Subgroups — #34030321

Order 18688.

490 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626346	C₂	1	8	yes	maximal, complemented
2	#22626266	C₂	1	2	yes	maximal
3	#22626263	C₂	1	1	yes	maximal, characteristic
4	#22626140	C₂	1	2	yes	maximal
5	C₂×#17989475	C₂	1	1	yes	maximal, characteristic
6	#22618866	C₂	1	1	yes	maximal, characteristic
7	C₂×#14556665	C₂²	1	2	yes	—
8	C₂²×#11924031	C₂²	1	1	yes	characteristic
9	C₂²×#11924031	C₂²	1	1	yes	characteristic
10	C₄⋊#11924031	C₂²	1	4	yes	—
11	C₄⋊#11924031	C₂²	1	8	yes	—
12	#17989475	C₂²	1	4	yes	—
13	#17989475	C₂²	1	4	yes	—
14	C₄×#11924031	C₂²	1	4	yes	—
15	#17988718	C₄	1	4	yes	—
16	Q₈×#10596831	C₂²	1	4	yes	—
17	#17988614	C₄	1	2	yes	—
18	#17988611	C₄	1	1	yes	characteristic
19	#17988543	C₂²	1	2	yes	—
20	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
21	#533×C₇₃	C₄	1	1	yes	characteristic
22	#14556665	—	2	4	no	—
23	C₂×#11924031	—	2	2	no	—
24	C₂×#11924031	Q₈	1	4	yes	complemented
25	C₂×#11924031	C₂³	1	2	yes	—
26	C₂×#11924031	C₂³	1	4	yes	—
27	C₂×#11924031	C₂³	1	2	yes	—
28	C₂×#11924012	C₂×C₄	1	2	yes	—
29	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
30	C₂²×#10596831	C₂³	1	1	yes	characteristic
31	C₄⋊#10596831	C₂×C₄	1	4	yes	—
32	C₄⋊#10596831	C₂×C₄	1	8	yes	—
33	#14556502	C₂×C₄	1	4	yes	—
34	C₄×#10596831	C₂×C₄	1	2	yes	—
35	C₄×#10596831	C₂³	1	4	yes	—
36	C₄×#10596831	C₂³	1	2	yes	—
37	C₄⋊Q₈×C₇₃	C₈	1	2	yes	—
38	#125×C₇₃	C₈	1	2	yes	—
39	C₂²⋊Q₈×C₇₃	C₈	1	4	yes	—
40	C₄×Q₈×C₇₃	C₂×C₄	1	4	yes	—
41	#117×C₇₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
43	#11924031	—	2	2	no	—
44	#11924031	—	2	4	no	—
45	#11924031	C₂×Q₈	1	4	yes	—
46	#11924012	—	2	4	no	—
47	C₂×#10596831	—	2	2	no	—
48	C₂×#10596831	C₂²×C₄	1	2	yes	—
49	C₂×#10596831	C₂²×C₄	1	4	yes	—
50	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
51	C₂×#10596831	C₂×Q₈	1	2	yes	—
52	C₂×#10596831	#41	1	2	yes	—
53	C₂×#10596831	C₂⁴	1	1	yes	characteristic
54	C₂×Q₈×C₇₃	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₄⋊C₄×C₇₃	C₂×C₈	1	4	yes	—
58	C₄⋊C₄×C₇₃	C₂×C₈	1	8	yes	—
59	C₄⋊C₄×C₇₃	C₂×C₈	1	4	yes	—
60	C₂²⋊C₄×C₇₃	C₂×C₈	1	4	yes	—
61	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
63	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
64	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
65	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
66	Q₈×C₇₃	—	4	4	no	—
67	C₂×C₄×C₇₃	—	4	2	no	—
68	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
69	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
70	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
71	C₂×C₄×C₇₃	C₂²×C₈	1	8	yes	—
72	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₇₃	#131	1	2	yes	—
74	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
75	#10596831	—	2	2	no	—
76	#10596831	—	2	4	no	—
77	#10596831	C₄×Q₈	1	2	yes	—
78	#10596831	C₂²×Q₈	1	1	yes	characteristic
79	#10596831	C₂×#41	1	1	yes	characteristic
80	C₂²×C₇₃	—	2	2	no	—
81	C₂²×C₇₃	C₈×Q₈	1	2	yes	—
82	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
83	C₄×C₇₃	—	2	2	no	—
84	C₄×C₇₃	—	2	2	no	—
85	C₄×C₇₃	—	2	8	no	—
86	C₄×C₇₃	C₈×Q₈	1	2	yes	—
87	C₄×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
88	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
89	#33475	—	73	1	no	maximal, Sylow
90	C₂×C₇₃	—	2	2	no	—
91	C₂×C₇₃	C₂×C₈×Q₈	1	1	yes	characteristic
92	C₂×C₇₃	#3407	1	2	yes	—
93	#2960	—	73	1	no	—
94	#2183	—	73	8	no	—
95	#2151	—	73	2	no	—
96	#2150	—	73	1	no	—
97	#2108	—	73	2	no	—
98	C₂×#346	—	73	1	no	—
99	C₇₃	#33475	1	1	yes	characteristic, complemented, Sylow
100	#533	#64408	1	1	yes	characteristic
101	C₂×M₃₂	—	73	2	no	—
102	C₂²×C₁₆	—	73	1	no	—
103	C₂²×C₁₆	—	73	1	no	—
104	#446	—	73	4	no	—
105	C₈×Q₈	—	73	4	no	—
106	#424	—	73	2	no	—
107	#423	—	73	1	no	—
108	#405	—	73	2	no	—
109	C₂×C₄×C₈	—	73	1	no	—
110	C₄⋊C₁₆	—	73	4	no	—
111	C₄⋊C₁₆	—	73	8	no	—
112	#346	—	73	4	no	—
113	#346	—	73	4	no	—
114	C₄×C₁₆	—	73	4	no	—
115	C₂×M₁₆	—	73	2	no	—
116	C₂²×C₈	—	73	1	no	—
117	C₂²×C₈	—	73	1	no	—
118	C₄⋊Q₈	#10596833	1	2	yes	—
119	#125	#10596833	1	2	yes	—
120	C₂²⋊Q₈	#10596833	1	4	yes	—
121	C₄×Q₈	C₂×#64408	1	4	yes	—
122	#117	C₂×#64408	1	2	yes	—
123	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
124	M₃₂	—	146	4	no	—
125	C₂×C₁₆	—	73	2	no	—
126	C₂×C₁₆	—	73	4	no	—
127	C₂×C₁₆	—	73	4	no	—
128	C₂×C₁₆	—	73	2	no	—
129	C₂×C₁₆	—	146	2	no	—
130	C₄⋊C₈	—	73	4	no	—
131	C₄⋊C₈	—	73	8	no	—
132	#97	—	73	4	no	—
133	C₄×C₈	—	73	2	no	—
134	C₄×C₈	—	73	4	no	—
135	C₄×C₈	—	73	2	no	—
136	C₂×Q₈	C₂×#10596833	1	4	yes	—
137	C₂²×C₄	C₂×#10596833	1	2	yes	—
138	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
139	M₁₆	—	146	4	no	—
140	C₂×C₈	—	73	2	no	—
141	C₂×C₈	—	73	2	no	—
142	C₂×C₈	—	73	1	no	—
143	C₂×C₈	—	73	1	no	—
144	C₂×C₈	—	73	2	no	—
145	C₂×C₈	—	73	4	no	—
146	C₂×C₈	—	146	2	no	—
147	C₄⋊C₄	C₂×#10596833	1	4	yes	—
148	C₄⋊C₄	C₂×#10596833	1	8	yes	—
149	C₄⋊C₄	C₂×#10596833	1	4	yes	—
150	C₂²⋊C₄	C₂×#10596833	1	4	yes	—
151	C₄²	C₂×#10596833	1	1	yes	characteristic
152	C₄²	C₂×#10596833	1	1	yes	characteristic
153	C₄²	C₂²×#64408	1	2	yes	—
154	C₄²	C₂²×#64408	1	4	yes	—
155	C₁₆	—	73	4	no	—
156	C₁₆	—	146	2	no	—
157	C₁₆	—	146	4	no	—
158	C₂³	C₂²×#10596833	1	1	yes	characteristic
159	Q₈	—	4	4	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	Q₈×#64408	1	2	yes	—
162	C₂×C₄	C₂²×#10596833	1	2	yes	—
163	C₂×C₄	C₂²×#10596833	1	2	yes	—
164	C₂×C₄	C₂²×#10596833	1	8	yes	—
165	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
166	C₂×C₄	#14556679	1	2	yes	—
167	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
168	C₈	—	73	1	no	—
169	C₈	—	73	1	no	—
170	C₈	—	73	2	no	—
171	C₈	—	146	2	no	—
172	C₈	—	146	4	no	—
173	C₂²	—	2	2	no	—
174	C₂²	Q₈×#10596833	1	2	yes	—
175	C₂²	C₂³×#10596833	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	2	8	no	—
179	C₄	Q₈×#10596833	1	2	yes	—
180	C₄	C₂×Q₈×#64408	1	1	yes	characteristic
181	C₄	C₂×#14556679	1	1	yes	characteristic
182	C₂	—	2	2	no	—
183	C₂	C₂×Q₈×#10596833	1	1	yes	characteristic
184	C₂	#22626898	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	#33475
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
24. C₂×#11924031	Q₈	semidirect
99. C₇₃	#33475	semidirect

Maximal subgroups

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

subgroup	index
#22626346	2
#22626266	2
#22626263	2
#22626140	2
C₂×#17989475	2
#22618866	2
#33475	73