Subgroups — #34030068

Order 18688.

706 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989645	C₂	1	2	yes	maximal
2	#22626266	C₂	1	6	yes	maximal, complemented
3	#22626263	C₂	1	3	yes	maximal, complemented
4	C₂×C₄⋊#11924031	C₂	1	3	yes	maximal, complemented
5	C₄×#14556598	C₂	1	1	yes	maximal, characteristic
6	#17989645	—	2	8	no	—
7	C₂×#14556665	C₂²	1	6	yes	—
8	C₂²×#11924031	C₂²	1	6	yes	—
9	C₄⋊#11924031	C₂²	1	1	yes	characteristic
10	C₄⋊#11924031	C₂²	1	3	yes	—
11	C₄⋊#11924031	C₂²	1	6	yes	—
12	C₄⋊#11924031	C₂²	1	6	yes	—
13	C₂×#14556598	C₄	1	1	yes	characteristic
14	C₂×#14556598	C₂²	1	1	yes	characteristic
15	#17988543	C₄	1	3	yes	—
16	#17988543	C₂²	1	3	yes	—
17	C₄×#11924012	C₄	1	3	yes	—
18	C₂×C₄×#10596831	C₂²	1	3	yes	—
19	C₄×#41×C₇₃	C₄	1	1	yes	characteristic
20	#14556665	—	2	12	no	—
21	C₂×#11924031	—	2	12	no	—
22	C₂×#11924031	C₂³	1	2	yes	—
23	C₂×#11924031	C₂³	1	6	yes	—
24	#14556598	—	2	4	no	—
25	#14556598	D₄	1	4	yes	—
26	#14556598	Q₈	1	4	yes	—
27	C₂×#11924012	C₂×C₄	1	3	yes	—
28	C₂×#11924012	C₂×C₄	1	3	yes	—
29	C₂²×#10596831	C₂×C₄	1	3	yes	—
30	C₂²×#10596831	C₂³	1	3	yes	—
31	#14556502	C₂×C₄	1	6	yes	—
32	C₄×#10596831	C₂×C₄	1	3	yes	—
33	C₄×#10596831	C₂×C₄	1	3	yes	—
34	C₄×#10596831	C₂³	1	1	yes	characteristic
35	C₄×#10596831	C₂³	1	3	yes	—
36	C₂×#41×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₄×Q₈×C₇₃	C₈	1	1	yes	characteristic
38	C₄×Q₈×C₇₃	C₈	1	1	yes	characteristic
39	C₄×D₄×C₇₃	C₈	1	3	yes	—
40	C₄×D₄×C₇₃	C₈	1	3	yes	—
41	#117×C₇₃	C₂×C₄	1	3	yes	—
42	C₂×C₄²×C₇₃	C₂×C₄	1	3	yes	—
43	#11924031	—	2	2	no	—
44	#11924031	—	2	6	no	—
45	#11924012	—	2	6	no	—
46	#11924012	C₄⋊C₄	1	12	yes	—
47	C₂×#10596831	—	2	6	no	—
48	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
49	C₂×#10596831	C₂²×C₄	1	3	yes	—
50	C₂×#10596831	C₂²×C₄	1	3	yes	—
51	C₂×#10596831	C₂×D₄	1	6	yes	—
52	C₂×#10596831	C₂×Q₈	1	6	yes	—
53	C₂×#10596831	C₂⁴	1	1	yes	characteristic
54	#41×C₇₃	C₄⋊C₄	1	4	yes	—
55	#41×C₇₃	M₁₆	1	4	yes	—
56	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
57	C₂×D₄×C₇₃	C₂×C₈	1	3	yes	—
58	C₂²×C₄×C₇₃	C₂×C₈	1	3	yes	—
59	C₂²×C₄×C₇₃	C₂×C₈	1	3	yes	—
60	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
61	C₄⋊C₄×C₇₃	C₂×C₈	1	3	yes	—
62	C₄⋊C₄×C₇₃	C₂×C₈	1	3	yes	—
63	C₂²⋊C₄×C₇₃	C₂×C₈	1	6	yes	—
64	C₄²×C₇₃	C₂×C₈	1	3	yes	—
65	C₄²×C₇₃	C₂×C₈	1	3	yes	—
66	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
67	C₄²×C₇₃	C₂²×C₄	1	3	yes	—
68	C₂³×C₇₃	C₂²×C₈	1	3	yes	—
69	Q₈×C₇₃	C₄⋊C₈	1	4	yes	—
70	D₄×C₇₃	C₄⋊C₈	1	12	yes	—
71	C₂×C₄×C₇₃	—	2	6	no	—
72	C₂×C₄×C₇₃	—	4	3	no	—
73	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	6	yes	—
74	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
76	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
77	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
78	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₂×M₁₆	1	6	yes	—
80	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
81	#10596831	—	2	1	no	—
82	#10596831	—	2	3	no	—
83	#10596831	C₂×C₄⋊C₄	1	6	yes	—
84	#10596831	C₂²×D₄	1	1	yes	characteristic
85	#10596831	C₂²×Q₈	1	1	yes	characteristic
86	C₂²×C₇₃	—	2	3	no	—
87	C₂²×C₇₃	—	2	3	no	—
88	C₂²×C₇₃	C₂×C₄⋊C₈	1	6	yes	—
89	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
90	C₄×C₇₃	—	2	3	no	—
91	C₄×C₇₃	—	2	1	no	—
92	C₄×C₇₃	—	2	3	no	—
93	C₄×C₇₃	C₂×C₄⋊C₈	1	6	yes	—
94	C₄×C₇₃	#504	1	2	yes	complemented
95	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
96	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
97	#33398	—	73	1	no	maximal, Sylow
98	C₂×C₇₃	—	2	6	no	—
99	C₂×C₇₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
100	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
101	C₂×C₇₃	#3407	1	1	yes	characteristic
102	C₂×#504	—	73	2	no	—
103	C₄×#131	—	73	1	no	—
104	#2151	—	73	6	no	—
105	#2150	—	73	3	no	—
106	C₂×C₄⋊C₁₆	—	73	3	no	—
107	C₇₃	#33398	1	1	yes	characteristic, complemented, Sylow
108	C₂×#131	—	73	1	no	—
109	C₂×#131	—	73	1	no	—
110	C₄×#41	#64408	1	1	yes	characteristic
111	#504	—	146	8	no	—
112	C₂×M₃₂	—	73	6	no	—
113	C₂²×C₁₆	—	73	6	no	—
114	#405	—	73	3	no	—
115	#405	—	73	3	no	—
116	C₄×M₁₆	—	73	3	no	—
117	C₂×C₄×C₈	—	73	3	no	—
118	C₄⋊C₁₆	—	73	1	no	—
119	C₄⋊C₁₆	—	73	3	no	—
120	C₄⋊C₁₆	—	73	6	no	—
121	C₄⋊C₁₆	—	73	6	no	—
122	C₂×#41	C₂×#64408	1	1	yes	characteristic
123	#131	—	73	4	no	—
124	#131	—	73	4	no	—
125	#131	—	146	4	no	—
126	C₂×M₁₆	—	73	3	no	—
127	C₂×M₁₆	—	73	3	no	—
128	C₂²×C₈	—	73	3	no	—
129	C₂²×C₈	—	73	3	no	—
130	C₄×Q₈	#10596833	1	1	yes	characteristic
131	C₄×Q₈	#10596833	1	1	yes	characteristic
132	C₄×D₄	#10596833	1	3	yes	—
133	C₄×D₄	#10596833	1	3	yes	—
134	#117	C₂×#64408	1	3	yes	—
135	C₂×C₄²	C₂×#64408	1	3	yes	—
136	M₃₂	—	146	12	no	—
137	C₂×C₁₆	—	73	2	no	—
138	C₂×C₁₆	—	73	6	no	—
139	C₂×C₁₆	—	146	12	no	—
140	#97	—	73	6	no	—
141	C₄×C₈	—	73	3	no	—
142	C₄×C₈	—	73	3	no	—
143	C₄×C₈	—	73	1	no	—
144	C₄×C₈	—	73	3	no	—
145	#41	C₄⋊#64408	1	4	yes	—
146	#41	#11924038	1	4	yes	—
147	C₂×Q₈	C₂×#10596833	1	1	yes	characteristic
148	C₂×D₄	C₂×#10596833	1	3	yes	—
149	C₂²×C₄	C₂×#10596833	1	3	yes	—
150	C₂²×C₄	C₂×#10596833	1	3	yes	—
151	C₂²×C₄	C₂²×#64408	1	3	yes	—
152	M₁₆	—	73	12	no	—
153	M₁₆	—	146	6	no	—
154	C₂×C₈	—	73	3	no	—
155	C₂×C₈	—	73	1	no	—
156	C₂×C₈	—	73	6	no	—
157	C₂×C₈	—	73	6	no	—
158	C₂×C₈	—	73	1	no	—
159	C₂×C₈	—	73	3	no	—
160	C₂×C₈	—	146	6	no	—
161	C₄⋊C₄	C₂×#10596833	1	3	yes	—
162	C₄⋊C₄	C₂×#10596833	1	3	yes	—
163	C₂²⋊C₄	C₂×#10596833	1	6	yes	—
164	C₄²	C₂×#10596833	1	3	yes	—
165	C₄²	C₂×#10596833	1	3	yes	—
166	C₄²	C₂²×#64408	1	1	yes	characteristic
167	C₄²	C₂²×#64408	1	3	yes	—
168	C₁₆	—	146	2	no	—
169	C₁₆	—	146	6	no	—
170	C₂³	C₂²×#10596833	1	3	yes	—
171	Q₈	C₄⋊#10596833	1	4	yes	—
172	D₄	C₄⋊#10596833	1	12	yes	—
173	C₂×C₄	—	2	6	no	—
174	C₂×C₄	—	4	3	no	—
175	C₂×C₄	C₂×C₄⋊#64408	1	6	yes	—
176	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
177	C₂×C₄	C₂²×#10596833	1	3	yes	—
178	C₂×C₄	C₂²×#10596833	1	3	yes	—
179	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
180	C₂×C₄	C₂²×#10596833	1	3	yes	—
181	C₂×C₄	C₂×#11924038	1	6	yes	—
182	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
183	C₈	—	73	1	no	—
184	C₈	—	73	1	no	—
185	C₈	—	73	6	no	—
186	C₈	—	146	1	no	—
187	C₈	—	146	3	no	—
188	C₂²	—	2	3	no	—
189	C₂²	—	2	3	no	—
190	C₂²	C₂×C₄⋊#10596833	1	6	yes	—
191	C₂²	C₂³×#10596833	1	1	yes	characteristic
192	C₄	—	2	3	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	3	no	—
195	C₄	C₂×C₄⋊#10596833	1	6	yes	—
196	C₄	#17989645	1	2	yes	complemented
197	C₄	C₂²×C₄⋊#64408	1	1	yes	characteristic
198	C₄	C₂²×#11924038	1	1	yes	characteristic
199	C₂	—	2	6	no	—
200	C₂	C₂²×C₄⋊#10596833	1	1	yes	characteristic
201	C₂	C₂×#17989645	1	1	yes	characteristic
202	C₂	#22626898	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	C₄×#41×C₇₃	C₄
prime radical	C₂×#41×C₇₃	C₂×C₄
cosocle	C₂×#10596831	C₂⁴
hypercenter	C₄×#41	#64408
hypocenter	C₇₃	#33398
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. #22626266	C₂	semidirect
3. #22626263	C₂	semidirect
4. C₂×C₄⋊#11924031	C₂	semidirect
94. C₄×C₇₃	#504	semidirect
107. C₇₃	#33398	semidirect
196. C₄	#17989645	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989645	2
#22626266	2
#22626263	2
C₂×C₄⋊#11924031	2
C₄×#14556598	2
#33398	73