈×Q₁₆	C₂×C₆₇	1	16	yes	direct product factor
114	C₂×#500	C₂×C₆₇	1	3	yes	—
115	C₂×#477	C₂×C₆₇	1	6	yes	—
116	C₂×C₈×Q₈	C₂×C₆₇	1	1	yes	characteristic
117	C₂×C₄×Q₁₆	C₂×C₆₇	1	3	yes	—
118	C₆₇	C₂×Q₈×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
119	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
120	Q₈²	—	2	8	no	—
121	Q₈²	C₂²×C₆₇	1	16	yes	—
122	C₂×C₄⋊Q₈	—	2	6	no	—
123	C₂×C₄⋊Q₈	C₂²×C₆₇	1	6	yes	—
124	C₂×C₄×Q₈	—	2	2	no	—
125	C₂×C₄×Q₈	C₂²×C₆₇	1	6	yes	—
126	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
127	#500	C₂²×C₆₇	1	24	yes	—
128	#477	C₂²×C₆₇	1	48	yes	—
129	C₈×Q₈	C₂²×C₆₇	1	8	yes	—
130	C₄×Q₁₆	C₂²×C₆₇	1	24	yes	—
131	C₂×#107	C₂²×C₆₇	1	6	yes	—
132	C₂×#107	C₂²×C₆₇	1	3	yes	—
133	C₂×C₄⋊C₈	C₂²×C₆₇	1	3	yes	—
134	C₂×#103	C₂²×C₆₇	1	6	yes	—
135	C₂×C₄×C₈	C₂²×C₆₇	1	3	yes	—
136	C₂²×Q₈	—	2	6	no	—
137	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
138	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
139	C₂×Q₁₆	Q₈×C₆₇	1	8	yes	direct product factor
140	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
141	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
142	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
143	C₄⋊Q₈	—	2	48	no	—
144	C₄⋊Q₈	—	2	12	no	—
145	C₄⋊Q₈	C₂³×C₆₇	1	24	yes	—
146	C₄×Q₈	—	2	16	no	—
147	C₄×Q₈	—	2	12	no	—
148	C₄×Q₈	D₄×C₆₇	1	4	yes	—
149	C₄×Q₈	C₂³×C₆₇	1	24	yes	—
150	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
151	C₂×C₄⋊C₄	—	2	6	no	—
152	C₂×C₄⋊C₄	—	2	6	no	—
153	C₂×C₄⋊C₄	D₄×C₆₇	1	3	yes	—
154	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
155	C₂×C₄²	—	2	6	no	—
156	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
157	#107	C₂³×C₆₇	1	12	yes	—
158	#107	C₂³×C₆₇	1	24	yes	—
159	C₄⋊C₈	C₂³×C₆₇	1	12	yes	—
160	#103	C₂³×C₆₇	1	24	yes	—
161	C₄×C₈	C₂³×C₆₇	1	12	yes	—
162	C₂×Q₈	—	2	2	no	—
163	C₂×Q₈	—	2	24	no	—
164	C₂×Q₈	—	2	4	no	—
165	C₂×Q₈	—	8	12	no	—
166	C₂×Q₈	Q₁₆×C₆₇	1	8	yes	direct product factor
167	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
168	C₂×Q₈	C₂×Q₈×C₆₇	1	8	yes	—
169	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	6	no	—
172	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
173	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
174	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
175	Q₁₆	C₂×Q₈×C₆₇	1	16	yes	direct product factor
176	C₂×C₈	—	2	6	no	—
177	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
178	C₂×C₈	C₂⁴×C₆₇	1	6	yes	—
179	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
180	C₄⋊C₄	—	2	24	no	—
181	C₄⋊C₄	—	2	24	no	—
182	C₄⋊C₄	—	2	6	no	—
183	C₄⋊C₄	C₂×D₄×C₆₇	1	12	yes	—
184	C₄⋊C₄	C₂⁴×C₆₇	1	12	yes	—
185	C₄²	—	2	24	no	—
186	C₄²	C₂×D₄×C₆₇	1	6	yes	—
187	C₄²	C₂⁴×C₆₇	1	6	yes	—
188	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
189	Q₈	—	2	8	no	—
190	Q₈	—	2	4	no	—
191	Q₈	—	8	24	no	—
192	Q₈	C₂×Q₁₆×C₆₇	1	16	yes	direct product factor
193	Q₈	C₂²×Q₈×C₆₇	1	8	yes	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	2	6	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	12	no	—
198	C₂×C₄	—	4	12	no	—
199	C₂×C₄	C₂×Q₁₆×C₆₇	1	12	yes	—
200	C₂×C₄	C₂²×D₄×C₆₇	1	6	yes	—
201	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
202	C₂×C₄	C₂²×D₄×C₆₇	1	6	yes	—
203	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
204	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
205	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
206	C₈	—	2	6	no	—
207	C₈	C₂²×Q₈×C₆₇	1	4	yes	—
208	C₂²	D₄×Q₈×C₆₇	1	2	yes	—
209	C₂²	C₂²×Q₁₆×C₆₇	1	2	yes	—
210	C₂²	#576×C₆₇	1	2	yes	—
211	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
212	C₄	—	2	6	no	—
213	C₄	—	2	8	no	—
214	C₄	—	4	12	no	—
215	C₄	D₄×Q₈×C₆₇	1	2	yes	—
216	C₄	C₂²×Q₁₆×C₆₇	1	12	yes	—
217	C₄	C₂³×Q₈×C₆₇	1	1	yes	characteristic
218	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
219	C₂	Q₈×Q₁₆×C₆₇	1	4	yes	direct product factor
220	C₂	C₂×D₄×Q₈×C₆₇	1	1	yes	characteristic
221	C₂	C₂³×Q₁₆×C₆₇	1	1	yes	characteristic
222	C₂	C₂×#576×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₈×Q₁₆×C₆₇	C₂	direct (N × H)
27. C₂×Q₁₆×C₆₇	Q₈	direct (N × H)
54. C₂×Q₈×C₆₇	Q₁₆	direct (N × H)
63. Q₁₆×C₆₇	C₂×Q₈	direct (N × H)
80. Q₈×C₆₇	C₂×Q₁₆	direct (N × H)
107. C₂×Q₈×Q₁₆	C₆₇	direct (N × H)
108. C₂×C₆₇	Q₈×Q₁₆	direct (N × H)
113. Q₈×Q₁₆	C₂×C₆₇	direct (N × H)
118. C₆₇	C₂×Q₈×Q₁₆	direct (N × H)
139. C₂×Q₁₆	Q₈×C₆₇	direct (N × H)
166. C₂×Q₈	Q₁₆×C₆₇	direct (N × H)
175. Q₁₆	C₂×Q₈×C₆₇	direct (N × H)
192. Q₈	C₂×Q₁₆×C₆₇	direct (N × H)
219. C₂	Q₈×Q₁₆×C₆₇	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×Q₈²×C₆₇	2
Q₈×Q₁₆×C₆₇	2
C₂×#500×C₆₇	2
C₂×#477×C₆₇	2
C₂×C₈×Q₈×C₆₇	2
C₂×C₄×Q₁₆×C₆₇	2
C₂×Q₈×Q₁₆	67

Subgroups — C2×Q8×Q16×C67 #30740606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×Q₁₆×C₆₇ #30740606