Subgroups — #34011766

Order 18688.

262 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	#22626288	C₂	1	2	yes	maximal
2	#22624960	C₂	1	2	yes	maximal, complemented
3	#22624953	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×#10596831	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#11924031	C₂²	1	4	yes	—
6	C₂²⋊#11924031	—	2	2	no	—
7	C₄×#11924031	—	2	2	no	—
8	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
9	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
10	C₈×#10596831	C₄	1	1	yes	characteristic
11	C₈×#10596831	C₄	1	1	yes	characteristic
12	C₈×#10596831	C₂²	1	1	yes	characteristic
13	C₈×#10596831	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
15	C₂×#11924031	—	2	4	no	—
16	C₂²×#10596831	—	2	1	no	—
17	C₂²×#10596831	D₄	1	1	yes	characteristic
18	C₂²×#10596831	Q₈	1	1	yes	characteristic
19	C₄×#10596831	—	2	1	no	—
20	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
21	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
22	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
23	C₄×#10596831	D₄	1	1	yes	characteristic
24	C₄×#10596831	Q₈	1	1	yes	characteristic
25	C₄×#10596831	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₇₃	C₈	1	1	yes	characteristic
27	C₂²×C₈×C₇₃	C₈	1	1	yes	characteristic
28	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
30	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
31	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	#11924031	—	4	4	no	—
34	C₂×#10596831	—	2	4	no	—
35	C₂×#10596831	—	2	1	no	—
36	C₂×#10596831	—	2	1	no	—
37	C₂×#10596831	—	2	2	no	—
38	C₂×#10596831	—	2	2	no	—
39	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
40	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
41	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
42	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
46	C₂×C₈×C₇₃	—	2	2	no	—
47	C₂×C₈×C₇₃	C₁₆	1	4	yes	complemented
48	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
51	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
52	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
54	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
55	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
56	C₂³×C₇₃	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₇₃	—	2	2	no	—
58	C₂×C₄×C₇₃	—	2	1	no	—
59	C₂×C₄×C₇₃	—	2	1	no	—
60	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
64	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
65	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
68	C₈×C₇₃	—	2	1	no	—
69	C₈×C₇₃	—	2	1	no	—
70	C₈×C₇₃	C₂×C₁₆	1	2	yes	—
71	C₈×C₇₃	C₂×C₁₆	1	2	yes	—
72	#10596831	—	2	2	no	—
73	#10596831	—	2	2	no	—
74	#10596831	—	2	1	no	—
75	#10596831	—	2	1	no	—
76	#10596831	#108	1	2	yes	—
77	#10596831	#135	1	2	yes	—
78	C₂²×C₇₃	—	2	1	no	—
79	C₂²×C₇₃	—	2	1	no	—
80	C₂²×C₇₃	C₄⋊C₁₆	1	2	yes	—
81	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
82	C₄×C₇₃	—	2	1	no	—
83	C₄×C₇₃	—	2	1	no	—
84	C₄×C₇₃	C₄⋊C₁₆	1	2	yes	—
85	C₄×C₇₃	#364	1	2	yes	—
86	C₄×C₇₃	#427	1	1	yes	characteristic
87	C₄×C₇₃	C₂×#108	1	1	yes	characteristic
88	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
89	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
90	#12012	—	73	1	no	maximal, Sylow
91	C₂×C₇₃	—	2	2	no	—
92	C₂×C₇₃	#1564	1	1	yes	characteristic
93	C₂×C₇₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
94	C₂×C₇₃	C₂×#364	1	1	yes	characteristic
95	#2162	—	73	2	no	—
96	C₂×C₈²	—	73	1	no	—
97	#1370	—	73	2	no	—
98	#1367	—	73	2	no	—
99	C₇₃	#12012	1	1	yes	characteristic, complemented, Sylow
100	C₂×C₄×C₈	—	73	1	no	—
101	C₂×C₄×C₈	—	73	1	no	—
102	C₂×C₄×C₈	#64408	1	1	yes	characteristic
103	C₄⋊C₁₆	—	73	4	no	—
104	C₂²⋊C₁₆	—	146	2	no	—
105	C₄×C₁₆	—	146	2	no	—
106	C₈²	—	73	1	no	—
107	C₈²	—	73	1	no	—
108	C₈²	—	73	1	no	—
109	C₈²	—	73	1	no	—
110	C₂²×C₈	—	73	1	no	—
111	C₂²×C₈	—	73	1	no	—
112	C₂²×C₈	—	146	1	no	—
113	C₂²×C₈	#10596833	1	1	yes	characteristic
114	C₂²×C₈	#10596833	1	1	yes	characteristic
115	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
116	C₂×C₁₆	—	146	4	no	—
117	C₄×C₈	—	73	1	no	—
118	C₄×C₈	—	73	1	no	—
119	C₄×C₈	—	73	1	no	—
120	C₄×C₈	—	73	1	no	—
121	C₄×C₈	—	73	1	no	—
122	C₄×C₈	—	73	1	no	—
123	C₄×C₈	—	146	1	no	—
124	C₄×C₈	#10596833	1	1	yes	characteristic
125	C₄×C₈	#10596833	1	1	yes	characteristic
126	C₄×C₈	C₂×#64408	1	1	yes	characteristic
127	C₄×C₈	C₂×#64408	1	1	yes	characteristic
128	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
129	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
130	C₂²×C₄	#11924038	1	1	yes	characteristic
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	73	1	no	—
133	C₂×C₈	—	73	1	no	—
134	C₂×C₈	—	73	1	no	—
135	C₂×C₈	—	73	1	no	—
136	C₂×C₈	—	146	2	no	—
137	C₂×C₈	—	146	2	no	—
138	C₂×C₈	—	146	4	no	—
139	C₂×C₈	—	146	1	no	—
140	C₂×C₈	—	146	1	no	—
141	C₂×C₈	#11924031	1	4	yes	complemented
142	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
143	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
144	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
145	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
146	C₄²	C₄⋊#64408	1	1	yes	characteristic
147	C₄²	C₂×#10596833	1	1	yes	characteristic
148	C₄²	#11924038	1	1	yes	characteristic
149	C₄²	C₂²×#64408	1	1	yes	characteristic
150	C₁₆	—	292	4	no	—
151	C₂³	C₄⋊#10596833	1	1	yes	characteristic
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	C₄⋊#10596833	1	1	yes	characteristic
156	C₂×C₄	C₄⋊#10596833	1	1	yes	characteristic
157	C₂×C₄	C₄⋊#10596833	1	1	yes	characteristic
158	C₂×C₄	C₂×#11924031	1	2	yes	—
159	C₂×C₄	#14556665	1	2	yes	—
160	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
161	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
162	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	73	2	no	—
166	C₈	—	73	2	no	—
167	C₈	—	146	2	no	—
168	C₈	—	146	2	no	—
169	C₈	—	146	1	no	—
170	C₈	—	146	1	no	—
171	C₈	C₂×#11924031	1	2	yes	—
172	C₈	C₂×#11924031	1	2	yes	—
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	C₄⋊#11924031	1	2	yes	—
176	C₂²	C₂×C₄⋊#10596833	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	C₄⋊#11924031	1	2	yes	—
180	C₄	#17989509	1	2	yes	—
181	C₄	#17989597	1	1	yes	characteristic
182	C₄	D₇₃⋊#108	1	1	yes	characteristic
183	C₄	C₂²×#11924031	1	1	yes	characteristic
184	C₄	C₂×#14556665	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	#22625478	1	1	yes	characteristic
187	C₂	C₂×C₄⋊#11924031	1	1	yes	characteristic
188	C₂	C₂×#17989509	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₄⋊#64408
commutator	C₄×C₇₃	C₂²×C₁₆
Frattini	C₄²	C₂²×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄⋊C₈
prime radical	C₂³×C₇₃	C₄⋊C₈
cosocle	C₄×#10596831	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	#12012
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22624960	C₂	semidirect
3. #22624953	C₂	semidirect
47. C₂×C₈×C₇₃	C₁₆	semidirect
99. C₇₃	#12012	semidirect
141. C₂×C₈	#11924031	semidirect

Maximal subgroups

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

subgroup	index
#22626288	2
#22624960	2
#22624953	2
C₂×C₈×#10596831	2
#12012	73