Subgroups — C₂×#22609648 #33083384

Order 18688.

2398 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556550	C₂	1	2	yes	maximal
2	#22609648	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17988350	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556550	—	2	4	no	—
6	C₂×#14556550	—	2	8	no	—
7	C₂×#14556550	C₄	1	16	yes	complemented
8	C₂×#14556550	C₂²	1	8	yes	—
9	C₂²×C₄⋊Dic₇₃	C₂²	1	4	yes	—
10	C₂²×C₂²⋊Dic₇₃	—	2	2	no	—
11	C₂²×C₄×Dic₇₃	—	2	2	no	—
12	#17988350	C₂²	1	16	yes	complemented
13	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₇₃	—	2	8	no	—
15	C₂³×C₈×C₇₃	C₄	1	8	yes	complemented
16	C₂³×C₈×C₇₃	C₂²	1	2	yes	—
17	C₂³×C₈×C₇₃	C₂²	1	4	yes	—
18	C₂³×Dic₇₃	—	2	4	no	—
19	#14556550	—	2	32	no	—
20	#14556550	—	2	16	no	—
21	#14556550	C₂×C₄	1	32	yes	complemented
22	C₂×C₄⋊Dic₇₃	—	2	8	no	—
23	C₂×C₄⋊Dic₇₃	—	2	4	no	—
24	C₂×C₄⋊Dic₇₃	C₂×C₄	1	16	yes	—
25	C₂×C₄⋊Dic₇₃	C₂³	1	8	yes	—
26	C₂×C₂²⋊Dic₇₃	—	2	16	no	—
27	C₂×C₂²⋊Dic₇₃	—	2	8	no	—
28	C₂×C₄×Dic₇₃	—	2	16	no	—
29	C₂×C₄×Dic₇₃	—	2	8	no	—
30	C₂⁵×C₇₃	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₇₃	C₂×C₄	1	4	yes	—
32	C₂³×C₄×C₇₃	C₂×C₄	1	8	yes	—
33	C₂³×C₄×C₇₃	D₄	1	2	yes	—
34	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₇₃	D₄	1	8	yes	—
36	C₂³×C₄×C₇₃	Q₈	1	4	yes	—
37	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
38	C₂³×C₄×C₇₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₇₃	—	2	32	no	—
40	C₂²×C₈×C₇₃	—	2	8	no	—
41	C₂²×C₈×C₇₃	—	2	16	no	—
42	C₂²×C₈×C₇₃	C₂×C₄	1	8	yes	—
43	C₂²×C₈×C₇₃	C₂×C₄	1	16	yes	complemented
44	C₂²×C₈×C₇₃	C₂³	1	4	yes	—
45	C₂²×Dic₇₃	—	2	16	no	—
46	C₂²×Dic₇₃	—	2	8	no	—
47	C₂²×Dic₇₃	—	4	16	no	—
48	C₄⋊Dic₇₃	—	2	16	no	—
49	C₄⋊Dic₇₃	—	2	8	no	—
50	C₄⋊Dic₇₃	C₂²×C₄	1	16	yes	—
51	C₂²⋊Dic₇₃	—	2	32	no	—
52	C₄×Dic₇₃	—	2	32	no	—
53	C₂⁴×C₇₃	—	2	8	no	—
54	C₂⁴×C₇₃	C₂²⋊C₄	1	8	yes	—
55	C₂⁴×C₇₃	C₄⋊C₄	1	4	yes	—
56	C₂⁴×C₇₃	C₂×D₄	1	2	yes	—
57	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₃	C₄²	1	16	yes	—
59	C₂²×C₄×C₇₃	C₂²⋊C₄	1	8	yes	—
60	C₂²×C₄×C₇₃	C₂²⋊C₄	1	16	yes	—
61	C₂²×C₄×C₇₃	C₂²⋊C₄	1	16	yes	—
62	C₂²×C₄×C₇₃	C₄⋊C₄	1	32	yes	—
63	C₂²×C₄×C₇₃	C₄⋊C₄	1	8	yes	—
64	C₂²×C₄×C₇₃	C₄⋊C₄	1	4	yes	—
65	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₇₃	C₂²×C₄	1	8	yes	—
67	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₇₃	C₂×D₄	1	8	yes	—
71	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₃	C₂×Q₈	1	4	yes	—
74	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
76	C₂×C₈×C₇₃	—	2	16	no	—
77	C₂×C₈×C₇₃	—	2	32	no	—
78	C₂×C₈×C₇₃	—	2	8	no	—
79	C₂×C₈×C₇₃	C₂²×C₄	1	8	yes	—
80	C₂³×C₇₃	—	2	8	no	—
81	C₂³×C₇₃	—	2	32	no	—
82	C₂³×C₇₃	—	2	16	no	—
83	C₂³×C₇₃	#95	1	16	yes	—
84	C₂³×C₇₃	C₂×C₂²⋊C₄	1	8	yes	—
85	C₂³×C₇₃	C₂×C₂²⋊C₄	1	4	yes	—
86	C₂³×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
87	C₂³×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
88	C₂³×C₇₃	#135	1	8	yes	—
89	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₇₃	#95	1	32	yes	—
91	C₂×C₄×C₇₃	#95	1	16	yes	—
92	C₂×C₄×C₇₃	C₂×C₄²	1	8	yes	—
93	C₂×C₄×C₇₃	C₂×C₄²	1	4	yes	—
94	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	8	yes	—
95	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	4	yes	—
96	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	8	yes	—
97	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	4	yes	—
98	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	16	yes	—
99	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	8	yes	—
100	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
102	C₂×C₄×C₇₃	C₂³×C₄	1	2	yes	—
103	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
105	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
106	C₂×Dic₇₃	—	2	16	no	—
107	C₂×Dic₇₃	—	4	16	no	—
108	C₂×Dic₇₃	—	4	32	no	—
109	C₈×C₇₃	—	2	8	no	—
110	C₂²×C₇₃	—	2	16	no	—
111	C₂²×C₇₃	—	2	8	no	—
112	C₂²×C₇₃	—	2	32	no	—
113	C₂²×C₇₃	C₂×#95	1	8	yes	—
114	C₂²×C₇₃	C₂×#95	1	4	yes	—
115	C₂²×C₇₃	#416	1	16	yes	—
116	C₂²×C₇₃	#427	1	8	yes	—
117	C₂²×C₇₃	C₂²×C₂²⋊C₄	1	2	yes	—
118	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₇₃	C₂×#135	1	4	yes	—
120	C₄×C₇₃	C₂×#95	1	8	yes	—
121	C₄×C₇₃	C₂×#95	1	4	yes	—
122	C₄×C₇₃	C₂²×C₄²	1	1	yes	characteristic
123	C₄×C₇₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
124	C₄×C₇₃	C₂²×C₄⋊C₄	1	2	yes	—
125	Dic₇₃	—	4	16	no	—
126	C₂×#1735	—	73	1	no	maximal, Sylow
127	C₂×C₇₃	—	2	8	no	—
128	C₂×C₇₃	#1735	1	8	yes	complemented
129	C₂×C₇₃	C₂²×#95	1	1	yes	characteristic
130	C₂×C₇₃	C₂×#416	1	4	yes	—
131	C₂×C₇₃	C₂×#427	1	2	yes	—
132	C₂⁴×C₈	D₇₃	1	1	yes	characteristic
133	C₂²×#117	—	73	2	no	—
134	#1735	—	73	8	no	—
135	C₂×#340	—	73	4	no	—
136	C₇₃	C₂×#1735	1	1	yes	characteristic, complemented, Sylow
137	C₂⁴×C₄	C₂×D₇₃	1	1	yes	characteristic
138	C₂³×C₈	—	2	8	no	—
139	C₂³×C₈	Dic₇₃	1	8	yes	complemented
140	C₂³×C₈	C₂×D₇₃	1	2	yes	—
141	C₂³×C₈	C₂×D₇₃	1	4	yes	—
142	C₂×#117	—	73	16	no	—
143	C₂×#117	—	73	8	no	—
144	C₂×#117	—	146	4	no	—
145	C₂×#117	—	146	8	no	—
146	C₂²×C₄⋊C₄	—	73	4	no	—
147	C₂²×C₂²⋊C₄	—	146	2	no	—
148	C₂²×C₄²	—	146	2	no	—
149	#340	—	73	16	no	—
150	C₂⁵	D₂₉₂	1	1	yes	characteristic
151	C₂³×C₄	—	146	4	no	—
152	C₂³×C₄	Dic₁₄₆	1	4	yes	—
153	C₂³×C₄	C₄×D₇₃	1	8	yes	—
154	C₂³×C₄	D₂₉₂	1	2	yes	—
155	C₂³×C₄	D₂₉₂	1	1	yes	characteristic
156	C₂³×C₄	C₂×Dic₇₃	1	4	yes	—
157	C₂³×C₄	C₂²⋊D₇₃	1	8	yes	—
158	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
159	C₂³×C₄	C₂²×D₇₃	1	2	yes	—
160	C₂²×C₈	—	2	32	no	—
161	C₂²×C₈	—	2	8	no	—
162	C₂²×C₈	—	2	16	no	—
163	C₂²×C₈	C₂×Dic₇₃	1	8	yes	—
164	C₂²×C₈	C₂×Dic₇₃	1	16	yes	complemented
165	C₂²×C₈	C₂²×D₇₃	1	4	yes	—
166	#117	—	73	32	no	—
167	#117	—	146	32	no	—
168	#117	—	146	16	no	—
169	C₂×C₄⋊C₄	—	73	16	no	—
170	C₂×C₄⋊C₄	—	73	8	no	—
171	C₂×C₄⋊C₄	—	146	8	no	—
172	C₂×C₄⋊C₄	—	146	4	no	—
173	C₂×C₂²⋊C₄	—	146	16	no	—
174	C₂×C₂²⋊C₄	—	146	8	no	—
175	C₂×C₄²	—	146	16	no	—
176	C₂×C₄²	—	146	8	no	—
177	C₂⁴	—	2	8	no	—
178	C₂⁴	#11924005	1	8	yes	—
179	C₂⁴	C₄⋊Dic₇₃	1	4	yes	—
180	C₂⁴	C₂×D₂₉₂	1	2	yes	—
181	C₂⁴	C₂×D₂₉₂	1	1	yes	characteristic
182	C₂²×C₄	—	292	16	no	—
183	C₂²×C₄	—	146	16	no	—
184	C₂²×C₄	—	146	8	no	—
185	C₂²×C₄	C₄×Dic₇₃	1	16	yes	—
186	C₂²×C₄	#11924005	1	8	yes	—
187	C₂²×C₄	#11924005	1	16	yes	—
188	C₂²×C₄	C₂²⋊Dic₇₃	1	16	yes	—
189	C₂²×C₄	#11924007	1	32	yes	—
190	C₂²×C₄	C₄⋊Dic₇₃	1	8	yes	—
191	C₂²×C₄	C₄⋊Dic₇₃	1	4	yes	—
192	C₂²×C₄	C₂×C₄×D₇₃	1	4	yes	—
193	C₂²×C₄	C₂×C₄×D₇₃	1	8	yes	—
194	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
195	C₂²×C₄	C₂²×Dic₇₃	1	4	yes	—
196	C₂²×C₄	C₂×C₂²⋊D₇₃	1	4	yes	—
197	C₂²×C₄	C₂×C₂²⋊D₇₃	1	8	yes	—
198	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
199	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
200	C₂²×C₄	C₂×Dic₁₄₆	1	4	yes	—
201	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
202	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
203	C₂×C₈	—	2	16	no	—
204	C₂×C₈	—	2	32	no	—
205	C₂×C₈	—	2	8	no	—
206	C₂×C₈	C₂²×Dic₇₃	1	8	yes	—
207	C₄⋊C₄	—	73	16	no	—
208	C₄⋊C₄	—	146	16	no	—
209	C₄⋊C₄	—	146	8	no	—
210	C₂²⋊C₄	—	146	32	no	—
211	C₄²	—	146	32	no	—
212	C₂³	—	2	16	no	—
213	C₂³	—	2	32	no	—
214	C₂³	—	2	8	no	—
215	C₂³	#14556498	1	16	yes	—
216	C₂³	C₂×#11924005	1	8	yes	—
217	C₂³	C₂×#11924005	1	4	yes	—
218	C₂³	C₂×C₄⋊Dic₇₃	1	4	yes	—
219	C₂³	C₂×C₄⋊Dic₇₃	1	2	yes	—
220	C₂³	#14556613	1	8	yes	—
221	C₂³	C₂²×D₂₉₂	1	1	yes	characteristic
222	C₂×C₄	—	292	32	no	—
223	C₂×C₄	—	292	16	no	—
224	C₂×C₄	—	146	16	no	—
225	C₂×C₄	#14556498	1	32	yes	—
226	C₂×C₄	#14556498	1	16	yes	—
227	C₂×C₄	C₂×C₄×Dic₇₃	1	8	yes	—
228	C₂×C₄	C₂×C₄×Dic₇₃	1	4	yes	—
229	C₂×C₄	C₂×#11924005	1	8	yes	—
230	C₂×C₄	C₂×#11924005	1	4	yes	—
231	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	8	yes	—
232	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	4	yes	—
233	C₂×C₄	C₂×#11924007	1	16	yes	—
234	C₂×C₄	C₂×#11924007	1	8	yes	—
235	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
236	C₂×C₄	C₂×C₄⋊Dic₇₃	1	4	yes	—
237	C₂×C₄	C₂²×C₄×D₇₃	1	2	yes	—
238	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₂²⋊D₇₃	1	2	yes	—
240	C₂×C₄	C₂²×Dic₁₄₆	1	1	yes	characteristic
241	C₈	—	2	8	no	—
242	C₂²	—	2	16	no	—
243	C₂²	—	2	8	no	—
244	C₂²	—	2	32	no	—
245	C₂²	C₂×#14556498	1	8	yes	—
246	C₂²	C₂×#14556498	1	4	yes	—
247	C₂²	#17988584	1	16	yes	—
248	C₂²	#17988625	1	8	yes	—
249	C₂²	C₂²×#11924005	1	2	yes	—
250	C₂²	C₂²×C₄⋊Dic₇₃	1	1	yes	characteristic
251	C₂²	C₂×#14556613	1	4	yes	—
252	C₄	—	292	16	no	—
253	C₄	C₂×#14556498	1	8	yes	—
254	C₄	C₂×#14556498	1	4	yes	—
255	C₄	C₂²×C₄×Dic₇₃	1	1	yes	characteristic
256	C₄	C₂²×C₂²⋊Dic₇₃	1	1	yes	characteristic
257	C₄	C₂²×#11924007	1	2	yes	—
258	C₂	—	2	8	no	—
259	C₂	#22609648	1	8	yes	direct product factor
260	C₂	C₂²×#14556498	1	1	yes	characteristic
261	C₂	C₂×#17988584	1	4	yes	—
262	C₂	C₂×#17988625	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	D₂₉₂
commutator	C₄×C₇₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂⁴×C₈×C₇₃	C₂
socle	C₂⁴×C₇₃	C₂×D₄
prime radical	C₂⁵×C₇₃	D₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂⁴×C₈	D₇₃
hypocenter	C₇₃	C₂×#1735
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
2. #22609648	C₂	direct (N × H)
3. C₂×#17988350	C₂	semidirect
7. C₂×#14556550	C₄	semidirect
12. #17988350	C₂²	semidirect
15. C₂³×C₈×C₇₃	C₄	semidirect
21. #14556550	C₂×C₄	semidirect
43. C₂²×C₈×C₇₃	C₂×C₄	semidirect
128. C₂×C₇₃	#1735	semidirect
136. C₇₃	C₂×#1735	semidirect
139. C₂³×C₈	Dic₇₃	semidirect
164. C₂²×C₈	C₂×Dic₇₃	semidirect
259. C₂	#22609648	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×#14556550	2
#22609648	2
C₂×#17988350	2
C₂⁴×C₈×C₇₃	2
C₂×#1735	73

Subgroups — C2×#22609648 #33083384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22609648 #33083384