Subgroups — #34030317

Order 18688.

506 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626346	C₂	1	4	yes	maximal, complemented
2	Q₈×#11924031	C₂	1	4	yes	maximal, complemented
3	#22626266	C₂	1	2	yes	maximal
4	#22626263	C₂	1	1	yes	maximal, characteristic
5	#22626140	C₂	1	2	yes	maximal
6	C₂×C₄×#11924031	C₂	1	1	yes	maximal, characteristic
7	#22618866	C₂	1	1	yes	maximal, characteristic
8	C₂×#14556665	C₂²	1	2	yes	—
9	C₂²×#11924031	C₂²	1	1	yes	characteristic
10	C₂²×#11924031	C₂²	1	1	yes	characteristic
11	C₄⋊#11924031	C₂²	1	4	yes	—
12	C₄⋊#11924031	C₂²	1	8	yes	—
13	#17989475	C₂²	1	4	yes	—
14	C₄×#11924031	C₂²	1	2	yes	—
15	C₄×#11924031	C₂²	1	4	yes	—
16	C₄×#11924031	C₂²	1	2	yes	—
17	#17988718	C₄	1	4	yes	—
18	Q₈×#10596831	C₂²	1	4	yes	—
19	#17988614	C₄	1	2	yes	—
20	#17988611	C₄	1	1	yes	characteristic
21	#17988543	C₂²	1	2	yes	—
22	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
23	#533×C₇₃	C₄	1	1	yes	characteristic
24	#14556665	—	2	4	no	—
25	C₂×#11924031	—	2	2	no	—
26	C₂×#11924031	Q₈	1	4	yes	complemented
27	C₂×#11924031	C₂³	1	2	yes	—
28	C₂×#11924031	C₂³	1	4	yes	—
29	C₂×#11924031	C₂³	1	1	yes	characteristic
30	C₂×#11924031	C₂³	1	1	yes	characteristic
31	C₂×#11924012	C₂×C₄	1	2	yes	—
32	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
33	C₂²×#10596831	C₂³	1	1	yes	characteristic
34	C₄⋊#10596831	C₂×C₄	1	4	yes	—
35	C₄⋊#10596831	C₂×C₄	1	8	yes	—
36	#14556502	C₂×C₄	1	4	yes	—
37	C₄×#10596831	C₂×C₄	1	2	yes	—
38	C₄×#10596831	C₂³	1	2	yes	—
39	C₄×#10596831	C₂³	1	4	yes	—
40	C₄⋊Q₈×C₇₃	C₈	1	2	yes	—
41	#125×C₇₃	C₈	1	2	yes	—
42	C₂²⋊Q₈×C₇₃	C₈	1	4	yes	—
43	C₄×Q₈×C₇₃	C₂×C₄	1	4	yes	—
44	#117×C₇₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
46	#11924031	—	2	2	no	—
47	#11924031	—	2	4	no	—
48	#11924031	C₂×Q₈	1	2	yes	—
49	#11924031	C₂×Q₈	1	2	yes	—
50	#11924012	—	2	4	no	—
51	C₂×#10596831	—	2	2	no	—
52	C₂×#10596831	C₂²×C₄	1	2	yes	—
53	C₂×#10596831	C₂²×C₄	1	4	yes	—
54	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
55	C₂×#10596831	C₂×Q₈	1	2	yes	—
56	C₂×#10596831	#41	1	2	yes	—
57	C₂×#10596831	C₂⁴	1	1	yes	characteristic
58	C₂×Q₈×C₇₃	C₂×C₈	1	4	yes	—
59	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
61	C₄⋊C₄×C₇₃	C₂×C₈	1	4	yes	—
62	C₄⋊C₄×C₇₃	C₂×C₈	1	8	yes	—
63	C₄⋊C₄×C₇₃	C₂×C₈	1	4	yes	—
64	C₂²⋊C₄×C₇₃	C₂×C₈	1	4	yes	—
65	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
66	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
67	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
68	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
69	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
70	Q₈×C₇₃	—	2	8	no	—
71	C₂×C₄×C₇₃	—	2	4	no	—
72	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
73	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
74	C₂×C₄×C₇₃	C₂²×C₈	1	8	yes	—
75	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₇₃	#131	1	2	yes	—
78	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
79	#10596831	—	2	2	no	—
80	#10596831	—	2	4	no	—
81	#10596831	C₄×Q₈	1	2	yes	—
82	#10596831	C₂²×Q₈	1	1	yes	characteristic
83	#10596831	C₂×#41	1	1	yes	characteristic
84	C₂²×C₇₃	—	2	2	no	—
85	C₂²×C₇₃	C₈×Q₈	1	2	yes	—
86	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
87	C₄×C₇₃	—	2	2	no	—
88	C₄×C₇₃	—	2	8	no	—
89	C₄×C₇₃	C₈×Q₈	1	2	yes	—
90	C₄×C₇₃	#504	1	4	yes	—
91	C₄×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
92	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
93	#33474	—	73	1	no	maximal, Sylow
94	C₂×C₇₃	—	2	2	no	—
95	C₂×C₇₃	C₂×C₈×Q₈	1	1	yes	characteristic
96	C₂×C₇₃	C₂×#504	1	2	yes	—
97	#2960	—	73	1	no	—
98	#2183	—	73	4	no	—
99	Q₈×C₁₆	—	73	4	no	—
100	#2151	—	73	2	no	—
101	#2150	—	73	1	no	—
102	#2108	—	73	2	no	—
103	C₂×C₄×C₁₆	—	73	1	no	—
104	C₇₃	#33474	1	1	yes	characteristic, complemented, Sylow
105	#533	#64408	1	1	yes	characteristic
106	C₂×M₃₂	—	73	2	no	—
107	C₂²×C₁₆	—	73	1	no	—
108	C₂²×C₁₆	—	73	1	no	—
109	#446	—	73	4	no	—
110	C₈×Q₈	—	73	4	no	—
111	#424	—	73	2	no	—
112	#423	—	73	1	no	—
113	#405	—	73	2	no	—
114	C₂×C₄×C₈	—	73	1	no	—
115	C₄⋊C₁₆	—	73	4	no	—
116	C₄⋊C₁₆	—	73	8	no	—
117	#346	—	73	4	no	—
118	C₄×C₁₆	—	73	2	no	—
119	C₄×C₁₆	—	73	2	no	—
120	C₄×C₁₆	—	73	4	no	—
121	C₂×M₁₆	—	73	2	no	—
122	C₂²×C₈	—	73	1	no	—
123	C₂²×C₈	—	73	1	no	—
124	C₄⋊Q₈	#10596833	1	2	yes	—
125	#125	#10596833	1	2	yes	—
126	C₂²⋊Q₈	#10596833	1	4	yes	—
127	C₄×Q₈	C₂×#64408	1	4	yes	—
128	#117	C₂×#64408	1	2	yes	—
129	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
130	M₃₂	—	146	4	no	—
131	C₂×C₁₆	—	73	1	no	—
132	C₂×C₁₆	—	73	1	no	—
133	C₂×C₁₆	—	73	4	no	—
134	C₂×C₁₆	—	73	2	no	—
135	C₂×C₁₆	—	73	4	no	—
136	C₂×C₁₆	—	146	2	no	—
137	C₄⋊C₈	—	73	4	no	—
138	C₄⋊C₈	—	73	8	no	—
139	#97	—	73	4	no	—
140	C₄×C₈	—	73	2	no	—
141	C₄×C₈	—	73	2	no	—
142	C₄×C₈	—	73	4	no	—
143	C₂×Q₈	C₂×#10596833	1	4	yes	—
144	C₂²×C₄	C₂×#10596833	1	2	yes	—
145	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
146	M₁₆	—	146	4	no	—
147	C₂×C₈	—	73	2	no	—
148	C₂×C₈	—	73	2	no	—
149	C₂×C₈	—	73	1	no	—
150	C₂×C₈	—	73	1	no	—
151	C₂×C₈	—	73	2	no	—
152	C₂×C₈	—	73	4	no	—
153	C₂×C₈	—	146	2	no	—
154	C₄⋊C₄	C₂×#10596833	1	4	yes	—
155	C₄⋊C₄	C₂×#10596833	1	8	yes	—
156	C₄⋊C₄	C₂×#10596833	1	4	yes	—
157	C₂²⋊C₄	C₂×#10596833	1	4	yes	—
158	C₄²	C₂×#10596833	1	1	yes	characteristic
159	C₄²	C₂×#10596833	1	1	yes	characteristic
160	C₄²	C₂²×#64408	1	2	yes	—
161	C₄²	C₂²×#64408	1	4	yes	—
162	C₁₆	—	73	2	no	—
163	C₁₆	—	73	2	no	—
164	C₁₆	—	146	2	no	—
165	C₁₆	—	146	4	no	—
166	C₂³	C₂²×#10596833	1	1	yes	characteristic
167	Q₈	—	2	8	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	Q₈×#64408	1	2	yes	—
170	C₂×C₄	C₂²×#10596833	1	2	yes	—
171	C₂×C₄	C₂²×#10596833	1	8	yes	—
172	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
173	C₂×C₄	C₂²×#10596833	1	2	yes	—
174	C₂×C₄	#14556679	1	2	yes	—
175	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
176	C₈	—	73	1	no	—
177	C₈	—	73	1	no	—
178	C₈	—	73	2	no	—
179	C₈	—	146	2	no	—
180	C₈	—	146	4	no	—
181	C₂²	—	2	2	no	—
182	C₂²	Q₈×#10596833	1	2	yes	—
183	C₂²	C₂³×#10596833	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	2	8	no	—
186	C₄	Q₈×#10596833	1	2	yes	—
187	C₄	#17989645	1	4	yes	—
188	C₄	C₂×Q₈×#64408	1	1	yes	characteristic
189	C₄	C₂×#14556679	1	1	yes	characteristic
190	C₂	—	2	2	no	—
191	C₂	C₂×Q₈×#10596833	1	1	yes	characteristic
192	C₂	C₂×#17989645	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	#533×C₇₃	C₄
prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂×#10596831	C₂⁴
hypercenter	#533	#64408
hypocenter	C₇₃	#33474
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #22626346	C₂	semidirect
2. Q₈×#11924031	C₂	semidirect
26. C₂×#11924031	Q₈	semidirect
104. C₇₃	#33474	semidirect

Maximal subgroups

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

subgroup	index
#22626346	2
Q₈×#11924031	2
#22626266	2
#22626263	2
#22626140	2
C₂×C₄×#11924031	2
#22618866	2
#33474	73