Finite Group Databasebeta

Subgroups — #34014542

Order 18688.

186 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 52 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	#22626332	C₂	1	1	yes	maximal, characteristic, complemented
2	#22626330	C₂	1	1	yes	maximal, characteristic, complemented
3	#22625095	C₂	1	1	yes	maximal, characteristic, complemented
4	#22625083	C₂	1	1	yes	maximal, characteristic, complemented
5	#22625082	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₂×#64408	C₂	1	1	yes	maximal, characteristic, complemented
7	#22612239	C₂	1	1	yes	maximal, characteristic
8	#17989631	—	2	1	no	—
9	#17989629	—	2	1	no	—
10	#17989518	C₂²	1	1	yes	characteristic
11	#17989515	C₂²	1	1	yes	characteristic
12	#17989515	C₂²	1	1	yes	characteristic
13	#17989502	—	2	1	no	—
14	#17989501	—	2	1	no	—
15	C₁₆×#64408	C₂²	1	1	yes	characteristic
16	#17989020	C₂²	1	1	yes	characteristic
17	#17989016	C₂²	1	1	yes	characteristic
18	#17988427	C₄	1	1	yes	characteristic, complemented
19	#17988426	C₄	1	1	yes	characteristic, complemented
20	#17988422	C₄	1	1	yes	characteristic, complemented
21	C₃₂×D₇₃	C₂²	1	1	yes	characteristic
22	SD₆₄×C₇₃	C₄	1	1	yes	characteristic, complemented
23	#14556679	—	4	1	no	—
24	#14556678	—	4	1	no	—
25	#14556664	D₄	1	1	yes	characteristic
26	#14556662	—	2	1	no	—
27	#14556662	—	2	1	no	—
28	#14556656	—	4	1	no	—
29	#14556655	—	4	1	no	—
30	C₈×#64408	D₄	1	1	yes	characteristic
31	#14556641	—	2	1	no	—
32	#14556614	—	2	1	no	—
33	#14556610	—	2	1	no	—
34	Dic₅₈₄	C₂×C₄	1	1	yes	characteristic
35	#14556535	—	2	1	no	—
36	#14556534	—	2	1	no	—
37	D₁₁₆₈	C₂×C₄	1	1	yes	characteristic
38	C₁₆×D₇₃	C₂³	1	1	yes	characteristic
39	#14556497	C₂×C₄	1	1	yes	characteristic
40	Q₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic
41	D₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic
43	#11924038	—	4	1	no	—
44	#11924038	—	4	1	no	—
45	#11924038	—	8	1	no	—
46	#11924037	—	8	1	no	—
47	#11924036	—	8	1	no	—
48	#11924036	D₈	1	1	yes	characteristic
49	C₂×#10596833	—	8	1	no	—
50	C₄×#64408	D₈	1	1	yes	characteristic
51	#11924031	—	2	1	no	—
52	#11924029	—	4	1	no	—
53	#11924027	—	4	1	no	—
54	Dic₂₉₂	—	2	1	no	—
55	#11924016	—	4	1	no	—
56	#11924015	—	4	1	no	—
57	D₅₈₄	—	2	1	no	—
58	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
59	#11924003	#41	1	1	yes	characteristic
60	Q₁₆×C₇₃	—	2	1	no	—
61	D₈×C₇₃	—	2	1	no	—
62	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×#64408	—	2	1	no	—
64	Q₈×C₇₃	—	4	1	no	—
65	D₄×C₇₃	—	4	1	no	—
66	C₂²⋊D₇₃	—	8	1	no	—
67	C₂×Dic₇₃	—	8	1	no	—
68	D₂₉₂	—	4	1	no	—
69	D₂₉₂	—	8	1	no	—
70	C₄×D₇₃	—	8	1	no	—
71	C₄×D₇₃	C₂×D₈	1	1	yes	characteristic
72	Dic₁₄₆	—	4	1	no	—
73	#10596833	—	8	1	no	—
74	#10596833	—	8	1	no	—
75	#10596833	D₁₆	1	1	yes	characteristic
76	#10596833	D₁₆	1	1	yes	characteristic
77	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
78	#10596831	#135	1	1	yes	characteristic
79	C₂²×C₇₃	—	8	1	no	—
80	C₂×D₇₃	—	8	1	no	—
81	C₂×D₇₃	#508	1	1	yes	characteristic
82	#64408	—	2	2	no	—
83	C₄×C₇₃	—	8	1	no	—
84	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
85	Dic₇₃	—	8	1	no	—
86	Dic₇₃	C₂×D₁₆	1	1	yes	characteristic
87	#13655	—	73	1	no	maximal, Sylow
88	C₂×C₇₃	—	16	1	no	—
89	C₂×C₇₃	C₄×D₁₆	1	1	yes	characteristic
90	D₇₃	—	2	1	no	—
91	D₇₃	—	16	1	no	—
92	#2262	—	73	1	no	—
93	#2178	—	73	1	no	—
94	#2178	—	73	1	no	—
95	#1425	—	73	1	no	—
96	#1417	—	73	1	no	—
97	#1417	—	73	1	no	—
98	C₄×C₃₂	—	73	1	no	—
99	C₇₃	#13655	1	1	yes	characteristic, complemented, Sylow
100	#508	—	73	1	no	—
101	#508	—	73	1	no	—
102	#443	—	146	1	no	—
103	#443	—	146	1	no	—
104	Q₆₄	—	73	1	no	—
105	SD₆₄	—	73	1	no	—
106	SD₆₄	#64408	1	1	yes	characteristic, complemented
107	D₃₂	—	73	1	no	—
108	C₂×C₃₂	—	73	1	no	—
109	C₂×C₃₂	—	73	1	no	—
110	#368	—	73	1	no	—
111	#368	—	73	1	no	—
112	#359	—	146	1	no	—
113	#359	—	146	1	no	—
114	C₄×C₁₆	—	73	1	no	—
115	#135	—	146	1	no	—
116	#135	—	146	1	no	—
117	#131	—	292	1	no	—
118	#131	—	292	1	no	—
119	Q₃₂	—	73	1	no	—
120	Q₃₂	C₂×#64408	1	1	yes	characteristic
121	SD₃₂	—	146	1	no	—
122	SD₃₂	—	146	1	no	—
123	D₁₆	—	73	1	no	—
124	D₁₆	C₂×#64408	1	1	yes	characteristic
125	C₂×C₁₆	—	73	1	no	—
126	C₂×C₁₆	—	73	1	no	—
127	#108	—	146	1	no	—
128	#108	—	146	1	no	—
129	#104	—	292	1	no	—
130	#104	—	292	1	no	—
131	C₄×C₈	—	73	1	no	—
132	C₃₂	—	73	1	no	—
133	C₃₂	—	146	1	no	—
134	C₃₂	C₂×#64408	1	1	yes	characteristic
135	#41	—	292	1	no	—
136	#41	—	292	1	no	—
137	Q₁₆	—	2	1	no	—
138	Q₁₆	—	146	1	no	—
139	SD₁₆	—	292	1	no	—
140	SD₁₆	—	292	1	no	—
141	D₈	—	2	1	no	—
142	D₈	—	146	1	no	—
143	M₁₆	—	292	1	no	—
144	M₁₆	—	292	1	no	—
145	M₁₆	—	584	1	no	—
146	M₁₆	—	584	1	no	—
147	C₂×C₈	—	584	1	no	—
148	C₂×C₈	—	584	1	no	—
149	C₂×C₈	—	73	1	no	—
150	C₂×C₈	—	73	1	no	—
151	C₄²	—	73	1	no	—
152	C₁₆	—	73	1	no	—
153	C₁₆	—	146	1	no	—
154	C₁₆	C₂²×#64408	1	1	yes	characteristic
155	Q₈	—	4	1	no	—
156	Q₈	—	292	1	no	—
157	D₄	—	4	1	no	—
158	D₄	—	292	1	no	—
159	D₄	—	584	1	no	—
160	D₄	—	584	1	no	—
161	C₂×C₄	—	584	1	no	—
162	C₂×C₄	—	584	1	no	—
163	C₂×C₄	—	73	1	no	—
164	C₂×C₄	—	146	1	no	—
165	C₈	—	584	1	no	—
166	C₈	—	584	1	no	—
167	C₈	—	73	1	no	—
168	C₈	—	73	1	no	—
169	C₈	—	73	1	no	—
170	C₈	D₄×#64408	1	1	yes	characteristic
171	C₂²	—	8	1	no	—
172	C₂²	—	584	1	no	—
173	C₂²	—	73	1	no	—
174	C₄	—	8	1	no	—
175	C₄	—	584	1	no	—
176	C₄	—	73	1	no	—
177	C₄	—	146	2	no	—
178	C₄	D₈×#64408	1	1	yes	characteristic
179	C₂	—	16	1	no	—
180	C₂	—	1168	1	no	—
181	C₂	—	146	1	no	—
182	C₂	D₁₆×#64408	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₆×#64408
commutator	C₁₆×C₇₃	C₂²×C₄
Frattini	C₁₆	C₂²×#64408
Fitting	SD₆₄×C₇₃	C₄
socle	C₂×C₇₃	C₄×D₁₆
prime radical	#22612239	C₂
cosocle	C₁₆×D₇₃	C₂³
hypercenter	SD₆₄	#64408
hypocenter	C₇₃	#13655
LCS	G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ SD₆₄

Complements

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

N	H ≅ G/N	type
1. #22626332	C₂	semidirect
2. #22626330	C₂	semidirect
3. #22625095	C₂	semidirect
4. #22625083	C₂	semidirect
5. #22625082	C₂	semidirect
6. C₃₂×#64408	C₂	semidirect
18. #17988427	C₄	semidirect
19. #17988426	C₄	semidirect
20. #17988422	C₄	semidirect
22. SD₆₄×C₇₃	C₄	semidirect
99. C₇₃	#13655	semidirect
106. SD₆₄	#64408	semidirect

Maximal subgroups

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

subgroup	index
#22626332	2
#22626330	2
#22625095	2
#22625083	2
#22625082	2
C₃₂×#64408	2
#22612239	2
#13655	73