Finite Group Databasebeta

Subgroups — #34012756

Order 18688.

894 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 894 conjugacy classes of subgroups — too many to draw. Only its 160 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626167	C₂	1	3	yes	maximal, complemented
2	#22626161	C₂	1	3	yes	maximal, complemented
3	#22624643	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556665	C₂²	1	3	yes	complemented
5	C₂²×#11924031	C₂²	1	1	yes	characteristic, complemented
6	#17989483	—	2	6	no	—
7	C₂²⋊#11924031	—	2	6	no	—
8	#17989354	—	2	6	no	—
9	#17989354	C₄	1	3	yes	—
10	C₂×#14556598	—	2	6	no	—
11	C₂×#14556598	C₂²	1	3	yes	—
12	#585×C₇₃	C₄	1	1	yes	characteristic
13	#14556665	—	2	6	no	—
14	C₂×#11924031	—	2	3	no	—
15	C₂×#11924031	—	2	3	no	—
16	#14556598	—	2	4	no	—
17	#14556598	—	2	12	no	—
18	#14556598	—	2	12	no	—
19	#14556598	—	2	6	no	—
20	#14556598	—	2	6	no	—
21	C₂×#11924012	—	2	12	no	—
22	C₂×#11924012	—	2	6	no	—
23	C₂×#11924012	C₂×C₄	1	3	yes	—
24	C₂×#11924012	D₄	1	6	yes	—
25	C₂²×#10596831	—	2	4	no	—
26	C₂²×#10596831	D₄	1	6	yes	—
27	C₂²×#10596831	C₂³	1	1	yes	characteristic
28	2₋¹⁺⁴×C₇₃	—	2	3	no	—
29	H(5,𝔽₂)×C₇₃	—	2	3	no	—
30	H(5,𝔽₂)×C₇₃	C₈	1	4	yes	—
31	C₂×#41×C₇₃	—	2	6	no	—
32	C₂×#41×C₇₃	C₂×C₄	1	3	yes	—
33	#11924031	—	4	1	no	—
34	#11924031	—	4	3	no	—
35	#11924012	—	2	12	no	—
36	#11924012	—	2	6	no	—
37	#11924012	—	2	12	no	—
38	C₂×#10596831	—	2	12	no	—
39	C₂×#10596831	—	2	6	no	—
40	C₂×#10596831	—	2	6	no	—
41	C₂×#10596831	C₂²⋊C₄	1	6	yes	—
42	C₂×#10596831	C₂×D₄	1	3	yes	—
43	C₂×#10596831	C₂×D₄	1	3	yes	—
44	#41×C₇₃	—	2	4	no	—
45	#41×C₇₃	—	2	12	no	—
46	#41×C₇₃	—	2	12	no	—
47	#41×C₇₃	—	2	6	no	—
48	#41×C₇₃	—	2	6	no	—
49	C₂×Q₈×C₇₃	—	2	6	no	—
50	C₂×Q₈×C₇₃	M₁₆	1	3	yes	—
51	C₂×D₄×C₇₃	—	2	6	no	—
52	C₂×D₄×C₇₃	—	2	12	no	—
53	C₂×D₄×C₇₃	C₂×C₈	1	6	yes	—
54	C₂×D₄×C₇₃	M₁₆	1	3	yes	—
55	C₂²×C₄×C₇₃	—	2	4	no	—
56	C₂²×C₄×C₇₃	C₂²⋊C₄	1	6	yes	—
57	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
58	C₂³×C₇₃	—	2	4	no	—
59	C₂³×C₇₃	C₂²⋊C₈	1	6	yes	—
60	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
61	Q₈×C₇₃	—	2	4	no	—
62	Q₈×C₇₃	—	2	6	no	—
63	D₄×C₇₃	—	2	12	no	—
64	D₄×C₇₃	—	2	12	no	—
65	D₄×C₇₃	—	2	6	no	—
66	C₂×C₄×C₇₃	—	2	12	no	—
67	C₂×C₄×C₇₃	—	2	6	no	—
68	C₂×C₄×C₇₃	—	2	6	no	—
69	C₂×C₄×C₇₃	C₂²⋊C₈	1	6	yes	—
70	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	3	yes	—
71	C₂×C₄×C₇₃	C₂×M₁₆	1	3	yes	—
72	#10596831	—	2	6	no	—
73	#10596831	#120	1	1	yes	characteristic
74	#10596831	#120	1	3	yes	—
75	C₂²×C₇₃	—	2	6	no	—
76	C₂²×C₇₃	—	4	1	no	—
77	C₂²×C₇₃	—	4	6	no	—
78	C₂²×C₇₃	—	8	4	no	—
79	C₂²×C₇₃	C₂×C₂²⋊C₈	1	3	yes	—
80	C₄×C₇₃	—	2	6	no	—
81	C₄×C₇₃	#379	1	1	yes	characteristic
82	C₄×C₇₃	#407	1	3	yes	—
83	#12684	—	73	1	no	maximal, Sylow
84	C₂×C₇₃	—	2	3	no	—
85	C₂×C₇₃	—	4	6	no	—
86	C₂×C₇₃	#1779	1	1	yes	characteristic
87	#3573	—	73	1	no	—
88	#2115	—	73	3	no	—
89	#2113	—	73	3	no	—
90	C₇₃	#12684	1	1	yes	characteristic, complemented, Sylow
91	#585	#64408	1	1	yes	characteristic
92	#568	—	73	3	no	—
93	#568	—	146	6	no	—
94	C₂×#131	—	73	3	no	—
95	C₂×#131	—	146	6	no	—
96	C₂×M₃₂	—	73	3	no	—
97	C₂²×C₁₆	—	73	1	no	—
98	#349	—	146	6	no	—
99	C₂²⋊C₁₆	—	146	6	no	—
100	2₋¹⁺⁴	—	2	3	no	—
101	H(5,𝔽₂)	—	2	3	no	—
102	H(5,𝔽₂)	#10596833	1	4	yes	—
103	C₂×#41	—	2	6	no	—
104	C₂×#41	C₂×#64408	1	3	yes	—
105	#131	—	146	6	no	—
106	#131	—	146	6	no	—
107	#131	—	146	12	no	—
108	#131	—	146	4	no	—
109	#131	—	146	12	no	—
110	C₂×M₁₆	—	73	6	no	—
111	C₂×M₁₆	—	73	3	no	—
112	C₂×M₁₆	—	146	12	no	—
113	C₂×M₁₆	—	146	6	no	—
114	C₂²×C₈	—	73	1	no	—
115	C₂²×C₈	—	73	6	no	—
116	C₂²×C₈	—	146	4	no	—
117	M₃₂	—	146	6	no	—
118	C₂×C₁₆	—	146	3	no	—
119	C₂×C₁₆	—	146	3	no	—
120	#41	—	2	4	no	—
121	#41	—	2	12	no	—
122	#41	—	2	6	no	—
123	#41	—	2	12	no	—
124	#41	—	2	6	no	—
125	C₂×Q₈	—	2	6	no	—
126	C₂×Q₈	#11924037	1	3	yes	—
127	C₂×D₄	—	2	6	no	—
128	C₂×D₄	—	2	12	no	—
129	C₂×D₄	C₂×#10596833	1	6	yes	—
130	C₂×D₄	#11924037	1	3	yes	—
131	C₂²×C₄	—	2	4	no	—
132	C₂²×C₄	C₂²⋊#64408	1	6	yes	—
133	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
134	M₁₆	—	146	12	no	—
135	M₁₆	—	146	12	no	—
136	M₁₆	—	146	6	no	—
137	C₂×C₈	—	73	3	no	—
138	C₂×C₈	—	73	3	no	—
139	C₂×C₈	—	73	6	no	—
140	C₂×C₈	—	146	6	no	—
141	C₂×C₈	—	146	6	no	—
142	C₂×C₈	—	146	12	no	—
143	C₁₆	—	292	1	no	—
144	C₁₆	—	292	3	no	—
145	C₂³	—	2	4	no	—
146	C₂³	C₂²⋊#10596833	1	6	yes	—
147	C₂³	C₂²×#10596833	1	1	yes	characteristic
148	Q₈	—	2	4	no	—
149	Q₈	—	2	6	no	—
150	D₄	—	2	12	no	—
151	D₄	—	2	12	no	—
152	D₄	—	2	6	no	—
153	C₂×C₄	—	2	12	no	—
154	C₂×C₄	—	2	6	no	—
155	C₂×C₄	—	2	6	no	—
156	C₂×C₄	C₂²⋊#10596833	1	6	yes	—
157	C₂×C₄	C₂×C₂²⋊#64408	1	3	yes	—
158	C₂×C₄	C₂×#11924037	1	3	yes	—
159	C₈	—	73	1	no	—
160	C₈	—	73	3	no	—
161	C₈	—	146	6	no	—
162	C₂²	—	2	6	no	—
163	C₂²	—	4	1	no	—
164	C₂²	—	4	6	no	—
165	C₂²	—	8	4	no	—
166	C₂²	C₂×C₂²⋊#10596833	1	3	yes	—
167	C₄	—	2	6	no	—
168	C₄	#17989528	1	1	yes	characteristic
169	C₄	#17989557	1	3	yes	—
170	C₂	—	2	3	no	—
171	C₂	—	4	6	no	—
172	C₂	#22625753	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17989528
commutator	C₂³×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting, prime radical	#585×C₇₃	C₄
socle	C₂×C₇₃	#1779
cosocle	C₂²×#10596831	C₂³
hypercenter	#585	#64408
hypocenter	C₇₃	#12684
LCS	G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #585

Complements

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

N	H ≅ G/N	type
1. #22626167	C₂	semidirect
2. #22626161	C₂	semidirect
4. C₂×#14556665	C₂²	semidirect
5. C₂²×#11924031	C₂²	semidirect
90. C₇₃	#12684	semidirect

Maximal subgroups

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

subgroup	index
#22626167	2
#22626161	2
#22624643	2
#12684	73