Subgroups — C₂×#22626266 #34030060

Order 18688.

734 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂²×C₄	C₂²×#64408
commutator	C₂²×C₇₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₂×#117×C₇₃	C₄
socle	C₂³×C₇₃	C₂²×C₈
prime radical	C₂⁴×C₇₃	C₂×C₈
cosocle	C₂×#10596831	C₂⁴
hypercenter	C₂×#117	#64408
hypocenter	C₇₃	C₂×#2151
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #22626266	C₂	direct (N × H)
4. C₂×C₄⋊#11924031	C₂	semidirect
12. C₄⋊#11924031	C₂²	semidirect
103. C₂×C₇₃	#2151	semidirect
111. C₇₃	C₂×#2151	semidirect
208. C₂	#22626266	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14556665	2
C₂³×#11924031	2
#22626266	2
C₂×C₄⋊#11924031	2
C₂×#17988543	2
C₂×#2151	73

Subgroups — C2×#22626266 #34030060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22626266 #34030060