Subgroups — C₂×#22612085 #33240596

Order 18688.

606 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₅₈₄	C₂	1	1	yes	maximal, characteristic
2	#22612085	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17988414	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#17988412	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊#11924003	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17988387	C₂	1	1	yes	maximal, characteristic
7	C₂×#17988387	C₂	1	1	yes	maximal, characteristic
8	C₂×#499×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₅₈₄	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₂₉₂	—	2	1	no	—
11	C₂²×#11924003	C₂²	1	1	yes	characteristic
12	C₂²×#11924003	C₂²	1	1	yes	characteristic
13	C₄×D₅₈₄	—	2	2	no	—
14	C₄×D₅₈₄	C₂²	1	4	yes	—
15	C₂×#14556525	C₂²	1	1	yes	characteristic
16	C₂×#14556513	—	2	1	no	—
17	#17988414	C₂²	1	4	yes	complemented
18	#17988412	C₂²	1	4	yes	complemented
19	C₄⋊#11924003	C₂²	1	4	yes	complemented
20	#17988387	C₂²	1	4	yes	—
21	#17988387	C₂²	1	4	yes	—
22	C₂×#125×C₇₃	—	2	1	no	—
23	#499×C₇₃	—	2	2	no	—
24	#499×C₇₃	C₂²	1	4	yes	complemented
25	C₂×#107×C₇₃	C₂²	1	1	yes	characteristic
26	C₂×#107×C₇₃	C₂²	1	1	yes	characteristic
27	C₂×#106×C₇₃	—	2	1	no	—
28	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
29	C₂²×D₂₉₂	—	2	1	no	—
30	C₂²×C₄×D₇₃	—	4	1	no	—
31	C₂×D₅₈₄	—	2	1	no	—
32	C₂×D₅₈₄	—	2	2	no	—
33	C₂×D₅₈₄	Q₈	1	4	yes	—
34	C₂×D₅₈₄	C₂³	1	2	yes	—
35	C₄×D₂₉₂	—	2	4	no	—
36	C₄×D₂₉₂	—	4	2	no	—
37	C₂×C₄⋊Dic₇₃	—	2	1	no	—
38	C₂×#11924005	—	4	1	no	—
39	C₂×#11924003	—	2	2	no	—
40	C₂×#11924003	—	2	2	no	—
41	C₂×#11924003	C₂³	1	2	yes	—
42	C₂×#11924003	C₂³	1	2	yes	—
43	#14556525	—	2	1	no	—
44	#14556525	C₂³	1	2	yes	—
45	#14556513	—	2	4	no	—
46	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
48	#125×C₇₃	—	2	4	no	—
49	#125×C₇₃	—	4	2	no	—
50	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
51	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
52	C₂×C₄⋊C₄×C₇₃	—	4	1	no	—
53	C₂×C₄⋊C₄×C₇₃	—	4	1	no	—
54	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
55	#107×C₇₃	—	2	1	no	—
56	#107×C₇₃	—	2	1	no	—
57	#107×C₇₃	C₂³	1	2	yes	—
58	#107×C₇₃	C₂³	1	2	yes	—
59	#106×C₇₃	—	2	4	no	—
60	C₄×C₈×C₇₃	D₄	1	2	yes	—
61	C₄×C₈×C₇₃	C₂³	1	2	yes	—
62	C₂³×D₇₃	—	4	1	no	—
63	C₂×D₂₉₂	—	2	4	no	—
64	C₂×D₂₉₂	—	2	2	no	—
65	C₂×D₂₉₂	—	4	1	no	—
66	C₂×D₂₉₂	—	4	2	no	—
67	C₂²×Dic₇₃	—	4	1	no	—
68	C₂×C₄×D₇₃	—	4	4	no	—
69	C₂×C₄×D₇₃	—	8	4	no	—
70	D₅₈₄	—	2	4	no	—
71	D₅₈₄	—	2	2	no	—
72	D₅₈₄	C₂×Q₈	1	4	yes	—
73	C₄⋊Dic₇₃	—	2	2	no	—
74	C₄⋊Dic₇₃	—	4	1	no	—
75	#11924005	—	4	4	no	—
76	#11924003	—	2	2	no	—
77	#11924003	—	2	2	no	—
78	C₂²×C₄×C₇₃	—	4	1	no	—
79	C₂²×C₄×C₇₃	—	4	1	no	—
80	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₇₃	—	2	2	no	—
84	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
87	C₂×C₈×C₇₃	#41	1	2	yes	—
88	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₇₃	—	2	2	no	—
90	C₄⋊C₄×C₇₃	—	2	2	no	—
91	C₄⋊C₄×C₇₃	—	4	1	no	—
92	C₄⋊C₄×C₇₃	—	4	1	no	—
93	C₄⋊C₄×C₇₃	—	4	4	no	—
94	C₄⋊C₄×C₇₃	—	4	4	no	—
95	C₄²×C₇₃	D₈	1	2	yes	—
96	C₄²×C₇₃	C₂×D₄	1	2	yes	—
97	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
98	C₂²×D₇₃	—	4	2	no	—
99	C₂²×D₇₃	—	4	4	no	—
100	C₂²×D₇₃	—	8	2	no	—
101	C₂²×D₇₃	—	8	2	no	—
102	C₂×C₄×C₇₃	—	2	2	no	—
103	C₂×C₄×C₇₃	—	2	2	no	—
104	C₂×C₄×C₇₃	—	4	2	no	—
105	C₂×C₄×C₇₃	—	4	2	no	—
106	C₂×C₄×C₇₃	—	8	2	no	—
107	C₂×C₄×C₇₃	—	8	2	no	—
108	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
111	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₇₃	#137	1	2	yes	—
113	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×Dic₇₃	—	4	2	no	—
115	C₂×Dic₇₃	—	8	2	no	—
116	D₂₉₂	—	2	4	no	—
117	D₂₉₂	—	4	2	no	—
118	D₂₉₂	—	4	4	no	—
119	C₄×D₇₃	—	8	8	no	—
120	C₈×C₇₃	—	2	2	no	—
121	C₈×C₇₃	C₂²⋊Q₈	1	2	yes	—
122	C₈×C₇₃	C₂²×Q₈	1	1	yes	characteristic
123	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
124	C₂²×C₇₃	#474	1	2	yes	—
125	C₂²×C₇₃	#508	1	2	yes	—
126	C₂²×C₇₃	#509	1	2	yes	—
127	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
128	C₂×D₇₃	—	4	4	no	—
129	C₂×D₇₃	—	8	2	no	—
130	C₂×D₇₃	—	8	2	no	—
131	C₂×D₇₃	—	8	8	no	—
132	C₄×C₇₃	—	2	2	no	—
133	C₄×C₇₃	—	2	2	no	—
134	C₄×C₇₃	—	8	2	no	—
135	C₄×C₇₃	—	8	2	no	—
136	C₄×C₇₃	#474	1	2	yes	—
137	C₄×C₇₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
138	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
139	Dic₇₃	—	8	2	no	—
140	C₂×#2228	—	73	1	no	maximal, Sylow
141	C₂×C₇₃	#2228	1	4	yes	complemented
142	C₂×C₇₃	C₂×#474	1	1	yes	characteristic
143	C₂×C₇₃	C₂×#508	1	1	yes	characteristic
144	C₂×C₇₃	C₂×#509	1	1	yes	characteristic
145	D₇₃	—	8	4	no	—
146	C₂×#499	D₇₃	1	1	yes	characteristic, complemented
147	C₂×C₄×D₈	—	73	1	no	—
148	#2228	—	73	8	no	—
149	C₂×#367	—	73	1	no	—
150	C₂×#366	—	73	1	no	—
151	C₂×C₄⋊C₁₆	—	73	1	no	—
152	C₂×#357	—	73	1	no	—
153	C₂×#357	—	73	1	no	—
154	C₇₃	C₂×#2228	1	1	yes	characteristic, complemented, Sylow
155	C₂²×D₈	—	73	1	no	—
156	C₂×#125	—	2	1	no	—
157	C₂×C₄×D₄	—	146	1	no	—
158	C₂²×C₁₆	—	73	1	no	—
159	C₂²×C₁₆	—	73	1	no	—
160	#499	—	2	2	no	—
161	#499	C₂×D₇₃	1	4	yes	complemented
162	C₄×D₈	—	73	4	no	—
163	C₄×D₈	—	146	2	no	—
164	C₂×#107	—	73	1	no	—
165	C₂×#107	C₂×D₇₃	1	1	yes	characteristic
166	C₂×#107	C₂×D₇₃	1	1	yes	characteristic
167	C₂×#106	—	2	1	no	—
168	C₂×#102	—	146	1	no	—
169	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
170	#367	—	73	4	no	—
171	#366	—	73	4	no	—
172	C₄⋊C₁₆	—	73	4	no	—
173	#357	—	73	4	no	—
174	#357	—	73	4	no	—
175	C₂²×D₄	—	146	1	no	—
176	C₂³×C₄	—	292	1	no	—
177	C₂×D₈	—	73	4	no	—
178	C₂×D₈	—	73	2	no	—
179	C₂×D₈	—	146	1	no	—
180	C₂×D₈	—	146	2	no	—
181	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
183	#125	—	2	4	no	—
184	#125	—	4	2	no	—
185	C₄×D₄	—	292	2	no	—
186	C₄×D₄	—	146	4	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	4	1	no	—
190	C₂×C₄⋊C₄	—	4	1	no	—
191	C₂×C₄⋊C₄	—	146	1	no	—
192	C₂×C₂²⋊C₄	—	292	1	no	—
193	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
194	C₂×C₁₆	—	73	2	no	—
195	C₂×C₁₆	—	73	2	no	—
196	C₂×C₁₆	—	146	2	no	—
197	C₂×C₁₆	—	146	2	no	—
198	#107	—	2	1	no	—
199	#107	—	2	1	no	—
200	#107	—	73	2	no	—
201	#107	—	146	1	no	—
202	#107	C₂²×D₇₃	1	2	yes	—
203	#107	C₂²×D₇₃	1	2	yes	—
204	#106	—	2	4	no	—
205	#102	—	146	4	no	—
206	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
207	C₄×C₈	C₂²×D₇₃	1	2	yes	—
208	C₂⁴	—	292	1	no	—
209	C₂×D₄	—	292	1	no	—
210	C₂×D₄	—	292	2	no	—
211	C₂×D₄	—	146	4	no	—
212	C₂×D₄	—	146	2	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	292	1	no	—
216	C₂²×C₄	—	292	4	no	—
217	C₂²×C₄	—	584	4	no	—
218	C₂²×C₄	#11924013	1	1	yes	characteristic
219	C₂²×C₄	#11924013	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
221	D₈	—	73	4	no	—
222	D₈	—	146	4	no	—
223	D₈	—	146	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
226	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
227	C₂×C₈	Q₈×D₇₃	1	2	yes	—
228	C₂×C₈	#11924029	1	2	yes	—
229	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₄⋊C₄	—	4	4	no	—
235	C₄⋊C₄	—	4	4	no	—
236	C₄⋊C₄	—	292	1	no	—
237	C₄⋊C₄	—	146	2	no	—
238	C₂²⋊C₄	—	292	4	no	—
239	C₄²	#11924013	1	2	yes	—
240	C₄²	C₂×C₂²⋊D₇₃	1	2	yes	—
241	C₁₆	—	146	2	no	—
242	C₁₆	—	146	2	no	—
243	C₂³	—	292	4	no	—
244	C₂³	—	292	2	no	—
245	C₂³	—	584	2	no	—
246	C₂³	—	584	2	no	—
247	C₂³	C₂×#11924013	1	1	yes	characteristic
248	D₄	—	292	4	no	—
249	D₄	—	292	2	no	—
250	D₄	—	146	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	—	292	2	no	—
258	C₂×C₄	—	584	8	no	—
259	C₂×C₄	—	584	2	no	—
260	C₂×C₄	#14556567	1	2	yes	—
261	C₂×C₄	C₂×#11924013	1	2	yes	—
262	C₂×C₄	C₂×#11924013	1	2	yes	—
263	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
264	C₂×C₄	#14556622	1	2	yes	—
265	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	#14556567	1	2	yes	—
268	C₈	C₂×Q₈×D₇₃	1	1	yes	characteristic
269	C₈	C₂×#11924029	1	1	yes	characteristic
270	C₂²	—	292	4	no	—
271	C₂²	—	584	2	no	—
272	C₂²	—	584	2	no	—
273	C₂²	—	584	8	no	—
274	C₂²	#17988864	1	2	yes	—
275	C₂²	#17989017	1	2	yes	—
276	C₂²	#17989022	1	2	yes	—
277	C₂²	C₂²×#11924013	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	2	no	—
282	C₄	—	584	2	no	—
283	C₄	#17988864	1	2	yes	—
284	C₄	C₂×#14556567	1	1	yes	characteristic
285	C₄	C₂×#14556622	1	1	yes	characteristic
286	C₂	—	584	4	no	—
287	C₂	#22612085	1	4	yes	direct product factor
288	C₂	C₂×#17988864	1	1	yes	characteristic
289	C₂	C₂×#17989017	1	1	yes	characteristic
290	C₂	C₂×#17989022	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11924013
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×#499×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₈
prime radical	C₂²×D₅₈₄	C₂²
hypercenter	C₂×#499	D₇₃
hypocenter	C₇₃	C₂×#2228
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
2. #22612085	C₂	direct (N × H)
3. C₂×#17988414	C₂	semidirect
4. C₂×#17988412	C₂	semidirect
5. C₂×C₄⋊#11924003	C₂	semidirect
8. C₂×#499×C₇₃	C₂	semidirect
17. #17988414	C₂²	semidirect
18. #17988412	C₂²	semidirect
19. C₄⋊#11924003	C₂²	semidirect
24. #499×C₇₃	C₂²	semidirect
141. C₂×C₇₃	#2228	semidirect
146. C₂×#499	D₇₃	semidirect
154. C₇₃	C₂×#2228	semidirect
161. #499	C₂×D₇₃	semidirect
287. C₂	#22612085	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×C₄×D₅₈₄	2
#22612085	2
C₂×#17988414	2
C₂×#17988412	2
C₂×C₄⋊#11924003	2
C₂×#17988387	2
C₂×#499×C₇₃	2
C₂×#2228	73

Subgroups — C2×#22612085 #33240596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22612085 #33240596