Subgroups — #34012628

Order 18688.

382 conjugacy classes of subgroups.

The subgroup lattice has 196 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	1	yes	maximal, characteristic
3	C₂×C₄×#11924031	C₂	1	1	yes	maximal, characteristic
4	#22624969	C₂	1	2	yes	maximal
5	#22609618	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924031	C₂²	1	2	yes	—
7	C₂²×#11924031	C₂²	1	2	yes	—
8	C₄⋊#11924031	C₄	1	8	yes	complemented
9	#17989475	—	2	2	no	—
10	C₄×#11924031	—	2	2	no	—
11	C₂×#14556502	C₄	1	1	yes	characteristic
12	C₂×#14556502	C₄	1	1	yes	characteristic
13	C₂×#14556502	C₄	1	1	yes	characteristic
14	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
15	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
16	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
17	#376×C₇₃	C₄	1	1	yes	characteristic
18	C₂×#11924031	—	2	2	no	—
19	C₂×#11924031	—	2	2	no	—
20	C₂×#11924031	—	2	2	no	—
21	C₂×#11924031	—	2	2	no	—
22	C₂×#11924031	C₂×C₄	1	4	yes	—
23	C₂×#11924031	C₂×C₄	1	4	yes	—
24	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
25	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
26	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
27	C₂²×#10596831	C₂³	1	1	yes	characteristic
28	#14556502	—	2	2	no	—
29	#14556502	—	2	2	no	—
30	#14556502	C₈	1	4	yes	—
31	C₄×#10596831	—	2	2	no	—
32	C₄×#10596831	C₂×C₄	1	4	yes	—
33	C₄×#10596831	D₄	1	2	yes	—
34	C₄×#10596831	Q₈	1	2	yes	—
35	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
38	#95×C₇₃	C₈	1	4	yes	—
39	#11924031	—	2	4	no	—
40	#11924031	—	2	4	no	—
41	C₂×#10596831	—	2	1	no	—
42	C₂×#10596831	—	2	1	no	—
43	C₂×#10596831	—	2	1	no	—
44	C₂×#10596831	—	2	1	no	—
45	C₂×#10596831	C₄²	1	2	yes	—
46	C₂×#10596831	C₄⋊C₄	1	2	yes	—
47	C₂×#10596831	C₂×C₈	1	2	yes	—
48	C₂×#10596831	C₂×C₈	1	2	yes	—
49	C₂×#10596831	M₁₆	1	2	yes	—
50	C₂×#10596831	C₂²×C₄	1	2	yes	—
51	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
52	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
53	C₂×#10596831	#41	1	2	yes	—
54	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₇₃	—	2	1	no	—
59	C₄²×C₇₃	—	2	1	no	—
60	C₄²×C₇₃	C₄²	1	2	yes	—
61	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
62	C₄²×C₇₃	C₂×C₈	1	2	yes	—
63	C₄²×C₇₃	M₁₆	1	2	yes	—
64	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₇₃	—	2	4	no	—
66	C₂×C₄×C₇₃	—	2	4	no	—
67	C₂×C₄×C₇₃	—	2	2	no	—
68	C₂×C₄×C₇₃	—	2	4	no	—
69	C₂×C₄×C₇₃	C₄×C₈	1	4	yes	—
70	C₂×C₄×C₇₃	C₄⋊C₈	1	4	yes	—
71	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
77	#10596831	—	2	2	no	—
78	#10596831	—	2	2	no	—
79	#10596831	C₄⋊C₈	1	4	yes	—
80	#10596831	C₄×D₄	1	2	yes	—
81	#10596831	C₄×Q₈	1	2	yes	—
82	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
83	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
84	C₂²×C₇₃	#431	1	1	yes	characteristic
85	C₂²×C₇₃	#504	1	4	yes	—
86	C₄×C₇₃	—	2	4	no	—
87	C₄×C₇₃	—	2	4	no	—
88	C₄×C₇₃	—	4	2	no	—
89	C₄×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
90	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
91	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
92	C₄×C₇₃	#431	1	1	yes	characteristic
93	#12654	—	73	1	no	maximal, Sylow
94	C₂×C₇₃	C₄×C₄⋊C₈	1	1	yes	characteristic
95	C₂×C₇₃	#2108	1	2	yes	—
96	C₂×C₇₃	#2151	1	2	yes	—
97	C₂×C₇₃	#2164	1	2	yes	—
98	C₂×C₄⋊C₁₆	—	73	2	no	—
99	C₂×#346	—	73	1	no	—
100	C₂×C₄×C₁₆	—	73	1	no	—
101	#1726	—	73	1	no	—
102	#1374	—	73	2	no	—
103	C₇₃	#12654	1	1	yes	characteristic, complemented, Sylow
104	C₂²×C₁₆	—	73	2	no	—
105	C₂²×C₁₆	—	73	2	no	—
106	C₂×#97	—	73	1	no	—
107	C₂×#97	—	73	1	no	—
108	C₂×#97	—	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	#376	#64408	1	1	yes	characteristic
113	C₄⋊C₁₆	—	73	8	no	—
114	#346	—	146	2	no	—
115	C₄×C₁₆	—	146	2	no	—
116	C₂²×C₈	—	73	1	no	—
117	C₂²×C₈	—	73	1	no	—
118	C₂²×C₈	—	73	1	no	—
119	C₂²×C₈	—	73	1	no	—
120	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
121	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
122	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
123	C₂×C₁₆	—	73	4	no	—
124	C₂×C₁₆	—	73	4	no	—
125	C₂×C₁₆	—	146	2	no	—
126	C₂×C₁₆	—	146	2	no	—
127	C₂×C₁₆	—	146	2	no	—
128	C₂×C₁₆	—	146	2	no	—
129	#97	—	73	4	no	—
130	#97	—	146	2	no	—
131	#97	—	146	2	no	—
132	C₄×C₈	—	73	4	no	—
133	C₄×C₈	—	73	2	no	—
134	C₄×C₈	—	73	2	no	—
135	C₄×C₈	—	146	2	no	—
136	#95	#10596833	1	4	yes	—
137	C₂²×C₄	C₂×#10596833	1	2	yes	—
138	C₂²×C₄	C₂×#10596833	1	2	yes	—
139	C₂²×C₄	C₂×#10596833	1	2	yes	—
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	1	no	—
156	C₄²	C₄×#64408	1	2	yes	—
157	C₄²	C₄⋊#64408	1	2	yes	—
158	C₄²	#11924036	1	2	yes	—
159	C₄²	#11924038	1	2	yes	—
160	C₁₆	—	146	4	no	—
161	C₁₆	—	146	4	no	—
162	C₂³	C₂²×#10596833	1	1	yes	characteristic
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	C₄×#10596833	1	4	yes	—
168	C₂×C₄	C₄⋊#10596833	1	4	yes	—
169	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
170	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
171	C₂×C₄	#14556670	1	1	yes	characteristic
172	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
173	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
174	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
175	C₈	—	73	2	no	—
176	C₈	—	73	2	no	—
177	C₈	—	73	4	no	—
178	C₈	—	146	2	no	—
179	C₈	—	146	2	no	—
180	C₂²	C₂×C₄×#10596833	1	1	yes	characteristic
181	C₂²	C₂×C₄⋊#10596833	1	1	yes	characteristic
182	C₂²	#17989603	1	1	yes	characteristic
183	C₂²	#17989645	1	4	yes	—
184	C₄	—	2	4	no	—
185	C₄	—	2	4	no	—
186	C₄	—	4	2	no	—
187	C₄	C₄×C₄⋊#64408	1	1	yes	characteristic
188	C₄	C₄×#11924038	1	1	yes	characteristic
189	C₄	#17989588	1	1	yes	characteristic
190	C₄	#17989606	1	1	yes	characteristic
191	C₂	C₄×C₄⋊#10596833	1	1	yes	characteristic
192	C₂	#22626140	1	2	yes	—
193	C₂	#22626266	1	2	yes	—
194	C₂	#22626296	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#64408
commutator	C₂²×C₇₃	C₂×C₄×C₈
Fitting	#376×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂²×#10596831	C₂³
hypercenter	#376	#64408
hypocenter	C₇₃	#12654
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
8. C₄⋊#11924031	C₄	semidirect
103. C₇₃	#12654	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
C₂×C₄×#11924031	2
#22624969	2
#22609618	2
#12654	73