Subgroups — #34000193

Order 18688.

262 conjugacy classes of subgroups.

The subgroup lattice has 166 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×C₈×#10596831	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11924031	—	2	2	no	—
4	C₄⋊#11924031	—	2	2	no	—
5	C₄⋊#11924031	C₄	1	4	yes	—
6	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
7	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
8	C₈×#10596831	—	2	2	no	—
9	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
10	C₂×#11924031	—	2	4	no	—
11	C₂×#11924031	—	4	2	no	—
12	C₂×#11924031	—	4	2	no	—
13	C₂²×#10596831	C₈	1	2	yes	—
14	C₂²×#10596831	D₄	1	1	yes	characteristic
15	C₂²×#10596831	Q₈	1	1	yes	characteristic
16	C₄×#10596831	—	2	1	no	—
17	C₄×#10596831	—	2	1	no	—
18	C₄×#10596831	C₂×C₄	1	2	yes	—
19	C₄×#10596831	D₄	1	2	yes	—
20	C₂²×C₈×C₇₃	—	2	1	no	—
21	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₇₃	—	2	1	no	—
23	C₄×C₈×C₇₃	C₈	1	2	yes	—
24	#11924031	—	4	4	no	—
25	C₂×#10596831	—	2	2	no	—
26	C₂×#10596831	—	2	1	no	—
27	C₂×#10596831	—	2	1	no	—
28	C₂×#10596831	—	2	2	no	—
29	C₂×#10596831	C₁₆	1	4	yes	complemented
30	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
31	C₂×#10596831	C₄⋊C₄	1	2	yes	—
32	C₂×#10596831	D₈	1	1	yes	characteristic
33	C₂×#10596831	SD₁₆	1	2	yes	—
34	C₂×#10596831	Q₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
38	C₂×C₈×C₇₃	—	2	4	no	—
39	C₂×C₈×C₇₃	—	2	2	no	—
40	C₄²×C₇₃	C₄²	1	1	yes	characteristic
41	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
43	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
44	C₂³×C₇₃	C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₇₃	—	2	2	no	—
46	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
53	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#108	1	1	yes	characteristic
55	C₂×C₄×C₇₃	#108	1	1	yes	characteristic
56	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
57	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
58	C₈×C₇₃	—	2	4	no	—
59	#10596831	—	2	2	no	—
60	#10596831	—	2	4	no	—
61	#10596831	#102	1	2	yes	—
62	#10596831	#103	1	2	yes	—
63	C₂²×C₇₃	#334	1	1	yes	characteristic
64	C₂²×C₇₃	#335	1	1	yes	characteristic
65	C₂²×C₇₃	#336	1	1	yes	characteristic
66	C₂²×C₇₃	C₄⋊C₁₆	1	2	yes	—
67	C₂²×C₇₃	#364	1	2	yes	—
68	C₄×C₇₃	—	2	1	no	—
69	C₄×C₇₃	—	2	1	no	—
70	C₄×C₇₃	#334	1	1	yes	characteristic
71	C₄×C₇₃	#335	1	1	yes	characteristic
72	C₄×C₇₃	#340	1	1	yes	characteristic
73	C₄×C₇₃	#341	1	1	yes	characteristic
74	C₄×C₇₃	C₄×C₁₆	1	1	yes	characteristic
75	C₄×C₇₃	#346	1	1	yes	characteristic
76	C₄×C₇₃	C₂²⋊C₁₆	1	2	yes	—
77	#7151	—	73	1	no	maximal, Sylow
78	C₂×C₇₃	#1277	1	1	yes	characteristic
79	C₂×C₇₃	#1367	1	2	yes	—
80	C₂×C₇₃	#1370	1	2	yes	—
81	C₂×C₇₃	#1374	1	1	yes	characteristic
82	C₂×C₇₃	#1376	1	1	yes	characteristic
83	C₂×C₄⋊C₁₆	—	73	2	no	—
84	C₂×C₈²	—	73	1	no	—
85	C₇₃	#7151	1	1	yes	characteristic, complemented, Sylow
86	C₂²×C₁₆	—	146	2	no	—
87	C₂×C₄×C₈	—	73	1	no	—
88	C₂×C₄×C₈	—	73	1	no	—
89	C₂×C₄×C₈	#64408	1	1	yes	characteristic
90	C₄⋊C₁₆	—	73	4	no	—
91	C₄⋊C₁₆	—	146	2	no	—
92	C₈²	—	146	2	no	—
93	C₂²×C₈	—	2	1	no	—
94	C₂²×C₈	—	73	1	no	—
95	C₂²×C₈	—	73	1	no	—
96	C₂²×C₈	—	73	2	no	—
97	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
98	C₂×C₁₆	—	292	2	no	—
99	C₂×C₁₆	—	292	2	no	—
100	C₂×C₁₆	—	146	4	no	—
101	C₄×C₈	—	2	1	no	—
102	C₄×C₈	—	73	2	no	—
103	C₄×C₈	—	73	2	no	—
104	C₄×C₈	—	146	1	no	—
105	C₄×C₈	—	146	1	no	—
106	C₄×C₈	#10596833	1	2	yes	—
107	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
108	C₂²×C₄	#11924036	1	1	yes	characteristic
109	C₂²×C₄	#11924037	1	1	yes	characteristic
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	73	2	no	—
113	C₂×C₈	—	73	2	no	—
114	C₂×C₈	—	73	1	no	—
115	C₂×C₈	—	73	1	no	—
116	C₂×C₈	—	73	2	no	—
117	C₂×C₈	—	73	4	no	—
118	C₂×C₈	—	146	2	no	—
119	C₂×C₈	—	146	1	no	—
120	C₂×C₈	—	146	1	no	—
121	C₂×C₈	—	146	2	no	—
122	C₄²	C₄×#64408	1	1	yes	characteristic
123	C₄²	C₂²⋊#64408	1	1	yes	characteristic
124	C₄²	C₂×#10596833	1	1	yes	characteristic
125	C₄²	#11924038	1	1	yes	characteristic
126	C₁₆	—	292	4	no	—
127	C₂³	#14556658	1	1	yes	characteristic
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
130	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
131	C₂×C₄	#14556646	1	1	yes	characteristic
132	C₂×C₄	C₂²⋊#10596833	1	1	yes	characteristic
133	C₂×C₄	#14556648	1	1	yes	characteristic
134	C₂×C₄	C₄⋊#10596833	1	1	yes	characteristic
135	C₂×C₄	#14556659	1	1	yes	characteristic
136	C₂×C₄	#14556660	1	1	yes	characteristic
137	C₂×C₄	#14556661	1	1	yes	characteristic
138	C₂×C₄	#14556662	1	1	yes	characteristic
139	C₂×C₄	#14556664	1	2	yes	—
140	C₂×C₄	#14556666	1	2	yes	—
141	C₈	—	2	4	no	—
142	C₈	—	73	2	no	—
143	C₈	—	73	2	no	—
144	C₈	—	146	2	no	—
145	C₈	—	146	4	no	—
146	C₂²	#17989449	1	1	yes	characteristic
147	C₂²	#17989451	1	1	yes	characteristic
148	C₂²	#17989453	1	1	yes	characteristic
149	C₂²	#17989508	1	2	yes	—
150	C₂²	#17989510	1	2	yes	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	#17989448	1	1	yes	characteristic
154	C₄	#17989450	1	1	yes	characteristic
155	C₄	#17989461	1	1	yes	characteristic
156	C₄	#17989463	1	1	yes	characteristic
157	C₄	#17989473	1	1	yes	characteristic
158	C₄	#17989476	1	1	yes	characteristic
159	C₄	#17989482	1	2	yes	—
160	C₂	#22624761	1	1	yes	characteristic
161	C₂	#22624954	1	2	yes	—
162	C₂	#22624961	1	2	yes	—
163	C₂	#22624970	1	1	yes	characteristic
164	C₂	#22624975	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄⋊#64408
commutator	C₄×C₇₃	C₄×C₁₆
Frattini	C₂×C₄²	C₂×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₄⋊C₈
cosocle	C₂×C₄×#10596831	C₂²
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	#7151
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
29. C₂×#10596831	C₁₆	semidirect
85. C₇₃	#7151	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×C₈×#10596831	2
#7151	73