Subgroups — C₂×#22611803 #33239282

Order 18688.

670 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₅₈₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₉₂	C₂	1	2	yes	maximal
3	#22611803	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17988415	C₂	1	1	yes	maximal, characteristic
5	C₂×#17988392	C₂	1	2	yes	maximal
6	C₂×#346×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₉₂	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₄₆	—	2	2	no	—
9	C₂×Dic₅₈₄	C₄	1	8	yes	complemented
10	C₂×Dic₅₈₄	C₂²	1	4	yes	—
11	C₄×Dic₂₉₂	—	2	4	no	—
12	C₄×Dic₂₉₂	C₂²	1	8	yes	—
13	C₂×#14556525	C₂²	1	2	yes	—
14	C₂×#14556516	—	2	2	no	—
15	#17988415	C₂²	1	4	yes	—
16	#17988392	C₂²	1	8	yes	—
17	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
20	#346×C₇₃	C₂²	1	4	yes	—
21	C₂²×Dic₁₄₆	—	2	2	no	—
22	C₂×Dic₂₉₂	—	2	2	no	—
23	C₂×Dic₂₉₂	—	2	4	no	—
24	C₂×Dic₂₉₂	C₂×C₄	1	8	yes	—
25	C₂×Dic₂₉₂	C₂³	1	4	yes	—
26	C₄×Dic₁₄₆	—	2	8	no	—
27	C₄×Dic₁₄₆	—	4	4	no	—
28	C₂×C₄⋊Dic₇₃	—	2	2	no	—
29	C₂×#11924007	—	4	2	no	—
30	C₂×C₄×Dic₇₃	—	4	2	no	—
31	Dic₅₈₄	C₂×C₄	1	16	yes	complemented
32	#14556525	—	2	2	no	—
33	#14556525	C₂³	1	4	yes	—
34	#14556516	—	2	8	no	—
35	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
38	C₂×C₁₆×C₇₃	—	2	2	no	—
39	C₂×C₁₆×C₇₃	C₂×C₄	1	4	yes	—
40	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
41	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
42	C₄×C₈×C₇₃	D₄	1	2	yes	—
43	C₄×C₈×C₇₃	C₂³	1	2	yes	—
44	C₂×Dic₁₄₆	—	2	8	no	—
45	C₂×Dic₁₄₆	—	2	4	no	—
46	C₂×Dic₁₄₆	—	4	2	no	—
47	C₂×Dic₁₄₆	—	4	4	no	—
48	C₂²×Dic₇₃	—	4	2	no	—
49	C₂²×Dic₇₃	—	4	2	no	—
50	Dic₂₉₂	—	2	8	no	—
51	Dic₂₉₂	—	2	4	no	—
52	Dic₂₉₂	C₂²×C₄	1	8	yes	—
53	C₄⋊Dic₇₃	—	2	4	no	—
54	C₄⋊Dic₇₃	—	4	2	no	—
55	#11924007	—	4	8	no	—
56	C₄×Dic₇₃	—	4	8	no	—
57	C₂²×C₄×C₇₃	D₈	1	2	yes	—
58	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₇₃	—	2	2	no	—
60	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₇₃	#41	1	2	yes	—
64	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
65	C₄²×C₇₃	D₈	1	2	yes	—
66	C₄²×C₇₃	C₂×D₄	1	2	yes	—
67	C₁₆×C₇₃	—	2	2	no	—
68	C₁₆×C₇₃	C₂²×C₄	1	4	yes	—
69	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₇₃	—	2	4	no	—
71	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₇₃	C₂×D₈	1	4	yes	—
73	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₇₃	#135	1	2	yes	—
75	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
76	C₂×Dic₇₃	—	4	4	no	—
77	C₂×Dic₇₃	—	4	8	no	—
78	C₂×Dic₇₃	—	4	4	no	—
79	C₂×Dic₇₃	—	8	4	no	—
80	Dic₁₄₆	—	2	8	no	—
81	Dic₁₄₆	—	4	4	no	—
82	Dic₁₄₆	—	4	8	no	—
83	C₈×C₇₃	—	2	2	no	—
84	C₈×C₇₃	C₄×D₄	1	2	yes	—
85	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
86	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
87	C₂²×C₇₃	C₄×D₈	1	2	yes	—
88	C₂²×C₇₃	#510	1	4	yes	—
89	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
90	C₄×C₇₃	—	2	4	no	—
91	C₄×C₇₃	C₄×D₈	1	2	yes	—
92	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
93	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
94	Dic₇₃	—	4	8	no	—
95	Dic₇₃	—	8	4	no	—
96	C₂×#2176	—	73	1	no	maximal, Sylow
97	C₂×C₇₃	#2176	1	4	yes	complemented
98	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
99	C₂×C₇₃	C₂×#510	1	2	yes	—
100	C₂²×Q₃₂	—	73	1	no	—
101	C₂×C₄×Q₁₆	—	73	2	no	—
102	#2176	—	73	8	no	—
103	C₂×#367	—	73	1	no	—
104	C₂×#358	—	73	2	no	—
105	C₂×#346	D₇₃	1	1	yes	characteristic
106	C₇₃	C₂×#2176	1	1	yes	characteristic, complemented, Sylow
107	C₂²×Q₁₆	—	73	2	no	—
108	C₂×C₄×Q₈	—	146	2	no	—
109	C₂×Q₃₂	—	73	8	no	—
110	C₂×Q₃₂	—	73	4	no	—
111	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
112	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
113	C₄×Q₁₆	—	73	8	no	—
114	C₄×Q₁₆	—	146	4	no	—
115	C₂×#107	—	73	2	no	—
116	C₂×#103	—	146	2	no	—
117	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
118	#367	—	73	4	no	—
119	#358	—	73	8	no	—
120	#346	C₂×D₇₃	1	4	yes	—
121	C₂²×Q₈	—	146	2	no	—
122	C₂×Q₁₆	—	73	8	no	—
123	C₂×Q₁₆	—	73	4	no	—
124	C₂×Q₁₆	—	146	2	no	—
125	C₂×Q₁₆	—	146	4	no	—
126	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
128	C₄×Q₈	—	292	4	no	—
129	C₄×Q₈	—	146	8	no	—
130	C₂×C₄⋊C₄	—	292	2	no	—
131	C₂×C₄⋊C₄	—	146	2	no	—
132	C₂×C₄²	—	292	2	no	—
133	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
134	Q₃₂	—	73	16	no	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	C₄×D₇₃	1	4	yes	—
137	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
138	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
139	#107	—	73	4	no	—
140	#107	—	146	2	no	—
141	#103	—	146	8	no	—
142	C₄×C₈	D₂₉₂	1	2	yes	—
143	C₄×C₈	C₂²×D₇₃	1	2	yes	—
144	C₂×Q₈	—	292	2	no	—
145	C₂×Q₈	—	292	4	no	—
146	C₂×Q₈	—	146	8	no	—
147	C₂×Q₈	—	146	4	no	—
148	C₂²×C₄	—	292	2	no	—
149	C₂²×C₄	—	292	2	no	—
150	C₂²×C₄	D₅₈₄	1	2	yes	—
151	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
152	Q₁₆	—	73	8	no	—
153	Q₁₆	—	146	8	no	—
154	Q₁₆	—	146	4	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
157	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
158	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
159	C₂×C₈	#11924028	1	2	yes	—
160	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
161	C₄⋊C₄	—	292	8	no	—
162	C₄⋊C₄	—	292	2	no	—
163	C₄⋊C₄	—	146	4	no	—
164	C₄²	—	292	8	no	—
165	C₄²	D₅₈₄	1	2	yes	—
166	C₄²	C₂×D₂₉₂	1	2	yes	—
167	C₁₆	—	2	2	no	—
168	C₁₆	C₂×C₄×D₇₃	1	4	yes	—
169	C₂³	C₂×D₅₈₄	1	1	yes	characteristic
170	Q₈	—	292	8	no	—
171	Q₈	—	292	4	no	—
172	Q₈	—	146	8	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	292	8	no	—
175	C₂×C₄	—	292	4	no	—
176	C₂×C₄	—	292	4	no	—
177	C₂×C₄	—	584	4	no	—
178	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
179	C₂×C₄	C₂×D₅₈₄	1	4	yes	—
180	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
181	C₂×C₄	#14556613	1	2	yes	—
182	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	C₄×D₂₉₂	1	2	yes	—
185	C₈	C₂²×C₄×D₇₃	1	1	yes	characteristic
186	C₈	C₂×#11924028	1	1	yes	characteristic
187	C₂²	C₄×D₅₈₄	1	2	yes	—
188	C₂²	#17989030	1	4	yes	—
189	C₂²	C₂²×D₅₈₄	1	1	yes	characteristic
190	C₄	—	2	4	no	—
191	C₄	—	292	8	no	—
192	C₄	—	584	4	no	—
193	C₄	C₄×D₅₈₄	1	2	yes	—
194	C₄	C₂×C₄×D₂₉₂	1	1	yes	characteristic
195	C₄	C₂×#14556613	1	1	yes	characteristic
196	C₂	#22611803	1	4	yes	direct product factor
197	C₂	C₂×C₄×D₅₈₄	1	1	yes	characteristic
198	C₂	C₂×#17989030	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₅₈₄
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×#346×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂×D₈
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₂×#346	D₇₃
hypocenter	C₇₃	C₂×#2176
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. #22611803	C₂	direct (N × H)
9. C₂×Dic₅₈₄	C₄	semidirect
31. Dic₅₈₄	C₂×C₄	semidirect
97. C₂×C₇₃	#2176	semidirect
106. C₇₃	C₂×#2176	semidirect
196. C₂	#22611803	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×Dic₅₈₄	2
C₂×C₄×Dic₂₉₂	2
#22611803	2
C₂×#17988415	2
C₂×#17988392	2
C₂×#346×C₇₃	2
C₂×#2176	73

Subgroups — C2×#22611803 #33239282

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22611803 #33239282