Subgroups — C₂×#22626310 #34030186

Order 18688.

806 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556664	C₂	1	2	yes	maximal, complemented
2	#22626310	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊#11924031	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#17989482	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×#11924031	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17988652	C₂	1	1	yes	maximal, characteristic
7	C₂×#14556664	—	2	8	no	—
8	C₂×#14556664	C₂²	1	8	yes	complemented
9	C₂²×#11924031	C₂²	1	2	yes	—
10	C₂²×#11924031	C₂²	1	1	yes	characteristic
11	C₂²×#11924031	C₂²	1	1	yes	characteristic
12	C₄⋊#11924031	C₂²	1	4	yes	complemented
13	#17989482	C₂²	1	8	yes	complemented
14	C₄×#11924031	C₂²	1	4	yes	complemented
15	C₂²×C₈×D₇₃	C₂²	1	2	yes	—
16	C₂×C₄×D₂₉₂	C₄	1	1	yes	characteristic
17	#17988652	C₄	1	4	yes	—
18	#17988652	C₂²	1	4	yes	—
19	C₂×#14556503	C₄	1	2	yes	—
20	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₇₃	C₄	1	1	yes	characteristic
22	#14556664	—	2	16	no	—
23	C₂×#11924031	—	2	4	no	—
24	C₂×#11924031	—	2	2	no	—
25	C₂×#11924031	D₄	1	4	yes	complemented
26	C₂×#11924031	C₂³	1	4	yes	—
27	C₂×#11924031	C₂³	1	2	yes	—
28	C₂×#11924031	C₂³	1	2	yes	—
29	C₂²×D₂₉₂	C₈	1	1	yes	characteristic
30	C₂²×C₄×D₇₃	C₂×C₄	1	2	yes	—
31	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
32	C₂²×#10596831	C₂³	1	1	yes	characteristic
33	C₂×C₈×D₇₃	—	2	4	no	—
34	C₂×C₈×D₇₃	—	2	4	no	—
35	C₂×C₈×D₇₃	C₂×C₄	1	4	yes	—
36	C₂×C₈×D₇₃	C₂³	1	4	yes	—
37	C₄×D₂₉₂	C₈	1	4	yes	—
38	C₄×D₂₉₂	C₂×C₄	1	4	yes	—
39	C₂×C₄⋊Dic₇₃	C₈	1	1	yes	characteristic
40	C₂×#11924005	C₈	1	2	yes	—
41	#14556503	C₂×C₄	1	8	yes	—
42	C₄×#10596831	C₂×C₄	1	2	yes	—
43	C₄×#10596831	C₂³	1	2	yes	—
44	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
46	C₄⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
47	#11924031	—	2	4	no	—
48	#11924031	—	2	2	no	—
49	#11924031	C₂×D₄	1	4	yes	complemented
50	C₂³×D₇₃	C₂×C₈	1	2	yes	—
51	C₂×D₂₉₂	C₁₆	1	8	yes	complemented
52	C₂×D₂₉₂	C₂×C₈	1	4	yes	—
53	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
54	C₂×C₄×D₇₃	—	2	4	no	—
55	C₂×C₄×D₇₃	—	2	4	no	—
56	C₂×C₄×D₇₃	C₂×C₈	1	4	yes	—
57	C₂×C₄×D₇₃	C₂²×C₄	1	4	yes	—
58	C₂×#10596831	—	2	2	no	—
59	C₂×#10596831	C₂²×C₄	1	2	yes	—
60	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
61	C₂×#10596831	C₂×D₄	1	2	yes	—
62	C₂×#10596831	#41	1	2	yes	—
63	C₂×#10596831	C₂⁴	1	1	yes	characteristic
64	C₈×D₇₃	—	2	8	no	—
65	C₈×D₇₃	—	2	4	no	—
66	C₈×D₇₃	—	2	4	no	—
67	C₄⋊Dic₇₃	C₂×C₈	1	4	yes	—
68	#11924005	C₂×C₈	1	8	yes	—
69	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₇₃	—	2	4	no	—
73	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
74	C₄²×C₇₃	C₂×C₈	1	2	yes	—
75	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
76	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
77	C₂²×D₇₃	—	2	4	no	—
78	C₂²×D₇₃	—	2	4	no	—
79	C₂²×D₇₃	C₂×C₁₆	1	8	yes	complemented
80	C₂²×D₇₃	C₂²×C₈	1	4	yes	—
81	C₂×C₄×C₇₃	—	2	2	no	—
82	C₂×C₄×C₇₃	C₂×C₁₆	1	4	yes	complemented
83	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
85	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
87	C₂×C₄×C₇₃	#131	1	2	yes	—
88	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
89	C₂×Dic₇₃	—	2	4	no	—
90	C₂×Dic₇₃	C₂²×C₈	1	4	yes	—
91	D₂₉₂	C₂×C₁₆	1	16	yes	complemented
92	C₄×D₇₃	—	2	4	no	—
93	C₄×D₇₃	—	2	4	no	—
94	C₄×D₇₃	—	2	8	no	—
95	C₈×C₇₃	—	2	4	no	—
96	#10596831	—	2	2	no	—
97	#10596831	C₄×D₄	1	2	yes	—
98	#10596831	C₂²×D₄	1	1	yes	characteristic
99	#10596831	C₂×#41	1	1	yes	characteristic
100	C₂²×C₇₃	C₈×D₄	1	2	yes	—
101	C₂²×C₇₃	C₂²×C₁₆	1	2	yes	—
102	C₂²×C₇₃	#504	1	2	yes	—
103	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
104	C₂×D₇₃	—	2	4	no	—
105	C₂×D₇₃	—	2	4	no	—
106	C₂×D₇₃	—	2	16	no	—
107	C₂×D₇₃	C₂²×C₁₆	1	8	yes	complemented
108	C₄×C₇₃	—	2	2	no	—
109	C₄×C₇₃	C₈×D₄	1	2	yes	—
110	C₄×C₇₃	C₂²×C₁₆	1	4	yes	complemented
111	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
112	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
113	Dic₇₃	—	2	4	no	—
114	C₂×D₄×C₁₆	—	73	1	no	maximal, Sylow
115	C₂×C₇₃	D₄×C₁₆	1	4	yes	complemented
116	C₂×C₇₃	C₂×C₈×D₄	1	1	yes	characteristic
117	C₂×C₇₃	C₂³×C₁₆	1	1	yes	characteristic
118	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
119	D₇₃	—	2	8	no	—
120	C₂³×C₁₆	—	73	2	no	—
121	C₂×C₈×D₄	—	73	1	no	—
122	D₄×C₁₆	—	73	8	no	—
123	C₂×C₄⋊C₁₆	—	73	1	no	—
124	C₂×C₂²⋊C₁₆	—	73	2	no	—
125	C₂×C₄×C₁₆	—	73	1	no	—
126	C₇₃	C₂×D₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
127	C₂³×C₈	—	73	2	no	—
128	C₂×C₄×D₄	—	73	1	no	—
129	C₂²×C₁₆	—	73	1	no	—
130	C₂²×C₁₆	—	73	1	no	—
131	C₂²×C₁₆	—	73	2	no	—
132	C₂²×C₁₆	—	73	8	no	—
133	C₂²×C₁₆	—	146	8	no	—
134	C₈×D₄	—	73	4	no	—
135	C₈×D₄	—	73	4	no	—
136	C₂×C₄⋊C₈	#64408	1	1	yes	characteristic
137	C₂×C₂²⋊C₈	—	73	2	no	—
138	C₂×C₄×C₈	—	73	1	no	—
139	C₄⋊C₁₆	—	73	4	no	—
140	C₂²⋊C₁₆	—	73	8	no	—
141	C₄×C₁₆	—	73	4	no	—
142	C₂²×D₄	—	73	1	no	—
143	C₂³×C₄	—	73	2	no	—
144	C₂²×C₈	—	73	1	no	—
145	C₂²×C₈	—	73	1	no	—
146	C₂²×C₈	—	73	4	no	—
147	C₂²×C₈	—	73	4	no	—
148	C₂²×C₈	—	146	4	no	—
149	C₂²×C₈	—	146	4	no	—
150	C₂²×C₈	C₂×#64408	1	2	yes	—
151	C₄×D₄	—	73	4	no	—
152	C₄×D₄	—	73	4	no	—
153	C₂×C₄⋊C₄	—	73	1	no	—
154	C₂×C₂²⋊C₄	—	73	2	no	—
155	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
156	C₂×C₁₆	—	73	2	no	—
157	C₂×C₁₆	—	73	4	no	—
158	C₂×C₁₆	—	73	2	no	—
159	C₂×C₁₆	—	73	4	no	—
160	C₂×C₁₆	—	146	16	no	—
161	C₂×C₁₆	—	146	4	no	—
162	C₂×C₁₆	—	146	2	no	—
163	C₄⋊C₈	C₂×#64408	1	4	yes	—
164	C₂²⋊C₈	—	73	8	no	—
165	C₄×C₈	—	73	2	no	—
166	C₄×C₈	—	73	2	no	—
167	C₂⁴	—	73	2	no	—
168	C₂×D₄	—	73	4	no	—
169	C₂×D₄	—	73	8	no	—
170	C₂²×C₄	—	73	2	no	—
171	C₂²×C₄	—	73	4	no	—
172	C₂²×C₄	—	73	4	no	—
173	C₂²×C₄	—	146	4	no	—
174	C₂²×C₄	—	146	4	no	—
175	C₂²×C₄	#11924036	1	1	yes	characteristic
176	C₂²×C₄	#11924036	1	1	yes	characteristic
177	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	73	1	no	—
180	C₂×C₈	—	73	1	no	—
181	C₂×C₈	—	73	2	no	—
182	C₂×C₈	—	73	2	no	—
183	C₂×C₈	—	73	2	no	—
184	C₂×C₈	—	146	4	no	—
185	C₂×C₈	—	146	4	no	—
186	C₂×C₈	—	146	8	no	—
187	C₂×C₈	—	146	2	no	—
188	C₂×C₈	C₂²×#64408	1	4	yes	—
189	C₄⋊C₄	—	73	4	no	—
190	C₂²⋊C₄	—	73	8	no	—
191	C₄²	#11924036	1	2	yes	—
192	C₄²	C₂²×#64408	1	2	yes	—
193	C₁₆	—	73	4	no	—
194	C₁₆	—	146	2	no	—
195	C₁₆	—	146	4	no	—
196	C₂³	—	73	8	no	—
197	C₂³	—	73	4	no	—
198	C₂³	—	146	4	no	—
199	C₂³	—	146	4	no	—
200	C₂³	C₂×#11924036	1	1	yes	characteristic
201	D₄	—	73	16	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	73	4	no	—
204	C₂×C₄	—	146	4	no	—
205	C₂×C₄	—	146	4	no	—
206	C₂×C₄	—	146	4	no	—
207	C₂×C₄	—	146	8	no	—
208	C₂×C₄	#14556664	1	4	yes	complemented
209	C₂×C₄	D₄×#64408	1	2	yes	—
210	C₂×C₄	C₂×#11924036	1	2	yes	—
211	C₂×C₄	C₂×#11924036	1	2	yes	—
212	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
213	C₂×C₄	#14556679	1	2	yes	—
214	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
215	C₈	—	2	4	no	—
216	C₈	—	73	1	no	—
217	C₈	—	73	1	no	—
218	C₈	—	73	2	no	—
219	C₈	—	146	2	no	—
220	C₂²	—	73	8	no	—
221	C₂²	—	146	16	no	—
222	C₂²	—	146	4	no	—
223	C₂²	—	146	4	no	—
224	C₂²	#17989615	1	2	yes	—
225	C₂²	C₂×#14556664	1	2	yes	—
226	C₂²	#17989646	1	2	yes	—
227	C₂²	C₂²×#11924036	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	146	4	no	—
230	C₄	#17989615	1	2	yes	—
231	C₄	C₂×#14556664	1	4	yes	complemented
232	C₄	C₂×D₄×#64408	1	1	yes	characteristic
233	C₄	C₂×#14556679	1	1	yes	characteristic
234	C₂	—	146	8	no	—
235	C₂	#22626310	1	4	yes	direct product factor
236	C₂	C₂×#17989615	1	1	yes	characteristic
237	C₂	C₂²×#14556664	1	1	yes	characteristic
238	C₂	C₂×#17989646	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#64408
commutator	C₂×C₇₃	C₂³×C₁₆
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₂×C₄⋊C₈×C₇₃	C₄
socle	C₂³×C₇₃	C₂²×C₈
prime radical	C₂²×D₂₉₂	C₈
cosocle	C₂×#10596831	C₂⁴
hypercenter	C₂×C₄⋊C₈	#64408
hypocenter	C₇₃	C₂×D₄×C₁₆
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#14556664	C₂	semidirect
2. #22626310	C₂	direct (N × H)
3. C₂×C₄⋊#11924031	C₂	semidirect
4. C₂×#17989482	C₂	semidirect
5. C₂×C₄×#11924031	C₂	semidirect
8. C₂×#14556664	C₂²	semidirect
12. C₄⋊#11924031	C₂²	semidirect
13. #17989482	C₂²	semidirect
14. C₄×#11924031	C₂²	semidirect
25. C₂×#11924031	D₄	semidirect
49. #11924031	C₂×D₄	semidirect
51. C₂×D₂₉₂	C₁₆	semidirect
79. C₂²×D₇₃	C₂×C₁₆	semidirect
82. C₂×C₄×C₇₃	C₂×C₁₆	semidirect
91. D₂₉₂	C₂×C₁₆	semidirect
107. C₂×D₇₃	C₂²×C₁₆	semidirect
110. C₄×C₇₃	C₂²×C₁₆	semidirect
115. C₂×C₇₃	D₄×C₁₆	semidirect
126. C₇₃	C₂×D₄×C₁₆	semidirect
208. C₂×C₄	#14556664	semidirect
231. C₄	C₂×#14556664	semidirect
235. C₂	#22626310	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×#14556664	2
#22626310	2
C₂×C₄⋊#11924031	2
C₂×#17989482	2
C₂×C₄×#11924031	2
C₂×#17988652	2
C₂×D₄×C₁₆	73

Subgroups — C2×#22626310 #34030186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22626310 #34030186