Subgroups — Q₈×#14556666 #34030331

Order 18688.

530 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626347	C₂	1	6	yes	maximal, complemented
2	Q₈×#11924031	C₂	1	2	yes	maximal, complemented
3	#22626264	C₂	1	3	yes	maximal
4	C₄×#14556666	C₂	1	3	yes	maximal
5	C₈×Q₈×D₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556666	C₂²	1	3	yes	—
7	C₂×#14556666	C₂²	1	1	yes	characteristic
8	#17989508	C₂²	1	6	yes	—
9	C₄⋊#11924031	C₂²	1	6	yes	—
10	#17989476	C₂²	1	6	yes	—
11	C₄×#11924031	C₂²	1	6	yes	—
12	C₄×Q₈×D₇₃	C₄	1	1	yes	characteristic
13	Q₈×#10596831	C₂²	1	1	yes	characteristic
14	C₈×Dic₁₄₆	C₄	1	3	yes	—
15	#17988711	C₂²	1	3	yes	—
16	C₄⋊C₈×D₇₃	C₄	1	3	yes	—
17	C₄×C₈×D₇₃	C₂²	1	3	yes	—
18	C₈×Q₈×C₇₃	C₄	1	1	yes	characteristic
19	#14556666	—	2	6	no	—
20	#14556666	Q₈	1	4	yes	direct product factor
21	C₂×#11924031	C₂³	1	6	yes	—
22	C₂×#11924031	C₂³	1	2	yes	—
23	C₂×Q₈×D₇₃	C₈	1	1	yes	characteristic
24	C₂×C₈×D₇₃	C₂×C₄	1	3	yes	—
25	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
26	Q₈×Dic₇₃	C₈	1	1	yes	characteristic
27	C₄×Dic₁₄₆	C₂×C₄	1	3	yes	—
28	#14556556	C₈	1	3	yes	—
29	C₄⋊C₄×D₇₃	C₈	1	3	yes	—
30	C₄²×D₇₃	C₂×C₄	1	3	yes	—
31	C₄⋊#10596831	C₂×C₄	1	3	yes	—
32	#14556520	C₂×C₄	1	6	yes	—
33	C₄×#10596831	C₂³	1	3	yes	—
34	C₈×Dic₇₃	C₂×C₄	1	3	yes	—
35	C₈×Dic₇₃	C₂³	1	3	yes	—
36	C₄×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₈×C₇₃	C₂×C₄	1	3	yes	—
38	C₄×C₈×C₇₃	C₂×C₄	1	3	yes	—
39	#11924031	—	2	6	no	—
40	#11924031	C₂×Q₈	1	4	yes	complemented
41	C₂×Dic₁₄₆	C₂×C₈	1	3	yes	—
42	Q₈×D₇₃	—	2	4	no	—
43	C₂×C₄×D₇₃	C₂×C₈	1	3	yes	—
44	C₂×C₄×D₇₃	C₂×C₈	1	3	yes	—
45	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×#10596831	C₂²×C₄	1	3	yes	—
47	C₂×#10596831	C₂⁴	1	1	yes	characteristic
48	C₈×D₇₃	—	2	6	no	—
49	C₈×D₇₃	C₂×Q₈	1	2	yes	—
50	C₈×D₇₃	#41	1	2	yes	—
51	C₄⋊Dic₇₃	C₂×C₈	1	3	yes	—
52	#11924007	C₂×C₈	1	6	yes	—
53	C₄×Dic₇₃	C₂×C₈	1	3	yes	—
54	C₄×Dic₇₃	C₂×C₈	1	3	yes	—
55	C₄×Dic₇₃	C₂²×C₄	1	3	yes	—
56	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
57	C₂×C₈×C₇₃	C₂²×C₄	1	3	yes	—
58	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
59	C₄⋊C₄×C₇₃	C₂×C₈	1	3	yes	—
60	C₄²×C₇₃	C₂²×C₄	1	3	yes	—
61	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
62	Q₈×C₇₃	M₃₂	1	4	yes	complemented
63	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
64	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
65	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
66	C₂×Dic₇₃	C₂²×C₈	1	3	yes	—
67	C₂×Dic₇₃	C₂²×C₈	1	3	yes	—
68	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
69	C₄×D₇₃	—	2	6	no	—
70	C₄×D₇₃	—	4	3	no	—
71	C₄×D₇₃	C₄×Q₈	1	2	yes	—
72	C₄×D₇₃	#131	1	2	yes	—
73	Dic₁₄₆	—	2	6	no	—
74	C₈×C₇₃	—	2	3	no	—
75	C₈×C₇₃	C₄×Q₈	1	2	yes	—
76	#10596831	—	2	3	no	—
77	#10596831	C₂²×Q₈	1	1	yes	characteristic
78	#10596831	C₂×#41	1	1	yes	characteristic
79	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
80	C₂×D₇₃	—	2	1	no	—
81	C₂×D₇₃	—	2	1	no	—
82	C₂×D₇₃	C₈×Q₈	1	2	yes	—
83	C₄×C₇₃	—	2	3	no	—
84	C₄×C₇₃	C₂×M₃₂	1	6	yes	—
85	C₄×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
86	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
87	Dic₇₃	—	2	3	no	—
88	Dic₇₃	—	2	3	no	—
89	Dic₇₃	C₈×Q₈	1	2	yes	—
90	Q₈×M₃₂	—	73	1	no	maximal, Sylow
91	C₂×C₇₃	C₂×C₈×Q₈	1	1	yes	characteristic
92	C₂×C₇₃	C₂²×M₃₂	1	1	yes	characteristic
93	C₂×C₇₃	#3407	1	1	yes	characteristic
94	D₇₃	—	2	2	no	—
95	C₂×C₈×Q₈	—	73	1	no	—
96	#2183	—	73	6	no	—
97	Q₈×C₁₆	—	73	2	no	—
98	#2150	—	73	3	no	—
99	C₄×M₃₂	—	73	3	no	—
100	C₇₃	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₄×Q₈	—	73	1	no	—
102	C₂×M₃₂	—	73	1	no	—
103	C₂×M₃₂	—	73	3	no	—
104	C₈×Q₈	—	73	1	no	—
105	C₈×Q₈	—	73	3	no	—
106	C₈×Q₈	—	73	3	no	—
107	C₈×Q₈	#64408	1	1	yes	characteristic
108	C₂×C₄⋊C₈	—	73	3	no	—
109	C₂×C₄×C₈	—	73	3	no	—
110	C₄⋊C₁₆	—	73	6	no	—
111	C₄⋊C₁₆	—	73	6	no	—
112	#346	—	73	6	no	—
113	C₄×C₁₆	—	73	6	no	—
114	C₂²×Q₈	—	73	1	no	—
115	C₂²×C₈	—	73	1	no	—
116	C₂²×C₈	—	73	3	no	—
117	C₄×Q₈	—	73	1	no	—
118	C₄×Q₈	—	73	3	no	—
119	C₄×Q₈	—	73	3	no	—
120	C₄×Q₈	C₂×#64408	1	1	yes	characteristic
121	C₂×C₄⋊C₄	—	73	3	no	—
122	C₂×C₄²	—	73	3	no	—
123	M₃₂	—	73	4	no	—
124	M₃₂	—	146	6	no	—
125	C₂×C₁₆	—	73	2	no	—
126	C₂×C₁₆	—	73	6	no	—
127	C₄⋊C₈	—	73	6	no	—
128	C₄⋊C₈	—	73	3	no	—
129	C₄⋊C₈	C₂×#64408	1	3	yes	—
130	C₄×C₈	—	73	3	no	—
131	C₄×C₈	—	73	3	no	—
132	C₄×C₈	—	73	3	no	—
133	C₄×C₈	C₂×#64408	1	3	yes	—
134	C₂×Q₈	—	73	3	no	—
135	C₂×Q₈	—	146	4	no	—
136	C₂×Q₈	#11924036	1	1	yes	characteristic
137	C₂²×C₄	—	73	3	no	—
138	C₂²×C₄	—	73	1	no	—
139	C₂²×C₄	—	73	3	no	—
140	C₂×C₈	—	73	1	no	—
141	C₂×C₈	—	73	2	no	—
142	C₂×C₈	—	73	2	no	—
143	C₂×C₈	—	73	3	no	—
144	C₂×C₈	—	146	6	no	—
145	C₂×C₈	C₂²×#64408	1	3	yes	—
146	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
147	C₄⋊C₄	—	73	6	no	—
148	C₄⋊C₄	—	73	3	no	—
149	C₄⋊C₄	#11924036	1	3	yes	—
150	C₄²	—	73	3	no	—
151	C₄²	—	73	3	no	—
152	C₄²	—	73	3	no	—
153	C₄²	C₂²×#64408	1	3	yes	—
154	C₁₆	—	73	4	no	—
155	C₁₆	—	146	6	no	—
156	C₂³	—	73	1	no	—
157	Q₈	—	146	6	no	—
158	Q₈	#14556666	1	4	yes	direct product factor
159	C₂×C₄	—	292	3	no	—
160	C₂×C₄	—	73	1	no	—
161	C₂×C₄	—	73	2	no	—
162	C₂×C₄	—	73	2	no	—
163	C₂×C₄	—	73	3	no	—
164	C₂×C₄	—	73	3	no	—
165	C₂×C₄	—	146	6	no	—
166	C₂×C₄	C₂×#11924036	1	3	yes	—
167	C₂×C₄	C₂×#11924036	1	3	yes	—
168	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
169	C₈	—	2	3	no	—
170	C₈	—	73	1	no	—
171	C₈	—	73	1	no	—
172	C₈	—	146	3	no	—
173	C₈	Q₈×#64408	1	2	yes	—
174	C₂²	—	73	2	no	—
175	C₂²	—	146	1	no	—
176	C₂²	—	146	1	no	—
177	C₂²	C₂²×#11924036	1	1	yes	characteristic
178	C₄	—	2	3	no	—
179	C₄	—	73	2	no	—
180	C₄	—	146	3	no	—
181	C₄	—	146	3	no	—
182	C₄	C₂×#14556666	1	6	yes	—
183	C₄	C₂×Q₈×#64408	1	1	yes	characteristic
184	C₄	D₇₃⋊#131	1	1	yes	characteristic
185	C₂	—	146	2	no	—
186	C₂	Q₈×#11924036	1	1	yes	characteristic
187	C₂	C₂²×#14556666	1	1	yes	characteristic
188	C₂	#22626899	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	C₈×Q₈×C₇₃	C₄
prime radical	C₂²×D₇₃	C₂²×C₈
cosocle	C₂×#10596831	C₂⁴
hypercenter	C₈×Q₈	#64408
hypocenter	C₇₃	Q₈×M₃₂
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. #22626347	C₂	semidirect
2. Q₈×#11924031	C₂	semidirect
20. #14556666	Q₈	direct (N × H)
40. #11924031	C₂×Q₈	semidirect
62. Q₈×C₇₃	M₃₂	semidirect
100. C₇₃	Q₈×M₃₂	semidirect
158. Q₈	#14556666	direct (N × H)

Maximal subgroups

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

subgroup	index
#22626347	2
Q₈×#11924031	2
#22626264	2
C₄×#14556666	2
C₈×Q₈×D₇₃	2
Q₈×M₃₂	73

Subgroups — Q8×#14556666 #34030331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14556666 #34030331