Subgroups — #34034856

Order 18688.

506 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627158	C₂	1	4	yes	maximal, complemented
2	#22627155	C₂	1	4	yes	maximal, complemented
3	#22627123	C₂	1	2	yes	maximal
4	#22627121	C₂	1	1	yes	maximal, characteristic
5	#22627093	C₂	1	2	yes	maximal
6	C₄×#14556686	C₂	1	1	yes	maximal, characteristic
7	#22626721	C₂	1	1	yes	maximal, characteristic
8	C₂×#14556688	C₂²	1	2	yes	—
9	C₂×#14556686	C₂²	1	1	yes	characteristic
10	C₂×#14556686	C₂²	1	1	yes	characteristic
11	#17989690	C₂²	1	4	yes	—
12	#17989690	C₂²	1	4	yes	—
13	C₄⋊#11924040	C₂²	1	4	yes	—
14	#17989681	C₂²	1	4	yes	—
15	C₄×#11924040	C₂²	1	2	yes	—
16	C₄×#11924040	C₂²	1	2	yes	—
17	C₄×#11924040	C₂²	1	4	yes	—
18	#17989639	C₄	1	4	yes	—
19	#17989636	C₂²	1	4	yes	—
20	#17989594	C₄	1	2	yes	—
21	#17989592	C₄	1	1	yes	characteristic
22	#17989552	C₂²	1	2	yes	—
23	C₄×#11924036	C₂²	1	1	yes	characteristic
24	#17989143	C₄	1	1	yes	characteristic
25	#14556688	—	2	4	no	—
26	#14556686	—	2	2	no	—
27	#14556686	Q₈	1	4	yes	complemented
28	C₂×#11924040	C₂³	1	2	yes	—
29	C₂×#11924040	C₂³	1	4	yes	—
30	C₂×#11924040	C₂³	1	1	yes	characteristic
31	C₂×#11924040	C₂³	1	1	yes	characteristic
32	C₂×#11924038	C₂×C₄	1	2	yes	—
33	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
34	C₂×#11924036	C₂³	1	1	yes	characteristic
35	#14556658	C₂×C₄	1	4	yes	—
36	#14556658	C₂×C₄	1	4	yes	—
37	C₄⋊#10596833	C₂×C₄	1	4	yes	—
38	#14556646	C₂×C₄	1	4	yes	—
39	C₄×#10596833	C₂×C₄	1	2	yes	—
40	C₄×#10596833	C₂³	1	2	yes	—
41	C₄×#10596833	C₂³	1	4	yes	—
42	#14556590	C₈	1	2	yes	—
43	#14556583	C₈	1	1	yes	characteristic
44	#14556569	C₈	1	4	yes	—
45	#14556556	C₂×C₄	1	4	yes	—
46	#14556549	C₂×C₄	1	2	yes	—
47	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
48	#125×C₇₃	C₈	1	1	yes	characteristic
49	#11924040	—	2	2	no	—
50	#11924040	—	2	4	no	—
51	#11924040	C₂×Q₈	1	2	yes	—
52	#11924040	C₂×Q₈	1	2	yes	—
53	#11924038	—	2	4	no	—
54	#11924036	—	2	2	no	—
55	#11924036	C₂×Q₈	1	2	yes	—
56	#11924036	#41	1	2	yes	—
57	C₂×#10596833	C₂²×C₄	1	2	yes	—
58	C₂×#10596833	C₂²×C₄	1	4	yes	—
59	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
60	C₂×#10596833	C₂⁴	1	1	yes	characteristic
61	C₂×Dic₁₄₆	C₂×C₈	1	4	yes	—
62	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
63	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
64	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
65	#11924007	C₂×C₈	1	4	yes	—
66	#11924007	C₂×C₈	1	4	yes	—
67	#11924005	C₂×C₈	1	4	yes	—
68	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
69	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
70	C₄×Dic₇₃	C₂²×C₄	1	4	yes	—
71	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
72	C₄⋊C₄×C₇₃	C₂×C₈	1	4	yes	—
73	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
74	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
76	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
77	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
78	C₂×Dic₇₃	C₂²×C₈	1	4	yes	—
79	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
80	C₂×Dic₇₃	C₂³×C₄	1	1	yes	characteristic
81	C₄×D₇₃	—	2	4	no	—
82	C₄×D₇₃	C₄×Q₈	1	2	yes	—
83	C₄×D₇₃	#131	1	2	yes	—
84	Dic₁₄₆	—	2	8	no	—
85	#10596833	—	2	2	no	—
86	#10596833	—	2	4	no	—
87	#10596833	C₄×Q₈	1	2	yes	—
88	#10596833	C₂²×Q₈	1	1	yes	characteristic
89	#10596833	C₂×#41	1	1	yes	characteristic
90	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
91	C₂×D₇₃	—	2	2	no	—
92	C₂×D₇₃	C₈×Q₈	1	2	yes	—
93	C₄×C₇₃	—	2	4	no	—
94	C₄×C₇₃	—	2	2	no	—
95	C₄×C₇₃	C₈×Q₈	1	2	yes	—
96	Dic₇₃	—	2	4	no	—
97	Dic₇₃	#504	1	4	yes	—
98	Dic₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
99	Dic₇₃	C₂×#131	1	1	yes	characteristic
100	#33474	—	73	1	no	maximal, Sylow
101	C₂×C₇₃	C₂×C₈×Q₈	1	1	yes	characteristic
102	C₂×C₇₃	C₂×#504	1	2	yes	—
103	D₇₃	—	2	2	no	—
104	#2960	—	73	1	no	—
105	#2183	—	73	4	no	—
106	Q₈×C₁₆	—	73	4	no	—
107	#2151	—	73	2	no	—
108	#2150	—	73	1	no	—
109	#2108	—	73	2	no	—
110	C₂×C₄×C₁₆	—	73	1	no	—
111	C₇₃	#33474	1	1	yes	characteristic, complemented, Sylow
112	#533	—	73	1	no	—
113	C₂×M₃₂	—	73	2	no	—
114	C₂²×C₁₆	—	73	1	no	—
115	C₂²×C₁₆	—	73	1	no	—
116	#446	—	73	4	no	—
117	C₈×Q₈	—	73	4	no	—
118	#424	—	73	2	no	—
119	#423	—	73	1	no	—
120	#405	—	73	2	no	—
121	C₂×C₄×C₈	—	73	1	no	—
122	C₄⋊C₁₆	—	73	4	no	—
123	C₄⋊C₁₆	—	73	4	no	—
124	C₄⋊C₁₆	—	73	4	no	—
125	#346	—	73	4	no	—
126	C₄×C₁₆	—	73	2	no	—
127	C₄×C₁₆	—	73	2	no	—
128	C₄×C₁₆	—	73	4	no	—
129	C₂×M₁₆	—	73	2	no	—
130	C₂²×C₈	—	73	1	no	—
131	C₂²×C₈	—	73	1	no	—
132	C₄⋊Q₈	—	73	2	no	—
133	#125	—	73	1	no	—
134	#125	#10596842	1	1	yes	characteristic
135	C₂²⋊Q₈	—	73	4	no	—
136	C₄×Q₈	—	73	4	no	—
137	#117	—	73	2	no	—
138	C₂×C₄²	—	73	1	no	—
139	M₃₂	—	146	4	no	—
140	C₂×C₁₆	—	73	1	no	—
141	C₂×C₁₆	—	73	1	no	—
142	C₂×C₁₆	—	73	4	no	—
143	C₂×C₁₆	—	73	2	no	—
144	C₂×C₁₆	—	73	4	no	—
145	C₂×C₁₆	—	146	2	no	—
146	C₄⋊C₈	—	73	4	no	—
147	C₄⋊C₈	—	73	4	no	—
148	C₄⋊C₈	—	73	4	no	—
149	#97	—	73	4	no	—
150	C₄×C₈	—	73	2	no	—
151	C₄×C₈	—	73	4	no	—
152	C₄×C₈	—	73	2	no	—
153	C₂×Q₈	—	73	4	no	—
154	C₂²×C₄	—	73	1	no	—
155	C₂²×C₄	—	73	2	no	—
156	M₁₆	—	146	4	no	—
157	C₂×C₈	—	73	1	no	—
158	C₂×C₈	—	73	1	no	—
159	C₂×C₈	—	73	2	no	—
160	C₂×C₈	—	73	2	no	—
161	C₂×C₈	—	73	2	no	—
162	C₂×C₈	—	73	4	no	—
163	C₂×C₈	—	146	2	no	—
164	C₄⋊C₄	—	73	4	no	—
165	C₄⋊C₄	—	73	4	no	—
166	C₄⋊C₄	—	73	2	no	—
167	C₄⋊C₄	C₂×#10596842	1	2	yes	—
168	C₄⋊C₄	C₂×#10596842	1	4	yes	—
169	C₂²⋊C₄	—	73	4	no	—
170	C₄²	—	73	1	no	—
171	C₄²	—	73	2	no	—
172	C₄²	—	73	4	no	—
173	C₄²	C₂×#10596842	1	1	yes	characteristic
174	C₁₆	—	73	2	no	—
175	C₁₆	—	73	2	no	—
176	C₁₆	—	146	2	no	—
177	C₁₆	—	146	4	no	—
178	C₂³	—	73	1	no	—
179	Q₈	—	146	8	no	—
180	C₂×C₄	—	73	1	no	—
181	C₂×C₄	—	73	2	no	—
182	C₂×C₄	—	73	2	no	—
183	C₂×C₄	—	73	2	no	—
184	C₂×C₄	—	73	4	no	—
185	C₂×C₄	—	146	4	no	—
186	C₂×C₄	C₂²×#10596842	1	2	yes	—
187	C₂×C₄	C₂²×#10596842	1	4	yes	—
188	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
189	C₈	—	73	1	no	—
190	C₈	—	73	1	no	—
191	C₈	—	73	2	no	—
192	C₈	—	146	2	no	—
193	C₈	—	146	4	no	—
194	C₂²	—	73	2	no	—
195	C₂²	—	146	2	no	—
196	C₂²	C₂³×#10596842	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	2	4	no	—
199	C₄	—	73	1	no	—
200	C₄	—	73	1	no	—
201	C₄	—	73	4	no	—
202	C₄	—	146	4	no	—
203	C₄	Q₈×#10596842	1	2	yes	—
204	C₂	—	146	2	no	—
205	C₂	C₂×Q₈×#10596842	1	1	yes	characteristic
206	C₂	#22627182	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10596842
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	#125×C₇₃	C₈
prime radical	C₂²×D₇₃	C₂²×C₈
cosocle	C₂×#10596833	C₂⁴
hypercenter	#125	#10596842
hypocenter	C₇₃	#33474
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #22627158	C₂	semidirect
2. #22627155	C₂	semidirect
27. #14556686	Q₈	semidirect
111. C₇₃	#33474	semidirect

Maximal subgroups

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

subgroup	index
#22627158	2
#22627155	2
#22627123	2
#22627121	2
#22627093	2
C₄×#14556686	2
#22626721	2
#33474	73