Subgroups — Q₈×#14556039 #31996121

Order 18176.

954 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₇₁⋊C₈	C₂	1	2	yes	maximal
2	#22588209	C₂	1	6	yes	maximal
3	#22588193	C₂	1	3	yes	maximal
4	C₄×#14556039	C₂	1	3	yes	maximal
5	C₄²×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₇₁⋊C₈	C₄	1	16	yes	complemented
7	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	6	yes	—
8	C₂×#14556039	C₂²	1	3	yes	—
9	C₂×#14556039	C₂²	1	6	yes	—
10	C₂×#14556039	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇₁⋊C₈	C₂²	1	6	yes	—
12	#17985754	C₂²	1	6	yes	—
13	C₂×C₄×Q₈×C₇₁	C₄	1	2	yes	—
14	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₇₁	C₄	1	6	yes	—
16	C₄×C₄⋊C₄×C₇₁	C₂²	1	3	yes	—
17	C₄³×C₇₁	C₂²	1	3	yes	—
18	C₂²×C₇₁⋊C₈	C₂³	1	6	yes	—
19	C₂²×C₇₁⋊C₈	C₂³	1	2	yes	—
20	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	24	yes	—
21	#14556039	—	2	12	no	—
22	#14556039	—	2	6	no	—
23	#14556039	Q₈	1	4	yes	direct product factor
24	C₄×C₇₁⋊C₈	C₂×C₄	1	24	yes	—
25	C₂²×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₄×Q₈×C₇₁	—	2	8	no	—
27	C₄×Q₈×C₇₁	C₂×C₄	1	8	yes	—
28	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	6	yes	—
29	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	3	yes	—
30	C₂×C₄²×C₇₁	C₂×C₄	1	6	yes	—
31	C₂×C₄²×C₇₁	C₂×C₄	1	6	yes	—
32	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
33	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
34	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
35	#95×C₇₁	C₂×C₄	1	6	yes	—
36	C₂×C₇₁⋊C₈	—	2	6	no	—
37	C₂×C₇₁⋊C₈	—	2	6	no	—
38	C₂×C₇₁⋊C₈	C₂²×C₄	1	12	yes	—
39	C₂×C₇₁⋊C₈	C₂²×C₄	1	4	yes	—
40	C₂×C₇₁⋊C₈	C₂×Q₈	1	4	yes	—
41	C₂×C₇₁⋊C₈	#41	1	4	yes	—
42	C₂×Q₈×C₇₁	C₄²	1	4	yes	—
43	C₂×Q₈×C₇₁	M₁₆	1	8	yes	—
44	C₂²×C₄×C₇₁	C₂²×C₄	1	6	yes	—
45	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
46	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
48	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
49	C₄⋊C₄×C₇₁	—	2	12	no	—
50	C₄⋊C₄×C₇₁	C₄²	1	12	yes	—
51	C₄²×C₇₁	—	2	12	no	—
52	C₄²×C₇₁	—	2	6	no	—
53	C₄²×C₇₁	—	4	6	no	—
54	C₄²×C₇₁	C₂²×C₄	1	12	yes	—
55	C₄²×C₇₁	C₂×Q₈	1	2	yes	—
56	C₄²×C₇₁	#41	1	2	yes	—
57	C₂³×C₇₁	C₂³×C₄	1	1	yes	characteristic
58	Q₈×C₇₁	#97	1	16	yes	complemented
59	C₂×C₄×C₇₁	—	2	2	no	—
60	C₂×C₄×C₇₁	—	2	2	no	—
61	C₂×C₄×C₇₁	—	2	6	no	—
62	C₂×C₄×C₇₁	—	2	6	no	—
63	C₂×C₄×C₇₁	—	2	6	no	—
64	C₂×C₄×C₇₁	—	2	6	no	—
65	C₂×C₄×C₇₁	C₂×C₄²	1	6	yes	—
66	C₂×C₄×C₇₁	C₂×C₄²	1	6	yes	—
67	C₂×C₄×C₇₁	C₄×Q₈	1	4	yes	—
68	C₂×C₄×C₇₁	C₂×M₁₆	1	12	yes	—
69	C₂×C₄×C₇₁	C₂³×C₄	1	2	yes	—
70	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₁	C₂×#41	1	2	yes	—
72	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
73	C₇₁⋊C₈	—	2	12	no	—
74	C₇₁⋊C₈	C₄×Q₈	1	8	yes	complemented
75	C₂²×C₇₁	C₂²×C₄²	1	1	yes	characteristic
76	C₂²×C₇₁	C₂×C₄×Q₈	1	1	yes	characteristic
77	C₂²×C₇₁	C₄×#41	1	1	yes	characteristic
78	C₂²×C₇₁	C₂²×M₁₆	1	2	yes	—
79	C₂²×C₇₁	#568	1	2	yes	—
80	C₄×C₇₁	—	2	6	no	—
81	C₄×C₇₁	—	2	4	no	—
82	C₄×C₇₁	—	4	6	no	—
83	C₄×C₇₁	C₂×#97	1	12	yes	—
84	C₄×C₇₁	C₂×C₄×Q₈	1	2	yes	—
85	C₄×C₇₁	C₄×#41	1	2	yes	—
86	Q₈×#97	—	71	1	no	maximal, Sylow
87	C₂×C₇₁	C₄²×Q₈	1	1	yes	characteristic
88	C₂×C₇₁	C₂²×#97	1	1	yes	characteristic
89	C₂×C₇₁	#2875	1	1	yes	characteristic
90	C₂×C₇₁	Q₈×M₁₆	1	2	yes	—
91	C₂×C₇₁	#2971	1	2	yes	—
92	C₂×C₈×Q₈	—	71	2	no	—
93	C₄²×Q₈	D₇₁	1	1	yes	characteristic
94	#1770	—	71	6	no	—
95	#1767	—	71	3	no	—
96	C₄×#97	—	71	3	no	—
97	C₇₁	Q₈×#97	1	1	yes	characteristic, complemented, Sylow
98	C₂×C₄×Q₈	Dic₇₁	1	2	yes	—
99	C₂×C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
100	C₈×Q₈	—	71	16	no	—
101	C₂×C₄⋊C₈	—	71	6	no	—
102	C₂×#97	—	71	1	no	—
103	C₂×#97	—	71	3	no	—
104	C₂×#97	—	71	6	no	—
105	C₂×C₄×C₈	—	71	6	no	—
106	C₄×C₄⋊C₄	Dic₇₁	1	6	yes	—
107	C₄×C₄⋊C₄	C₂×D₇₁	1	3	yes	—
108	C₄³	C₂×D₇₁	1	3	yes	—
109	#336	—	71	6	no	—
110	C₂²×Q₈	C₂×Dic₇₁	1	1	yes	characteristic
111	C₂²×C₈	—	71	2	no	—
112	C₂²×C₈	—	71	6	no	—
113	C₄×Q₈	—	2	8	no	—
114	C₄×Q₈	C₄×D₇₁	1	8	yes	—
115	C₂×C₄⋊C₄	C₂×Dic₇₁	1	6	yes	—
116	C₂×C₄⋊C₄	C₂×Dic₇₁	1	3	yes	—
117	C₂×C₄²	C₂×Dic₇₁	1	6	yes	—
118	C₂×C₄²	C₂×Dic₇₁	1	6	yes	—
119	C₂×C₄²	C₂²×D₇₁	1	3	yes	—
120	C₂×C₄²	C₂²×D₇₁	1	3	yes	—
121	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
122	C₄⋊C₈	—	71	24	no	—
123	#97	—	71	4	no	—
124	#97	—	142	12	no	—
125	#97	—	142	6	no	—
126	C₄×C₈	—	71	24	no	—
127	#95	C₂×Dic₇₁	1	6	yes	—
128	C₂×Q₈	C₄×Dic₇₁	1	4	yes	—
129	C₂×Q₈	#11765849	1	8	yes	—
130	C₂²×C₄	C₂²×Dic₇₁	1	6	yes	—
131	C₂²×C₄	C₂²×Dic₇₁	1	3	yes	—
132	C₂²×C₄	C₂²×Dic₇₁	1	2	yes	—
133	C₂²×C₄	C₂²×Dic₇₁	1	3	yes	—
134	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
135	C₂×C₈	—	71	4	no	—
136	C₂×C₈	—	71	4	no	—
137	C₂×C₈	—	71	4	no	—
138	C₂×C₈	—	71	12	no	—
139	C₂×C₈	—	142	6	no	—
140	C₂×C₈	—	142	6	no	—
141	C₄⋊C₄	—	2	12	no	—
142	C₄⋊C₄	C₄×Dic₇₁	1	12	yes	—
143	C₄²	—	2	6	no	—
144	C₄²	—	2	12	no	—
145	C₄²	—	4	6	no	—
146	C₄²	C₂×C₄×D₇₁	1	12	yes	—
147	C₄²	Q₈×D₇₁	1	2	yes	—
148	C₄²	#11765866	1	2	yes	—
149	C₂³	C₂³×Dic₇₁	1	1	yes	characteristic
150	Q₈	#14556039	1	16	yes	direct product factor
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	6	no	—
154	C₂×C₄	—	2	6	no	—
155	C₂×C₄	—	2	6	no	—
156	C₂×C₄	—	2	6	no	—
157	C₂×C₄	C₂×C₄×Dic₇₁	1	6	yes	—
158	C₂×C₄	C₂×C₄×Dic₇₁	1	6	yes	—
159	C₂×C₄	Q₈×Dic₇₁	1	4	yes	—
160	C₂×C₄	C₂×#11765849	1	12	yes	—
161	C₂×C₄	C₂²×C₄×D₇₁	1	2	yes	—
162	C₂×C₄	C₂×Q₈×D₇₁	1	1	yes	characteristic
163	C₂×C₄	#41×D₇₁	1	2	yes	—
164	C₂×C₄	C₂×#11765866	1	1	yes	characteristic
165	C₈	—	71	8	no	—
166	C₈	—	142	12	no	—
167	C₂²	C₂²×C₄×Dic₇₁	1	1	yes	characteristic
168	C₂²	C₂×Q₈×Dic₇₁	1	1	yes	characteristic
169	C₂²	#41×Dic₇₁	1	1	yes	characteristic
170	C₂²	C₂²×#11765849	1	2	yes	—
171	C₂²	#17986766	1	2	yes	—
172	C₄	—	2	6	no	—
173	C₄	—	2	4	no	—
174	C₄	—	4	6	no	—
175	C₄	C₂×#14556039	1	12	yes	—
176	C₄	C₄×Q₈×D₇₁	1	2	yes	—
177	C₄	C₄×#11765866	1	2	yes	—
178	C₂	C₄×Q₈×Dic₇₁	1	1	yes	characteristic
179	C₂	C₂²×#14556039	1	1	yes	characteristic
180	C₂	#22596768	1	1	yes	characteristic
181	C₂	Q₈×#11765849	1	2	yes	—
182	C₂	#22597360	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₁
commutator	C₂²×C₇₁	C₂²×C₄²
Fitting	C₄²×Q₈×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂³×C₄
cosocle	C₂²×C₄×C₇₁	C₂⁴
hypercenter	C₄²×Q₈	D₇₁
hypocenter	C₇₁	Q₈×#97
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
6. Q₈×C₇₁⋊C₈	C₄	semidirect
23. #14556039	Q₈	direct (N × H)
58. Q₈×C₇₁	#97	semidirect
74. C₇₁⋊C₈	C₄×Q₈	semidirect
97. C₇₁	Q₈×#97	semidirect
150. Q₈	#14556039	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×Q₈×C₇₁⋊C₈	2
#22588209	2
#22588193	2
C₄×#14556039	2
C₄²×Q₈×C₇₁	2
Q₈×#97	71

Subgroups — Q8×#14556039 #31996121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14556039 #31996121