Subgroups — C₂×#22620455 #33955600

Order 18688.

1902 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989144	C₂	1	2	yes	maximal
2	#22620455	C₂	1	16	yes	maximal, direct product factor
3	C₂×#17988996	C₂	1	8	yes	maximal, complemented
4	C₂×#17988625	C₂	1	4	yes	maximal
5	C₂²×#97×C₇₃	C₂	1	1	yes	maximal, characteristic
6	#17989144	—	2	8	no	—
7	#17989144	C₂²	1	16	yes	—
8	C₂×#14556591	C₂²	1	4	yes	—
9	C₂×#14556583	C₂²	1	4	yes	—
10	C₂×#14556570	—	2	4	no	—
11	C₂×C₄×Dic₁₄₆	—	2	4	no	—
12	C₂×#14556550	C₂²	1	4	yes	—
13	#17988996	C₂²	1	64	yes	complemented
14	#17988625	C₂²	1	32	yes	—
15	C₂×#14556525	C₂²	1	8	yes	—
16	C₂×#14556523	C₂²	1	8	yes	—
17	C₂³×C₈×C₇₃	C₂²	1	2	yes	—
18	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×#97×C₇₃	C₂²	1	4	yes	—
20	C₂×#97×C₇₃	C₂²	1	8	yes	—
21	C₂²×Dic₁₄₆	—	2	4	no	—
22	#14556591	—	2	8	no	—
23	#14556591	C₂³	1	16	yes	—
24	#14556583	—	2	8	no	—
25	#14556583	C₂³	1	16	yes	—
26	#14556570	—	2	32	no	—
27	C₄×Dic₁₄₆	—	2	32	no	—
28	#14556550	—	2	8	no	—
29	#14556550	C₂³	1	16	yes	—
30	C₂×C₄⋊Dic₇₃	—	2	4	no	—
31	C₂×C₄⋊Dic₇₃	C₂³	1	8	yes	—
32	C₂×#11924007	—	2	8	no	—
33	C₂×C₂²⋊Dic₇₃	—	2	4	no	—
34	C₂×C₄×Dic₇₃	—	2	4	no	—
35	#14556525	C₂³	1	32	yes	—
36	#14556523	C₂³	1	32	yes	—
37	C₂³×C₄×C₇₃	D₄	1	2	yes	—
38	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₇₃	Q₈	1	16	yes	complemented
40	C₂²×C₈×C₇₃	C₂³	1	4	yes	—
41	C₂²×C₈×C₇₃	C₂³	1	8	yes	—
42	C₂×C₄²×C₇₃	D₄	1	4	yes	—
43	C₂×C₄²×C₇₃	D₄	1	2	yes	—
44	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
45	C₂×C₄²×C₇₃	C₂³	1	4	yes	—
46	#97×C₇₃	C₂³	1	16	yes	—
47	C₂×Dic₁₄₆	—	2	16	no	—
48	C₂×Dic₁₄₆	—	4	16	no	—
49	C₂²×Dic₇₃	—	2	8	no	—
50	C₄⋊Dic₇₃	—	2	16	no	—
51	C₄⋊Dic₇₃	—	2	8	no	—
52	C₄⋊Dic₇₃	C₂⁴	1	16	yes	—
53	#11924007	—	2	32	no	—
54	C₂²⋊Dic₇₃	—	2	16	no	—
55	C₄×Dic₇₃	—	2	16	no	—
56	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₇₃	—	4	4	no	—
58	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
59	C₂²×C₄×C₇₃	C₂×D₄	1	8	yes	—
60	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₃	C₂×Q₈	1	8	yes	—
63	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
64	C₂²×C₄×C₇₃	C₂⁴	1	2	yes	—
65	C₂×C₈×C₇₃	C₂×Q₈	1	16	yes	—
66	C₂×C₈×C₇₃	C₂×Q₈	1	32	yes	complemented
67	C₂×C₈×C₇₃	C₂⁴	1	8	yes	—
68	C₄²×C₇₃	C₂×D₄	1	8	yes	—
69	C₄²×C₇₃	C₂×D₄	1	4	yes	—
70	C₄²×C₇₃	C₂⁴	1	4	yes	—
71	C₂³×C₇₃	—	2	4	no	—
72	C₂³×C₇₃	C₄⋊Q₈	1	4	yes	—
73	C₂³×C₇₃	C₂²×D₄	1	2	yes	—
74	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₄×C₇₃	—	2	8	no	—
76	C₂×C₄×C₇₃	—	4	8	no	—
77	C₂×C₄×C₇₃	C₄⋊Q₈	1	8	yes	—
78	C₂×C₄×C₇₃	C₄⋊Q₈	1	4	yes	—
79	C₂×C₄×C₇₃	C₂²×D₄	1	8	yes	—
80	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
81	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₇₃	C₂²×Q₈	1	4	yes	—
83	C₂×C₄×C₇₃	C₂²×Q₈	1	8	yes	—
84	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
85	C₂×Dic₇₃	—	2	16	no	—
86	C₂×Dic₇₃	—	4	16	no	—
87	Dic₁₄₆	—	4	32	no	—
88	C₈×C₇₃	C₂²×Q₈	1	16	yes	—
89	C₂²×C₇₃	—	2	8	no	—
90	C₂²×C₇₃	—	2	4	no	—
91	C₂²×C₇₃	C₂×C₄⋊Q₈	1	4	yes	—
92	C₂²×C₇₃	C₂×C₄⋊Q₈	1	2	yes	—
93	C₂²×C₇₃	#575	1	4	yes	—
94	C₂²×C₇₃	C₂³×D₄	1	1	yes	characteristic
95	C₄×C₇₃	—	2	8	no	—
96	C₄×C₇₃	C₂×C₄⋊Q₈	1	4	yes	—
97	C₄×C₇₃	C₂×C₄⋊Q₈	1	2	yes	—
98	C₄×C₇₃	C₂³×Q₈	1	2	yes	—
99	Dic₇₃	—	4	16	no	—
100	C₂×#3162	—	73	1	no	maximal, Sylow
101	C₂×C₇₃	—	2	4	no	—
102	C₂×C₇₃	#3162	1	4	yes	complemented
103	C₂×C₇₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₇₃	C₂×#575	1	2	yes	—
105	C₂×#533	—	73	2	no	—
106	#3162	—	73	16	no	—
107	C₂×#501	—	73	8	no	—
108	C₂×#427	—	73	4	no	—
109	C₂²×#97	D₇₃	1	1	yes	characteristic
110	C₇₃	C₂×#3162	1	1	yes	characteristic, complemented, Sylow
111	C₂³×C₈	C₂×D₇₃	1	2	yes	—
112	#533	—	73	16	no	—
113	#533	—	146	8	no	—
114	C₂×C₄⋊Q₈	—	73	4	no	—
115	C₂×#125	—	73	4	no	—
116	C₂×C₂²⋊Q₈	—	146	4	no	—
117	C₂×C₄×Q₈	—	146	4	no	—
118	C₂×#117	—	73	4	no	—
119	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
120	#501	—	73	64	no	—
121	#427	—	73	32	no	—
122	C₂×#107	—	73	8	no	—
123	C₂×#106	—	73	8	no	—
124	C₂×#97	C₂×D₇₃	1	4	yes	—
125	C₂×#97	C₂×D₇₃	1	8	yes	—
126	C₂²×Q₈	—	146	4	no	—
127	C₂³×C₄	D₂₉₂	1	2	yes	—
128	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
129	C₂²×C₈	Dic₁₄₆	1	16	yes	complemented
130	C₂²×C₈	C₂²×D₇₃	1	4	yes	—
131	C₂²×C₈	C₂²×D₇₃	1	8	yes	—
132	C₄⋊Q₈	—	73	16	no	—
133	C₄⋊Q₈	—	146	8	no	—
134	#125	—	73	16	no	—
135	#125	—	146	8	no	—
136	C₂²⋊Q₈	—	146	32	no	—
137	C₄×Q₈	—	146	32	no	—
138	#117	—	73	16	no	—
139	#117	—	146	8	no	—
140	C₂×C₄⋊C₄	—	73	8	no	—
141	C₂×C₄⋊C₄	—	146	8	no	—
142	C₂×C₄⋊C₄	—	146	4	no	—
143	C₂×C₂²⋊C₄	—	146	4	no	—
144	C₂×C₄²	—	146	4	no	—
145	C₂×C₄²	D₂₉₂	1	4	yes	—
146	C₂×C₄²	D₂₉₂	1	2	yes	—
147	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
148	C₂×C₄²	C₂²×D₇₃	1	4	yes	—
149	#107	—	73	32	no	—
150	#106	—	73	32	no	—
151	#97	C₂²×D₇₃	1	16	yes	—
152	C₂⁴	C₂×D₂₉₂	1	1	yes	characteristic
153	C₂×Q₈	—	292	16	no	—
154	C₂×Q₈	—	146	16	no	—
155	C₂²×C₄	—	4	4	no	—
156	C₂²×C₄	—	146	8	no	—
157	C₂²×C₄	C₂×D₂₉₂	1	4	yes	—
158	C₂²×C₄	C₂×D₂₉₂	1	8	yes	—
159	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
160	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
161	C₂²×C₄	C₂×Dic₁₄₆	1	8	yes	—
162	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
163	C₂²×C₄	C₂³×D₇₃	1	2	yes	—
164	C₂×C₈	C₂×Dic₁₄₆	1	16	yes	—
165	C₂×C₈	C₂×Dic₁₄₆	1	32	yes	complemented
166	C₂×C₈	C₂³×D₇₃	1	8	yes	—
167	C₄⋊C₄	—	73	16	no	—
168	C₄⋊C₄	—	146	32	no	—
169	C₄⋊C₄	—	146	8	no	—
170	C₄⋊C₄	—	146	16	no	—
171	C₂²⋊C₄	—	146	16	no	—
172	C₄²	—	146	16	no	—
173	C₄²	C₂×D₂₉₂	1	8	yes	—
174	C₄²	C₂×D₂₉₂	1	4	yes	—
175	C₄²	C₂³×D₇₃	1	4	yes	—
176	C₂³	—	2	4	no	—
177	C₂³	#14556591	1	4	yes	—
178	C₂³	C₂²×D₂₉₂	1	2	yes	—
179	C₂³	C₂²×D₂₉₂	1	1	yes	characteristic
180	Q₈	—	292	32	no	—
181	C₂×C₄	—	2	8	no	—
182	C₂×C₄	—	4	8	no	—
183	C₂×C₄	—	292	16	no	—
184	C₂×C₄	—	146	16	no	—
185	C₂×C₄	#14556591	1	8	yes	—
186	C₂×C₄	#14556591	1	4	yes	—
187	C₂×C₄	C₂²×D₂₉₂	1	8	yes	—
188	C₂×C₄	C₂²×D₂₉₂	1	2	yes	—
189	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
190	C₂×C₄	C₂²×Dic₁₄₆	1	4	yes	—
191	C₂×C₄	C₂²×Dic₁₄₆	1	8	yes	—
192	C₂×C₄	C₂⁴×D₇₃	1	1	yes	characteristic
193	C₈	C₂²×Dic₁₄₆	1	16	yes	—
194	C₂²	—	2	8	no	—
195	C₂²	—	2	4	no	—
196	C₂²	C₂×#14556591	1	4	yes	—
197	C₂²	C₂×#14556591	1	2	yes	—
198	C₂²	#17989387	1	4	yes	—
199	C₂²	C₂³×D₂₉₂	1	1	yes	characteristic
200	C₄	—	2	8	no	—
201	C₄	—	292	16	no	—
202	C₄	C₂×#14556591	1	2	yes	—
203	C₄	C₂×#14556591	1	4	yes	—
204	C₄	C₂³×Dic₁₄₆	1	2	yes	—
205	C₂	—	2	4	no	—
206	C₂	#22620455	1	4	yes	direct product factor
207	C₂	C₂²×#14556591	1	1	yes	characteristic
208	C₂	C₂×#17989387	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₉₂
commutator, cosocle	C₂×C₄×C₇₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇₃
Fitting	C₂²×#97×C₇₃	C₂
socle	C₂³×C₇₃	C₂²×D₄
prime radical	C₂⁴×C₇₃	C₂×D₄
hypercenter	C₂²×#97	D₇₃
hypocenter	C₇₃	C₂×#3162
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. #22620455	C₂	direct (N × H)
3. C₂×#17988996	C₂	semidirect
13. #17988996	C₂²	semidirect
39. C₂²×C₈×C₇₃	Q₈	semidirect
66. C₂×C₈×C₇₃	C₂×Q₈	semidirect
102. C₂×C₇₃	#3162	semidirect
110. C₇₃	C₂×#3162	semidirect
129. C₂²×C₈	Dic₁₄₆	semidirect
165. C₂×C₈	C₂×Dic₁₄₆	semidirect
206. C₂	#22620455	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17989144	2
#22620455	2
C₂×#17988996	2
C₂×#17988625	2
C₂²×#97×C₇₃	2
C₂×#3162	73

Subgroups — C2×#22620455 #33955600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22620455 #33955600