Subgroups — #34011778

Order 18688.

246 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	1	yes	maximal, characteristic
2	C₂×C₄⋊#11924031	C₂	1	1	yes	maximal, characteristic
3	#22624960	C₂	1	2	yes	maximal, complemented
4	#22624953	C₂	1	2	yes	maximal, complemented
5	M₁₆×#10596831	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924031	—	2	1	no	—
7	C₄⋊#11924031	—	2	1	no	—
8	C₄⋊#11924031	C₂²	1	1	yes	characteristic
9	C₄⋊#11924031	C₂²	1	2	yes	—
10	C₄⋊#11924031	C₂²	1	1	yes	characteristic
11	C₂²⋊#11924031	—	2	1	no	—
12	C₄×#11924031	—	2	1	no	—
13	#17988640	C₄	1	1	yes	characteristic
14	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
15	#17988308	C₄	1	2	yes	—
16	C₈×#10596831	C₂²	1	2	yes	—
17	C₄×M₁₆×C₇₃	C₄	1	1	yes	characteristic
18	C₂×#11924031	—	2	2	no	—
19	C₂×#11924031	—	2	1	no	—
20	C₂×#11924031	—	2	1	no	—
21	C₂×#11924031	—	4	2	no	—
22	C₂²×#10596831	D₄	1	1	yes	characteristic
23	C₂²×#10596831	Q₈	1	1	yes	characteristic
24	C₄⋊#10596831	—	2	1	no	—
25	C₂²⋊#10596831	—	2	1	no	—
26	C₄×#10596831	C₂×C₄	1	2	yes	—
27	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
28	C₄×#10596831	D₄	1	1	yes	characteristic
29	C₄×#10596831	Q₈	1	1	yes	characteristic
30	C₄×#10596831	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₇₃	C₈	1	1	yes	characteristic
32	C₂×M₁₆×C₇₃	C₈	1	1	yes	characteristic
33	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
34	#97×C₇₃	C₈	1	2	yes	—
35	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
36	#11924031	—	4	1	no	—
37	#11924031	—	4	2	no	—
38	#11924031	—	4	1	no	—
39	C₂×#10596831	—	2	1	no	—
40	C₂×#10596831	—	2	1	no	—
41	C₂×#10596831	—	2	2	no	—
42	C₂×#10596831	—	4	1	no	—
43	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
44	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
45	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
46	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
50	M₁₆×C₇₃	—	2	2	no	—
51	M₁₆×C₇₃	C₁₆	1	4	yes	complemented
52	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
53	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
54	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
55	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
56	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
57	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
58	C₂³×C₇₃	C₄⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₇₃	—	2	1	no	—
60	C₂×C₄×C₇₃	—	2	1	no	—
61	C₂×C₄×C₇₃	—	4	1	no	—
62	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
66	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
67	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
70	C₈×C₇₃	—	2	2	no	—
71	C₈×C₇₃	C₂×C₁₆	1	4	yes	complemented
72	#10596831	—	2	1	no	—
73	#10596831	—	2	1	no	—
74	#10596831	—	2	1	no	—
75	#10596831	—	4	2	no	—
76	#10596831	AGL(1,ℤ/8)	1	1	yes	characteristic
77	#10596831	#137	1	1	yes	characteristic
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	—	2	1	no	—
85	C₄×C₇₃	C₄⋊C₁₆	1	2	yes	—
86	C₄×C₇₃	#428	1	1	yes	characteristic
87	C₄×C₇₃	#430	1	1	yes	characteristic
88	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
89	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
90	#12015	—	73	1	no	maximal, Sylow
91	C₂×C₇₃	—	2	2	no	—
92	C₂×C₇₃	#1565	1	1	yes	characteristic
93	C₂×C₇₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
94	C₂×C₇₃	#2153	1	1	yes	characteristic
95	#2162	—	73	1	no	—
96	C₂×C₄⋊C₁₆	—	73	1	no	—
97	C₈×M₁₆	—	73	1	no	—
98	#1370	—	73	2	no	—
99	#1367	—	73	2	no	—
100	C₇₃	#12015	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₁₆	—	146	1	no	—
102	#431	—	73	1	no	—
103	C₄×M₁₆	#64408	1	1	yes	characteristic
104	C₂×C₄×C₈	—	73	1	no	—
105	C₄⋊C₁₆	—	73	1	no	—
106	C₄⋊C₁₆	—	73	2	no	—
107	C₄⋊C₁₆	—	73	1	no	—
108	C₄⋊C₁₆	—	146	1	no	—
109	C₂²⋊C₁₆	—	146	1	no	—
110	C₄×C₁₆	—	146	1	no	—
111	#322	—	73	2	no	—
112	C₈²	—	73	2	no	—
113	C₂×M₁₆	#10596833	1	1	yes	characteristic
114	C₂×M₁₆	#10596833	1	1	yes	characteristic
115	C₂²×C₈	—	73	1	no	—
116	C₂²×C₈	—	73	1	no	—
117	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
118	C₂×C₁₆	—	292	2	no	—
119	C₂×C₁₆	—	146	1	no	—
120	C₂×C₁₆	—	146	1	no	—
121	C₂×C₁₆	—	146	2	no	—
122	C₄⋊C₈	—	146	1	no	—
123	C₂²⋊C₈	—	146	1	no	—
124	#97	#10596833	1	2	yes	—
125	C₄×C₈	—	73	1	no	—
126	C₄×C₈	—	73	1	no	—
127	C₄×C₈	—	73	1	no	—
128	C₄×C₈	—	73	1	no	—
129	C₄×C₈	—	73	2	no	—
130	C₄×C₈	C₂×#64408	1	2	yes	—
131	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
132	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
133	C₂²×C₄	#11924038	1	1	yes	characteristic
134	M₁₆	—	2	2	no	—
135	M₁₆	#11924031	1	4	yes	complemented
136	C₂×C₈	—	292	1	no	—
137	C₂×C₈	—	73	1	no	—
138	C₂×C₈	—	73	1	no	—
139	C₂×C₈	—	73	1	no	—
140	C₂×C₈	—	73	1	no	—
141	C₂×C₈	—	146	2	no	—
142	C₂×C₈	—	146	1	no	—
143	C₂×C₈	—	146	1	no	—
144	C₂×C₈	C₂×#10596833	1	2	yes	—
145	C₂×C₈	C₂×#10596833	1	2	yes	—
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	2	no	—
151	C₁₆	—	292	1	no	—
152	C₁₆	—	292	1	no	—
153	C₂³	C₄⋊#10596833	1	1	yes	characteristic
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	C₄⋊#10596833	1	1	yes	characteristic
158	C₂×C₄	C₄⋊#10596833	1	1	yes	characteristic
159	C₂×C₄	C₄⋊#10596833	1	1	yes	characteristic
160	C₂×C₄	C₂×#11924031	1	2	yes	—
161	C₂×C₄	#14556665	1	2	yes	—
162	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
163	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
164	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
165	C₈	—	2	2	no	—
166	C₈	—	292	2	no	—
167	C₈	—	73	1	no	—
168	C₈	—	73	1	no	—
169	C₈	—	146	1	no	—
170	C₈	—	146	1	no	—
171	C₈	—	146	1	no	—
172	C₈	C₂×#11924031	1	4	yes	complemented
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	—	2	1	no	—
180	C₄	C₄⋊#11924031	1	2	yes	—
181	C₄	#17989598	1	1	yes	characteristic
182	C₄	#17989602	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	#22625482	1	1	yes	characteristic
187	C₂	C₂×C₄⋊#11924031	1	1	yes	characteristic
188	C₂	#22626272	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#64408
commutator	C₄×C₇₃	C₂²×C₁₆
Frattini	C₄²	C₂²×#64408
Fitting	C₄×M₁₆×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄⋊C₈
prime radical	C₂³×C₇₃	C₄⋊C₈
cosocle	C₄×#10596831	C₂³
hypercenter	C₄×M₁₆	#64408
hypocenter	C₇₃	#12015
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22624960	C₂	semidirect
4. #22624953	C₂	semidirect
51. M₁₆×C₇₃	C₁₆	semidirect
71. C₈×C₇₃	C₂×C₁₆	semidirect
100. C₇₃	#12015	semidirect
135. M₁₆	#11924031	semidirect
172. C₈	C₂×#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
C₂×C₄⋊#11924031	2
#22624960	2
#22624953	2
M₁₆×#10596831	2
#12015	73