Subgroups — Q₈×#14523975 #30874658

Order 17152.

838 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₄⋊Dic₆₇	C₂	1	2	yes	maximal
2	#22560973	C₂	1	6	yes	maximal
3	#22560366	C₂	1	3	yes	maximal
4	C₄×#14523975	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×Dic₆₇	—	2	2	no	—
7	C₂×#14523975	C₂²	1	3	yes	—
8	C₂×#14523975	C₂²	1	6	yes	—
9	C₂×#14523975	C₂²	1	1	yes	characteristic
10	#16863295	—	2	6	no	—
11	#16863266	C₂²	1	6	yes	—
12	C₄×C₄⋊Dic₆₇	C₂²	1	6	yes	—
13	#16863159	C₂²	1	6	yes	—
14	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₆₇	C₄	1	8	yes	complemented
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	3	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
18	Q₈×Dic₆₇	—	4	8	no	—
19	C₂×C₄⋊Dic₆₇	—	2	6	no	—
20	C₂×C₄⋊Dic₆₇	—	2	6	no	—
21	C₂×C₄⋊Dic₆₇	C₂³	1	6	yes	—
22	C₂×C₄⋊Dic₆₇	C₂³	1	2	yes	—
23	C₂×C₄×Dic₆₇	—	2	6	no	—
24	#14523975	—	2	6	no	—
25	#14523975	—	4	6	no	—
26	#14523975	Q₈	1	4	yes	direct product factor
27	C₆₇⋊#95	—	2	6	no	—
28	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
30	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
31	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
32	C₄×Q₈×C₆₇	Q₈	1	4	yes	—
33	C₂×C₄⋊C₄×C₆₇	D₄	1	3	yes	—
34	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
35	C₄⋊C₈×C₆₇	C₂×C₄	1	12	yes	—
36	C₄×C₈×C₆₇	C₂×C₄	1	12	yes	—
37	C₂²×Dic₆₇	—	2	2	no	—
38	C₂²×Dic₆₇	—	2	6	no	—
39	C₄⋊Dic₆₇	—	2	2	no	—
40	C₄⋊Dic₆₇	—	2	6	no	—
41	C₄⋊Dic₆₇	—	2	6	no	—
42	C₄⋊Dic₆₇	—	4	12	no	—
43	C₄⋊Dic₆₇	—	8	6	no	—
44	C₄⋊Dic₆₇	C₂×Q₈	1	4	yes	—
45	C₄×Dic₆₇	—	4	12	no	—
46	C₂×Q₈×C₆₇	C₄⋊C₄	1	4	yes	—
47	C₂×Q₈×C₆₇	SD₁₆	1	8	yes	—
48	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
49	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
50	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
51	C₂×C₈×C₆₇	—	2	3	no	—
52	C₂×C₈×C₆₇	—	2	3	no	—
53	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
54	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
55	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
56	C₂×C₈×C₆₇	#41	1	2	yes	—
57	C₄⋊C₄×C₆₇	C₄⋊C₄	1	12	yes	—
58	C₄²×C₆₇	C₂²×C₄	1	6	yes	—
59	C₄²×C₆₇	C₂×Q₈	1	6	yes	—
60	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
61	Q₈×C₆₇	#106	1	16	yes	complemented
62	C₂×C₄×C₆₇	—	2	6	no	—
63	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
64	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
65	C₂×C₄×C₆₇	C₂×SD₁₆	1	12	yes	—
66	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
70	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
71	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
72	C₂×Dic₆₇	—	2	4	no	—
73	C₂×Dic₆₇	—	4	2	no	—
74	C₂×Dic₆₇	—	4	2	no	—
75	C₂×Dic₆₇	—	4	6	no	—
76	C₂×Dic₆₇	—	4	6	no	—
77	C₂×Dic₆₇	—	4	6	no	—
78	C₈×C₆₇	—	2	6	no	—
79	C₈×C₆₇	C₄×Q₈	1	4	yes	complemented
80	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
82	C₂²×C₆₇	#550	1	1	yes	characteristic
83	C₂²×C₆₇	C₂²×SD₁₆	1	2	yes	—
84	C₂²×C₆₇	#577	1	2	yes	—
85	C₄×C₆₇	—	2	6	no	—
86	C₄×C₆₇	C₂×#106	1	12	yes	—
87	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
88	C₄×C₆₇	#520	1	1	yes	characteristic
89	C₄×C₆₇	#557	1	1	yes	characteristic
90	C₄×C₆₇	Q₈²	1	1	yes	characteristic
91	Dic₆₇	—	4	4	no	—
92	Dic₆₇	—	8	6	no	—
93	Q₈×#106	—	67	1	no	maximal, Sylow
94	C₂×C₆₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₆₇	C₂²×#106	1	1	yes	characteristic
96	C₂×C₆₇	#2912	1	1	yes	characteristic
97	C₂×C₆₇	Q₈×SD₁₆	1	2	yes	—
98	C₂×C₆₇	#3392	1	2	yes	—
99	C₂×C₈×Q₈	D₆₇	1	1	yes	characteristic
100	Q₈×C₄⋊C₄	—	67	2	no	—
101	#1948	—	67	6	no	—
102	#1844	—	67	3	no	—
103	C₄×#106	—	67	3	no	—
104	C₆₇	Q₈×#106	1	1	yes	characteristic, complemented, Sylow
105	C₂×C₄×Q₈	—	134	2	no	—
106	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
107	C₈×Q₈	Dic₆₇	1	8	yes	complemented
108	C₂×#106	—	67	6	no	—
109	C₂×#106	—	67	1	no	—
110	C₂×#106	—	67	3	no	—
111	C₂×C₄⋊C₈	C₂×D₆₇	1	3	yes	—
112	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
113	#389	—	134	6	no	—
114	#384	—	67	6	no	—
115	C₄×C₄⋊C₄	—	67	6	no	—
116	#340	—	67	6	no	—
117	C₂²×Q₈	D₂₆₈	1	1	yes	characteristic
118	C₂²×C₈	C₂²×D₆₇	1	3	yes	—
119	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
120	C₄×Q₈	—	268	8	no	—
121	C₄×Q₈	Dic₁₃₄	1	4	yes	—
122	C₄×Q₈	C₂×Dic₆₇	1	4	yes	—
123	C₂×C₄⋊C₄	—	67	2	no	—
124	C₂×C₄⋊C₄	—	67	6	no	—
125	C₂×C₄⋊C₄	—	134	6	no	—
126	C₂×C₄⋊C₄	—	134	6	no	—
127	C₂×C₄⋊C₄	D₂₆₈	1	3	yes	—
128	C₂×C₄²	—	134	6	no	—
129	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
130	#106	—	268	6	no	—
131	#106	—	67	4	no	—
132	#106	—	134	6	no	—
133	C₄⋊C₈	C₂×Dic₆₇	1	12	yes	—
134	C₄×C₈	C₂×Dic₆₇	1	12	yes	—
135	#95	—	134	6	no	—
136	C₂×Q₈	C₄⋊Dic₆₇	1	4	yes	—
137	C₂×Q₈	#11761617	1	8	yes	—
138	C₂²×C₄	—	134	6	no	—
139	C₂²×C₄	—	134	2	no	—
140	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
141	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
142	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
143	C₂×C₈	—	2	3	no	—
144	C₂×C₈	—	2	3	no	—
145	C₂×C₈	C₂²×Dic₆₇	1	6	yes	—
146	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
147	C₂×C₈	Q₈×D₆₇	1	2	yes	—
148	C₂×C₈	#11761629	1	2	yes	—
149	C₄⋊C₄	—	268	12	no	—
150	C₄⋊C₄	—	536	6	no	—
151	C₄⋊C₄	—	67	4	no	—
152	C₄⋊C₄	—	134	6	no	—
153	C₄⋊C₄	—	134	6	no	—
154	C₄⋊C₄	—	134	2	no	—
155	C₄⋊C₄	C₄⋊Dic₆₇	1	12	yes	—
156	C₄²	—	268	12	no	—
157	C₄²	C₂²×Dic₆₇	1	6	yes	—
158	C₄²	C₂×Dic₁₃₄	1	6	yes	—
159	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
160	Q₈	#14523975	1	16	yes	direct product factor
161	C₂×C₄	—	2	6	no	—
162	C₂×C₄	—	268	2	no	—
163	C₂×C₄	—	268	2	no	—
164	C₂×C₄	—	268	6	no	—
165	C₂×C₄	—	268	6	no	—
166	C₂×C₄	—	268	6	no	—
167	C₂×C₄	—	134	4	no	—
168	C₂×C₄	C₂×C₄⋊Dic₆₇	1	6	yes	—
169	C₂×C₄	C₂×C₄⋊Dic₆₇	1	6	yes	—
170	C₂×C₄	C₂×#11761617	1	12	yes	—
171	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
172	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
173	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
174	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
175	C₂×C₄	#14524089	1	1	yes	characteristic
176	C₂×C₄	#14524090	1	1	yes	characteristic
177	C₈	—	2	6	no	—
178	C₈	Q₈×Dic₆₇	1	4	yes	complemented
179	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
180	C₂²	Q₈×D₂₆₈	1	1	yes	characteristic
181	C₂²	#16864068	1	1	yes	characteristic
182	C₂²	C₂²×#11761617	1	2	yes	—
183	C₂²	#16864208	1	2	yes	—
184	C₄	—	2	6	no	—
185	C₄	—	268	4	no	—
186	C₄	—	536	6	no	—
187	C₄	C₂×#14523975	1	12	yes	—
188	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
189	C₄	#16863879	1	1	yes	characteristic
190	C₄	#16864125	1	1	yes	characteristic
191	C₄	Q₈×Dic₁₃₄	1	1	yes	characteristic
192	C₂	Q₈×C₄⋊Dic₆₇	1	1	yes	characteristic
193	C₂	C₂²×#14523975	1	1	yes	characteristic
194	C₂	#22568804	1	1	yes	characteristic
195	C₂	Q₈×#11761617	1	2	yes	—
196	C₂	#22573191	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×C₈×Q₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×C₈×Q₈	D₆₇
hypocenter	C₆₇	Q₈×#106
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
15. C₈×Q₈×C₆₇	C₄	semidirect
26. #14523975	Q₈	direct (N × H)
61. Q₈×C₆₇	#106	semidirect
79. C₈×C₆₇	C₄×Q₈	semidirect
104. C₆₇	Q₈×#106	semidirect
107. C₈×Q₈	Dic₆₇	semidirect
160. Q₈	#14523975	direct (N × H)
178. C₈	Q₈×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₄⋊Dic₆₇	2
#22560973	2
#22560366	2
C₄×#14523975	2
C₂×C₈×Q₈×C₆₇	2
Q₈×#106	67

Subgroups — Q8×#14523975 #30874658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14523975 #30874658