Subgroups — C₂×#22611713 #33239011

Order 18688.

694 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11924003	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988625	C₂	1	1	yes	maximal, characteristic
3	#22611713	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17988414	C₂	1	2	yes	maximal, complemented
5	C₂×#17988412	C₂	1	2	yes	maximal, complemented
6	C₂×#427×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#14556550	—	2	1	no	—
8	C₂²×#11924003	C₄	1	8	yes	complemented
9	C₂²×#11924003	C₂²	1	1	yes	characteristic
10	C₂²×#11924003	C₂²	1	1	yes	characteristic
11	C₂²×#11924003	C₂²	1	4	yes	—
12	#17988625	—	2	2	no	—
13	#17988625	C₂²	1	4	yes	—
14	C₂×#14556525	C₂²	1	2	yes	—
15	C₂×#14556523	—	2	1	no	—
16	#17988414	C₂²	1	8	yes	complemented
17	#17988412	C₂²	1	8	yes	complemented
18	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×#117×C₇₃	—	2	1	no	—
20	#427×C₇₃	—	2	2	no	—
21	#427×C₇₃	C₂²	1	4	yes	—
22	C₂×#107×C₇₃	C₂²	1	2	yes	—
23	C₂×#106×C₇₃	—	2	1	no	—
24	#14556550	—	2	4	no	—
25	#14556550	—	4	2	no	—
26	C₂×C₄⋊Dic₇₃	—	2	2	no	—
27	C₂×C₂²⋊Dic₇₃	—	4	1	no	—
28	C₂×C₄×Dic₇₃	—	4	1	no	—
29	C₂×#11924003	C₂×C₄	1	4	yes	—
30	C₂×#11924003	C₂×C₄	1	4	yes	—
31	C₂×#11924003	C₂×C₄	1	16	yes	complemented
32	C₂×#11924003	C₂³	1	2	yes	—
33	C₂×#11924003	C₂³	1	2	yes	—
34	#14556525	—	2	2	no	—
35	#14556525	C₂³	1	4	yes	—
36	#14556523	—	2	4	no	—
37	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
39	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₇₃	D₄	1	2	yes	—
41	C₂²×C₈×C₇₃	Q₈	1	4	yes	—
42	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
44	#117×C₇₃	—	2	4	no	—
45	#117×C₇₃	—	4	2	no	—
46	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
47	C₂×C₂²⋊C₄×C₇₃	—	4	1	no	—
48	C₂×C₄²×C₇₃	—	4	1	no	—
49	#107×C₇₃	—	2	2	no	—
50	#107×C₇₃	C₂³	1	4	yes	—
51	#106×C₇₃	—	2	4	no	—
52	C₂²×Dic₇₃	—	4	2	no	—
53	C₄⋊Dic₇₃	—	2	4	no	—
54	C₄⋊Dic₇₃	—	4	2	no	—
55	C₂²⋊Dic₇₃	—	4	4	no	—
56	C₄×Dic₇₃	—	4	4	no	—
57	#11924003	C₂²×C₄	1	4	yes	—
58	#11924003	C₂²×C₄	1	4	yes	—
59	C₂⁴×C₇₃	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₇₃	—	4	2	no	—
61	C₂²×C₄×C₇₃	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₇₃	D₈	1	2	yes	—
63	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₇₃	Q₁₆	1	4	yes	—
65	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
68	C₂×C₈×C₇₃	C₄⋊C₄	1	8	yes	—
69	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
71	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₇₃	C₂×Q₈	1	4	yes	—
75	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
76	C₄⋊C₄×C₇₃	—	2	4	no	—
77	C₄⋊C₄×C₇₃	—	4	2	no	—
78	C₂²⋊C₄×C₇₃	—	4	4	no	—
79	C₄²×C₇₃	—	4	4	no	—
80	C₂³×C₇₃	—	2	4	no	—
81	C₂³×C₇₃	#107	1	4	yes	—
82	C₂³×C₇₃	C₂×D₈	1	2	yes	—
83	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₇₃	—	4	4	no	—
85	C₂×C₄×C₇₃	—	8	4	no	—
86	C₂×C₄×C₇₃	#107	1	8	yes	—
87	C₂×C₄×C₇₃	#107	1	4	yes	—
88	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
89	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
91	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₇₃	C₂×Q₁₆	1	4	yes	—
93	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
94	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
95	C₂×Dic₇₃	—	4	4	no	—
96	C₂×Dic₇₃	—	8	4	no	—
97	C₈×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₈×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
99	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
100	C₈×C₇₃	C₂²×Q₈	1	1	yes	characteristic
101	C₂²×C₇₃	—	2	4	no	—
102	C₂²×C₇₃	—	2	8	no	—
103	C₂²×C₇₃	C₂×#107	1	4	yes	—
104	C₂²×C₇₃	C₂×#107	1	2	yes	—
105	C₂²×C₇₃	#508	1	4	yes	—
106	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
107	C₄×C₇₃	—	8	4	no	—
108	C₄×C₇₃	C₂×#107	1	4	yes	—
109	C₄×C₇₃	C₂×#107	1	2	yes	—
110	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
111	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
112	Dic₇₃	—	8	4	no	—
113	C₂×#2158	—	73	1	no	maximal, Sylow
114	C₂×C₇₃	—	2	4	no	—
115	C₂×C₇₃	#2158	1	4	yes	complemented
116	C₂×C₇₃	C₂²×#107	1	1	yes	characteristic
117	C₂×C₇₃	C₂×#508	1	2	yes	—
118	C₂³×C₁₆	—	73	1	no	—
119	C₂×#427	—	73	1	no	—
120	C₂×#427	D₇₃	1	1	yes	characteristic
121	#2158	—	73	8	no	—
122	C₂×#367	—	73	2	no	—
123	C₂×#366	—	73	2	no	—
124	C₇₃	C₂×#2158	1	1	yes	characteristic, complemented, Sylow
125	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
126	C₂×#117	—	2	1	no	—
127	C₂×#117	—	146	1	no	—
128	C₂²×C₁₆	—	73	8	no	—
129	C₂²×C₁₆	—	73	1	no	—
130	C₂²×C₁₆	—	73	1	no	—
131	C₂²×C₁₆	—	73	4	no	—
132	#427	—	2	2	no	—
133	#427	—	73	4	no	—
134	#427	—	146	2	no	—
135	#427	C₂×D₇₃	1	4	yes	—
136	C₂×#107	—	73	2	no	—
137	C₂×#107	C₂×D₇₃	1	2	yes	—
138	C₂×#106	—	2	1	no	—
139	C₂×#106	—	146	1	no	—
140	#367	—	73	8	no	—
141	#366	—	73	8	no	—
142	C₂³×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
143	C₂²×C₈	Dic₁₄₆	1	4	yes	—
144	C₂²×C₈	C₄×D₇₃	1	4	yes	—
145	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
147	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
148	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
149	#117	—	2	4	no	—
150	#117	—	4	2	no	—
151	#117	—	292	2	no	—
152	#117	—	146	4	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	146	2	no	—
155	C₂×C₂²⋊C₄	—	4	1	no	—
156	C₂×C₂²⋊C₄	—	292	1	no	—
157	C₂×C₄²	—	4	1	no	—
158	C₂×C₄²	—	292	1	no	—
159	C₂×C₁₆	—	73	4	no	—
160	C₂×C₁₆	—	73	4	no	—
161	C₂×C₁₆	—	73	16	no	—
162	C₂×C₁₆	—	73	2	no	—
163	C₂×C₁₆	—	73	2	no	—
164	#107	—	2	2	no	—
165	#107	—	73	4	no	—
166	#107	—	146	2	no	—
167	#107	C₂²×D₇₃	1	4	yes	—
168	#106	—	2	4	no	—
169	#106	—	146	4	no	—
170	C₂⁴	#11924013	1	1	yes	characteristic
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	—	292	2	no	—
173	C₂²×C₄	#11924007	1	4	yes	—
174	C₂²×C₄	#11924013	1	2	yes	—
175	C₂²×C₄	#11924013	1	1	yes	characteristic
176	C₂²×C₄	#11924018	1	4	yes	—
177	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
179	C₂×C₈	#11924007	1	4	yes	—
180	C₂×C₈	#11924007	1	8	yes	—
181	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
182	C₂×C₈	C₂×C₄×D₇₃	1	4	yes	—
183	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
184	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
185	C₂×C₈	C₂×Dic₁₄₆	1	2	yes	—
186	C₂×C₈	C₂×Dic₁₄₆	1	4	yes	—
187	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	292	2	no	—
191	C₄⋊C₄	—	146	4	no	—
192	C₂²⋊C₄	—	4	4	no	—
193	C₂²⋊C₄	—	292	4	no	—
194	C₄²	—	4	4	no	—
195	C₄²	—	292	4	no	—
196	C₁₆	—	73	4	no	—
197	C₁₆	—	73	4	no	—
198	C₂³	—	2	4	no	—
199	C₂³	#14556524	1	4	yes	—
200	C₂³	C₂×#11924013	1	2	yes	—
201	C₂³	C₂×#11924013	1	1	yes	characteristic
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	8	4	no	—
204	C₂×C₄	—	292	4	no	—
205	C₂×C₄	—	584	4	no	—
206	C₂×C₄	#14556524	1	8	yes	—
207	C₂×C₄	#14556524	1	4	yes	—
208	C₂×C₄	C₂×#11924007	1	4	yes	—
209	C₂×C₄	C₂×#11924007	1	2	yes	—
210	C₂×C₄	C₂×#11924013	1	2	yes	—
211	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
212	C₂×C₄	C₂×#11924018	1	4	yes	—
213	C₂×C₄	C₂×#11924018	1	2	yes	—
214	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
215	C₈	C₂×#11924007	1	4	yes	—
216	C₈	C₂×#11924007	1	2	yes	—
217	C₈	C₂²×C₄×D₇₃	1	1	yes	characteristic
218	C₈	C₂²×Dic₁₄₆	1	1	yes	characteristic
219	C₂²	—	2	8	no	—
220	C₂²	—	2	4	no	—
221	C₂²	C₂×#14556524	1	4	yes	—
222	C₂²	C₂×#14556524	1	2	yes	—
223	C₂²	#17989017	1	4	yes	—
224	C₂²	C₂²×#11924013	1	1	yes	characteristic
225	C₄	—	8	4	no	—
226	C₄	—	584	4	no	—
227	C₄	C₂×#14556524	1	2	yes	—
228	C₄	C₂×#14556524	1	4	yes	—
229	C₄	C₂²×#11924007	1	1	yes	characteristic
230	C₄	C₂²×#11924018	1	1	yes	characteristic
231	C₂	—	2	4	no	—
232	C₂	#22611713	1	4	yes	direct product factor
233	C₂	C₂²×#14556524	1	1	yes	characteristic
234	C₂	C₂×#17989017	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	#11924013
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×#427×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₈
prime radical	C₂⁴×C₇₃	D₈
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₂×#427	D₇₃
hypocenter	C₇₃	C₂×#2158
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
3. #22611713	C₂	direct (N × H)
4. C₂×#17988414	C₂	semidirect
5. C₂×#17988412	C₂	semidirect
8. C₂²×#11924003	C₄	semidirect
16. #17988414	C₂²	semidirect
17. #17988412	C₂²	semidirect
31. C₂×#11924003	C₂×C₄	semidirect
115. C₂×C₇₃	#2158	semidirect
124. C₇₃	C₂×#2158	semidirect
232. C₂	#22611713	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11924003	2
C₂×#17988625	2
#22611713	2
C₂×#17988414	2
C₂×#17988412	2
C₂×#427×C₇₃	2
C₂×#2158	73

Subgroups — C2×#22611713 #33239011

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22611713 #33239011