Subgroups — #34012623

Order 18688.

414 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11924031	C₂	1	2	yes	maximal
2	C₂×#17989475	C₂	1	2	yes	maximal
3	#22624969	C₂	1	2	yes	maximal
4	C₄²×#10596831	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924031	C₂²	1	4	yes	—
6	C₄⋊#11924031	C₄	1	8	yes	complemented
7	#17989475	—	2	4	no	—
8	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
9	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
10	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
11	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
12	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
13	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
14	C₄³×C₇₃	C₄	1	1	yes	characteristic
15	C₂×#11924031	—	2	4	no	—
16	C₂×#11924031	—	2	4	no	—
17	C₂×#11924031	C₂×C₄	1	8	yes	—
18	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
19	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
20	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
21	C₂²×#10596831	C₂³	1	1	yes	characteristic
22	C₄×#10596831	—	2	2	no	—
23	C₄×#10596831	—	2	2	no	—
24	C₄×#10596831	—	2	2	no	—
25	C₄×#10596831	C₈	1	4	yes	—
26	C₄×#10596831	C₂×C₄	1	4	yes	—
27	C₄×#10596831	D₄	1	2	yes	—
28	C₄×#10596831	Q₈	1	2	yes	—
29	C₂×C₄²×C₇₃	C₈	1	2	yes	—
30	C₂×C₄²×C₇₃	C₈	1	2	yes	—
31	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
34	#11924031	—	2	8	no	—
35	C₂×#10596831	—	2	1	no	—
36	C₂×#10596831	—	2	1	no	—
37	C₂×#10596831	—	2	1	no	—
38	C₂×#10596831	—	2	1	no	—
39	C₂×#10596831	C₄²	1	2	yes	—
40	C₂×#10596831	C₄⋊C₄	1	2	yes	—
41	C₂×#10596831	C₂×C₈	1	2	yes	—
42	C₂×#10596831	C₂×C₈	1	2	yes	—
43	C₂×#10596831	M₁₆	1	2	yes	—
44	C₂×#10596831	C₂²×C₄	1	2	yes	—
45	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
46	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
47	C₂×#10596831	#41	1	2	yes	—
48	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₇₃	—	2	4	no	—
54	C₄²×C₇₃	—	2	4	no	—
55	C₄²×C₇₃	—	2	1	no	—
56	C₄²×C₇₃	—	2	1	no	—
57	C₄²×C₇₃	C₄²	1	2	yes	—
58	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
59	C₄²×C₇₃	C₂×C₈	1	2	yes	—
60	C₄²×C₇₃	M₁₆	1	2	yes	—
61	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
62	C₂×C₄×C₇₃	—	2	2	no	—
63	C₂×C₄×C₇₃	—	2	2	no	—
64	C₂×C₄×C₇₃	—	2	2	no	—
65	C₂×C₄×C₇₃	—	2	2	no	—
66	C₂×C₄×C₇₃	—	2	2	no	—
67	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
68	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
69	C₂×C₄×C₇₃	C₄⋊C₈	1	4	yes	—
70	C₂×C₄×C₇₃	M₃₂	1	4	yes	—
71	C₂×C₄×C₇₃	M₃₂	1	4	yes	—
72	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
78	#10596831	—	2	2	no	—
79	#10596831	—	2	2	no	—
80	#10596831	C₄⋊C₈	1	4	yes	—
81	#10596831	C₄×D₄	1	2	yes	—
82	#10596831	C₄×Q₈	1	2	yes	—
83	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
84	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
85	C₂²×C₇₃	#431	1	1	yes	characteristic
86	C₂²×C₇₃	C₂×M₃₂	1	2	yes	—
87	C₂²×C₇₃	C₂×M₃₂	1	2	yes	—
88	C₄×C₇₃	—	2	2	no	—
89	C₄×C₇₃	—	2	4	no	—
90	C₄×C₇₃	—	2	4	no	—
91	C₄×C₇₃	#346	1	4	yes	complemented
92	C₄×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
93	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
94	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
95	C₄×C₇₃	#431	1	1	yes	characteristic
96	#12653	—	73	1	no	maximal, Sylow
97	C₂×C₇₃	C₄×C₄⋊C₈	1	1	yes	characteristic
98	C₂×C₇₃	C₂×#346	1	1	yes	characteristic
99	C₂×C₇₃	C₄×M₃₂	1	1	yes	characteristic
100	C₂×C₇₃	#2150	1	2	yes	—
101	C₂×C₇₃	#2163	1	2	yes	—
102	C₂×C₄⋊C₁₆	—	73	2	no	—
103	C₂×#346	—	73	2	no	—
104	C₄²×C₈	—	73	1	no	—
105	#1374	—	73	2	no	—
106	C₇₃	#12653	1	1	yes	characteristic, complemented, Sylow
107	C₂²×C₁₆	—	73	4	no	—
108	C₂×C₄×C₈	—	73	1	no	—
109	C₂×C₄×C₈	—	73	1	no	—
110	C₂×C₄×C₈	—	73	1	no	—
111	C₂×C₄×C₈	—	73	1	no	—
112	C₂×C₄×C₈	—	73	1	no	—
113	C₂×C₄×C₈	—	73	1	no	—
114	C₄³	#64408	1	1	yes	characteristic
115	C₄⋊C₁₆	—	73	8	no	—
116	#346	—	146	4	no	—
117	C₂²×C₈	—	73	1	no	—
118	C₂²×C₈	—	73	1	no	—
119	C₂²×C₈	—	73	1	no	—
120	C₂²×C₈	—	73	1	no	—
121	C₂×C₄²	#10596833	1	2	yes	—
122	C₂×C₄²	#10596833	1	2	yes	—
123	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
124	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
125	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
126	C₂×C₁₆	—	73	8	no	—
127	C₂×C₁₆	—	146	4	no	—
128	C₂×C₁₆	—	146	4	no	—
129	C₄×C₈	—	73	4	no	—
130	C₄×C₈	—	73	2	no	—
131	C₄×C₈	—	73	2	no	—
132	C₄×C₈	—	73	4	no	—
133	C₄×C₈	—	146	2	no	—
134	C₄×C₈	—	146	2	no	—
135	C₄×C₈	—	146	2	no	—
136	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
137	C₂²×C₄	C₂×#10596833	1	2	yes	—
138	C₂²×C₄	C₂×#10596833	1	2	yes	—
139	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
140	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
141	C₂×C₈	—	73	2	no	—
142	C₂×C₈	—	73	2	no	—
143	C₂×C₈	—	73	2	no	—
144	C₂×C₈	—	73	2	no	—
145	C₂×C₈	—	73	1	no	—
146	C₂×C₈	—	73	1	no	—
147	C₂×C₈	—	73	2	no	—
148	C₂×C₈	—	73	2	no	—
149	C₂×C₈	—	73	2	no	—
150	C₂×C₈	—	146	1	no	—
151	C₂×C₈	—	146	1	no	—
152	C₂×C₈	—	146	1	no	—
153	C₂×C₈	—	146	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	2	4	no	—
156	C₄²	—	2	4	no	—
157	C₄²	—	2	1	no	—
158	C₄²	C₄×#64408	1	2	yes	—
159	C₄²	C₄⋊#64408	1	2	yes	—
160	C₄²	#11924036	1	2	yes	—
161	C₄²	#11924038	1	2	yes	—
162	C₁₆	—	146	8	no	—
163	C₂³	C₂²×#10596833	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	C₄×#10596833	1	2	yes	—
170	C₂×C₄	C₄×#10596833	1	2	yes	—
171	C₂×C₄	C₄⋊#10596833	1	4	yes	—
172	C₂×C₄	#14556665	1	4	yes	—
173	C₂×C₄	#14556665	1	4	yes	—
174	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
176	C₂×C₄	#14556670	1	1	yes	characteristic
177	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
178	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
179	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
180	C₈	—	73	2	no	—
181	C₈	—	73	2	no	—
182	C₈	—	73	4	no	—
183	C₈	—	146	2	no	—
184	C₈	—	146	2	no	—
185	C₂²	C₂×C₄×#10596833	1	1	yes	characteristic
186	C₂²	C₂×C₄⋊#10596833	1	1	yes	characteristic
187	C₂²	#17989603	1	1	yes	characteristic
188	C₂²	C₂×#14556665	1	2	yes	—
189	C₂²	C₂×#14556665	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	2	4	no	—
192	C₄	—	2	4	no	—
193	C₄	#17989475	1	4	yes	complemented
194	C₄	C₄×C₄⋊#64408	1	1	yes	characteristic
195	C₄	C₄×#11924038	1	1	yes	characteristic
196	C₄	#17989588	1	1	yes	characteristic
197	C₄	#17989606	1	1	yes	characteristic
198	C₂	C₄×C₄⋊#10596833	1	1	yes	characteristic
199	C₂	C₂×#17989475	1	1	yes	characteristic
200	C₂	C₄×#14556665	1	1	yes	characteristic
201	C₂	#22626263	1	2	yes	—
202	C₂	#22626292	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#64408
commutator	C₂²×C₇₃	C₂×C₄×C₈
Fitting	C₄³×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂²×#10596831	C₂³
hypercenter	C₄³	#64408
hypocenter	C₇₃	#12653
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
6. C₄⋊#11924031	C₄	semidirect
91. C₄×C₇₃	#346	semidirect
106. C₇₃	#12653	semidirect
193. C₄	#17989475	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11924031	2
C₂×#17989475	2
#22624969	2
C₄²×#10596831	2
#12653	73