Subgroups — C₂×#22609471 #33068632

Order 18688.

702 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988934	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988882	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17988787	C₂	1	1	yes	maximal, characteristic
4	#22609471	C₂	1	8	yes	maximal, direct product factor
5	C₂×#17988340	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17988335	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#17988330	C₂	1	1	yes	maximal, characteristic
8	C₂×#446×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11924015	—	2	1	no	—
10	C₂×#14556591	C₂²	1	1	yes	characteristic
11	C₂×#14556588	C₂²	1	1	yes	characteristic
12	#17988934	—	2	2	no	—
13	#17988934	C₂²	1	4	yes	—
14	#17988882	—	2	2	no	—
15	#17988882	C₂²	1	4	yes	complemented
16	#17988787	—	2	2	no	—
17	#17988787	C₂²	1	4	yes	—
18	C₂×#14556524	—	2	1	no	—
19	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
21	C₂×#14556515	—	2	1	no	—
22	C₂×#14556513	—	2	1	no	—
23	C₂×#14556513	—	2	1	no	—
24	C₂×#14556511	—	2	1	no	—
25	#17988340	C₂²	1	4	yes	complemented
26	#17988335	C₂²	1	4	yes	complemented
27	#17988330	C₂²	1	4	yes	—
28	C₂×C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
29	#446×C₇₃	—	2	2	no	—
30	#446×C₇₃	C₂²	1	4	yes	complemented
31	C₂×C₄⋊C₈×C₇₃	—	2	1	no	—
32	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
33	C₂×#97×C₇₃	—	2	1	no	—
34	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
35	C₂²×Dic₁₄₆	—	2	1	no	—
36	C₂²×D₂₉₂	—	2	1	no	—
37	C₂²×D₂₉₂	—	2	1	no	—
38	C₂²×D₂₉₂	—	2	1	no	—
39	C₂×#11924015	—	2	4	no	—
40	C₂×#11924015	—	4	4	no	—
41	C₂²×#10596831	—	2	1	no	—
42	C₂²×#10596831	—	2	1	no	—
43	#14556591	—	2	1	no	—
44	#14556591	—	2	1	no	—
45	#14556591	—	2	1	no	—
46	#14556591	C₂³	1	2	yes	—
47	#14556588	—	2	1	no	—
48	#14556588	—	2	1	no	—
49	#14556588	—	2	1	no	—
50	#14556588	C₂³	1	2	yes	—
51	C₂×C₄⋊Dic₇₃	—	2	1	no	—
52	C₂×C₄⋊Dic₇₃	—	2	1	no	—
53	#14556524	—	2	4	no	—
54	C₄⋊#10596831	—	2	1	no	—
55	C₄⋊#10596831	—	2	1	no	—
56	C₄⋊#10596831	C₂³	1	2	yes	—
57	C₄⋊#10596831	C₂³	1	2	yes	—
58	#14556515	—	2	4	no	—
59	#14556513	—	2	4	no	—
60	#14556513	—	2	4	no	—
61	#14556511	—	2	4	no	—
62	C₂²×Q₈×C₇₃	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₇₃	—	2	1	no	—
64	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
65	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
66	C₄×Q₈×C₇₃	—	2	1	no	—
67	C₄×Q₈×C₇₃	—	2	1	no	—
68	C₄×Q₈×C₇₃	D₄	1	2	yes	—
69	C₄×Q₈×C₇₃	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
71	C₂×C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₇₃	—	2	1	no	—
73	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
74	C₄⋊C₈×C₇₃	—	2	4	no	—
75	C₄⋊C₈×C₇₃	—	2	1	no	—
76	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
77	#97×C₇₃	—	2	4	no	—
78	C₄×C₈×C₇₃	D₄	1	2	yes	—
79	C₄×C₈×C₇₃	C₂³	1	2	yes	—
80	C₂³×D₇₃	—	4	1	no	—
81	C₂×Dic₁₄₆	—	2	2	no	—
82	C₂×Dic₁₄₆	—	4	1	no	—
83	C₂×Dic₁₄₆	—	8	2	no	—
84	C₂×D₂₉₂	—	2	2	no	—
85	C₂×D₂₉₂	—	2	2	no	—
86	C₂×D₂₉₂	—	2	2	no	—
87	C₂×D₂₉₂	—	4	2	no	—
88	C₂×D₂₉₂	—	4	2	no	—
89	C₂×D₂₉₂	—	4	2	no	—
90	C₂×D₂₉₂	—	4	2	no	—
91	C₂×D₂₉₂	—	4	1	no	—
92	C₂×D₂₉₂	—	4	1	no	—
93	C₂×D₂₉₂	—	4	1	no	—
94	C₂×D₂₉₂	—	8	2	no	—
95	C₂²×Dic₇₃	—	4	1	no	—
96	#11924015	—	4	8	no	—
97	C₂×#10596831	—	2	2	no	—
98	C₂×#10596831	—	2	2	no	—
99	C₂×#10596831	—	4	2	no	—
100	C₂×#10596831	—	4	2	no	—
101	C₄⋊Dic₇₃	—	2	2	no	—
102	C₄⋊Dic₇₃	—	2	2	no	—
103	C₄⋊Dic₇₃	—	4	1	no	—
104	C₄⋊Dic₇₃	—	4	1	no	—
105	C₂×Q₈×C₇₃	—	2	1	no	—
106	C₂×Q₈×C₇₃	—	2	2	no	—
107	C₂×Q₈×C₇₃	—	2	2	no	—
108	C₂×Q₈×C₇₃	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₇₃	—	2	1	no	—
110	C₂²×C₄×C₇₃	—	2	1	no	—
111	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
114	C₂×C₈×C₇₃	—	2	2	no	—
115	C₂×C₈×C₇₃	—	2	2	no	—
116	C₂×C₈×C₇₃	—	2	2	no	—
117	C₂×C₈×C₇₃	—	4	2	no	—
118	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
119	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
120	C₄⋊C₄×C₇₃	—	2	2	no	—
121	C₄⋊C₄×C₇₃	—	2	1	no	—
122	C₄⋊C₄×C₇₃	—	2	2	no	—
123	C₄⋊C₄×C₇₃	C₂×D₄	1	2	yes	—
124	C₄²×C₇₃	—	2	2	no	—
125	C₄²×C₇₃	—	2	2	no	—
126	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₇₃	#41	1	1	yes	characteristic
129	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
130	C₂³×C₇₃	#121	1	1	yes	characteristic
131	C₂²×D₇₃	—	4	2	no	—
132	C₂²×D₇₃	—	8	2	no	—
133	C₂²×D₇₃	—	8	2	no	—
134	C₂²×D₇₃	—	8	2	no	—
135	Q₈×C₇₃	—	2	2	no	—
136	Q₈×C₇₃	—	2	2	no	—
137	Q₈×C₇₃	—	4	2	no	—
138	C₂×C₄×C₇₃	—	2	2	no	—
139	C₂×C₄×C₇₃	—	2	2	no	—
140	C₂×C₄×C₇₃	—	2	2	no	—
141	C₂×C₄×C₇₃	—	4	2	no	—
142	C₂×C₄×C₇₃	—	4	2	no	—
143	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
144	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
145	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
146	C₂×C₄×C₇₃	#135	1	2	yes	—
147	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₄×C₇₃	#137	1	2	yes	—
149	C₂×C₄×C₇₃	#137	1	2	yes	—
150	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
153	C₂×Dic₇₃	—	4	2	no	—
154	C₂×Dic₇₃	—	8	2	no	—
155	D₂₉₂	—	4	2	no	—
156	D₂₉₂	—	4	2	no	—
157	D₂₉₂	—	4	2	no	—
158	D₂₉₂	—	4	2	no	—
159	D₂₉₂	—	8	2	no	—
160	D₂₉₂	—	8	2	no	—
161	D₂₉₂	—	8	2	no	—
162	D₂₉₂	—	8	2	no	—
163	Dic₁₄₆	—	8	2	no	—
164	Dic₁₄₆	—	8	2	no	—
165	C₈×C₇₃	—	2	2	no	—
166	C₈×C₇₃	—	2	2	no	—
167	C₈×C₇₃	—	4	2	no	—
168	#10596831	—	4	2	no	—
169	#10596831	—	4	2	no	—
170	C₂²×C₇₃	#464	1	2	yes	—
171	C₂²×C₇₃	#468	1	2	yes	—
172	C₂²×C₇₃	#472	1	2	yes	—
173	C₂²×C₇₃	C₂×#121	1	1	yes	characteristic
174	C₂×D₇₃	—	8	2	no	—
175	C₂×D₇₃	—	8	2	no	—
176	C₂×D₇₃	—	8	2	no	—
177	C₂×D₇₃	—	16	4	no	—
178	C₄×C₇₃	—	2	1	no	—
179	C₄×C₇₃	—	2	1	no	—
180	C₄×C₇₃	—	4	2	no	—
181	C₄×C₇₃	—	4	2	no	—
182	C₄×C₇₃	#464	1	2	yes	—
183	C₄×C₇₃	#468	1	2	yes	—
184	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
185	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
187	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
188	Dic₇₃	—	8	2	no	—
189	C₂×#1698	—	73	1	no	maximal, Sylow
190	C₂×C₇₃	#1698	1	4	yes	complemented
191	C₂×C₇₃	C₂×#464	1	1	yes	characteristic
192	C₂×C₇₃	C₂×#468	1	1	yes	characteristic
193	C₂×C₇₃	C₂×#472	1	1	yes	characteristic
194	D₇₃	—	16	2	no	—
195	C₂×#486	—	73	1	no	—
196	C₂×#477	—	73	1	no	—
197	C₂×#460	—	73	1	no	—
198	C₂×#446	D₇₃	1	1	yes	characteristic, complemented
199	#1698	—	73	8	no	—
200	C₂×#335	—	73	1	no	—
201	C₂×#333	—	73	1	no	—
202	C₂×#331	—	73	1	no	—
203	C₇₃	C₂×#1698	1	1	yes	characteristic, complemented, Sylow
204	C₂²×SD₁₆	—	146	1	no	—
205	C₂×C₄⋊Q₈	—	73	1	no	—
206	C₂×#127	—	73	1	no	—
207	C₂×C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
208	#486	—	73	4	no	—
209	#486	—	146	2	no	—
210	#477	—	73	4	no	—
211	#477	—	146	2	no	—
212	#460	—	73	4	no	—
213	#460	—	146	2	no	—
214	#446	—	2	2	no	—
215	#446	C₂×D₇₃	1	4	yes	complemented
216	C₂×#107	—	146	1	no	—
217	C₂×C₄⋊C₈	—	2	1	no	—
218	C₂×C₄⋊C₈	—	73	1	no	—
219	C₂×C₄⋊C₈	—	73	1	no	—
220	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
221	C₂×#103	—	146	1	no	—
222	C₂×#102	—	146	1	no	—
223	C₂×#102	—	146	1	no	—
224	C₂×#102	—	146	1	no	—
225	C₂×#97	—	2	1	no	—
226	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
227	#335	—	73	4	no	—
228	#333	—	73	4	no	—
229	#331	—	73	4	no	—
230	C₂²×Q₈	—	146	1	no	—
231	C₂²×Q₈	D₂₉₂	1	1	yes	characteristic
232	C₂²×D₄	—	146	1	no	—
233	C₂²×D₄	—	146	1	no	—
234	C₂²×D₄	—	146	1	no	—
235	C₂×SD₁₆	—	292	4	no	—
236	C₂×SD₁₆	—	146	4	no	—
237	C₂²×C₈	—	2	1	no	—
238	C₂²×C₈	—	146	1	no	—
239	C₂²×C₈	—	146	1	no	—
240	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
241	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
242	C₄⋊Q₈	—	73	2	no	—
243	C₄⋊Q₈	—	146	1	no	—
244	C₄⋊Q₈	—	146	1	no	—
245	C₄⋊Q₈	—	146	1	no	—
246	#127	—	73	2	no	—
247	#127	—	146	1	no	—
248	#127	—	146	1	no	—
249	#127	—	146	1	no	—
250	C₄×Q₈	—	2	1	no	—
251	C₄×Q₈	—	2	1	no	—
252	C₄×Q₈	D₂₉₂	1	2	yes	—
253	C₄×Q₈	C₂²×D₇₃	1	2	yes	—
254	C₂×C₄⋊C₄	—	2	1	no	—
255	C₂×C₄⋊C₄	—	146	1	no	—
256	C₂×C₄⋊C₄	—	146	1	no	—
257	C₂×C₄⋊C₄	D₂₉₂	1	1	yes	characteristic
258	C₂×C₄²	—	2	1	no	—
259	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
260	#107	—	146	4	no	—
261	C₄⋊C₈	—	2	4	no	—
262	C₄⋊C₈	—	2	1	no	—
263	C₄⋊C₈	—	73	2	no	—
264	C₄⋊C₈	—	73	2	no	—
265	C₄⋊C₈	—	146	1	no	—
266	C₄⋊C₈	—	146	1	no	—
267	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
268	#103	—	146	4	no	—
269	#102	—	146	4	no	—
270	#102	—	146	4	no	—
271	#102	—	146	4	no	—
272	#97	—	2	4	no	—
273	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
274	C₄×C₈	C₂²×D₇₃	1	2	yes	—
275	C₂⁴	—	292	1	no	—
276	C₂×Q₈	—	2	1	no	—
277	C₂×Q₈	—	2	2	no	—
278	C₂×Q₈	—	2	2	no	—
279	C₂×Q₈	—	292	1	no	—
280	C₂×Q₈	—	584	2	no	—
281	C₂×Q₈	—	146	2	no	—
282	C₂×Q₈	C₂×D₂₉₂	1	2	yes	—
283	C₂×D₄	—	292	1	no	—
284	C₂×D₄	—	292	1	no	—
285	C₂×D₄	—	292	1	no	—
286	C₂×D₄	—	292	2	no	—
287	C₂×D₄	—	292	2	no	—
288	C₂×D₄	—	292	2	no	—
289	C₂×D₄	—	292	2	no	—
290	C₂×D₄	—	584	2	no	—
291	C₂×D₄	—	146	2	no	—
292	C₂×D₄	—	146	2	no	—
293	C₂×D₄	—	146	2	no	—
294	C₂²×C₄	—	2	1	no	—
295	C₂²×C₄	—	2	1	no	—
296	C₂²×C₄	—	292	1	no	—
297	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
298	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
299	C₂²×C₄	#11924028	1	1	yes	characteristic
300	SD₁₆	—	292	8	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	4	2	no	—
305	C₂×C₈	—	292	2	no	—
306	C₂×C₈	—	292	2	no	—
307	C₂×C₈	—	146	2	no	—
308	C₂×C₈	—	146	2	no	—
309	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
310	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	292	1	no	—
315	C₄⋊C₄	—	292	1	no	—
316	C₄⋊C₄	—	146	2	no	—
317	C₄⋊C₄	—	146	2	no	—
318	C₄⋊C₄	C₂×D₂₉₂	1	2	yes	—
319	C₄²	—	2	2	no	—
320	C₄²	—	2	2	no	—
321	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
322	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
323	C₄²	#11924028	1	1	yes	characteristic
324	C₄²	C₂³×D₇₃	1	1	yes	characteristic
325	C₂³	—	292	2	no	—
326	C₂³	—	584	2	no	—
327	C₂³	—	584	2	no	—
328	C₂³	—	584	2	no	—
329	C₂³	#14556565	1	1	yes	characteristic
330	Q₈	—	2	2	no	—
331	Q₈	—	2	2	no	—
332	Q₈	—	4	2	no	—
333	Q₈	—	584	2	no	—
334	Q₈	—	584	2	no	—
335	D₄	—	292	2	no	—
336	D₄	—	292	2	no	—
337	D₄	—	292	2	no	—
338	D₄	—	292	2	no	—
339	D₄	—	584	2	no	—
340	D₄	—	584	2	no	—
341	D₄	—	584	2	no	—
342	D₄	—	584	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	292	2	no	—
349	C₂×C₄	—	584	2	no	—
350	C₂×C₄	#14556565	1	1	yes	characteristic
351	C₂×C₄	#14556565	1	1	yes	characteristic
352	C₂×C₄	#14556565	1	1	yes	characteristic
353	C₂×C₄	#14556611	1	2	yes	—
354	C₂×C₄	#14556619	1	2	yes	—
355	C₂×C₄	#14556622	1	2	yes	—
356	C₂×C₄	#14556624	1	2	yes	—
357	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
358	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
359	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
360	C₈	—	2	2	no	—
361	C₈	—	2	2	no	—
362	C₈	—	4	2	no	—
363	C₈	—	292	2	no	—
364	C₈	—	292	2	no	—
365	C₂²	—	584	2	no	—
366	C₂²	—	584	2	no	—
367	C₂²	—	584	2	no	—
368	C₂²	—	1168	4	no	—
369	C₂²	#17988809	1	2	yes	—
370	C₂²	#17988833	1	2	yes	—
371	C₂²	#17988857	1	2	yes	—
372	C₂²	C₂×#14556565	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	4	2	no	—
376	C₄	—	4	2	no	—
377	C₄	—	584	2	no	—
378	C₄	#17988809	1	2	yes	—
379	C₄	#17988833	1	2	yes	—
380	C₄	C₂×#14556611	1	1	yes	characteristic
381	C₄	C₂×#14556619	1	1	yes	characteristic
382	C₄	C₂×#14556622	1	1	yes	characteristic
383	C₄	C₂×#14556624	1	1	yes	characteristic
384	C₂	—	1168	2	no	—
385	C₂	#22609471	1	4	yes	direct product factor
386	C₂	C₂×#17988809	1	1	yes	characteristic
387	C₂	C₂×#17988833	1	1	yes	characteristic
388	C₂	C₂×#17988857	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556565
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₂×#446×C₇₃	C₂
socle	C₂³×C₇₃	#121
prime radical	C₂×#14556588	C₂²
hypercenter	C₂×#446	D₇₃
hypocenter	C₇₃	C₂×#1698
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
2. C₂×#17988882	C₂	semidirect
4. #22609471	C₂	direct (N × H)
5. C₂×#17988340	C₂	semidirect
6. C₂×#17988335	C₂	semidirect
8. C₂×#446×C₇₃	C₂	semidirect
15. #17988882	C₂²	semidirect
25. #17988340	C₂²	semidirect
26. #17988335	C₂²	semidirect
30. #446×C₇₃	C₂²	semidirect
190. C₂×C₇₃	#1698	semidirect
198. C₂×#446	D₇₃	semidirect
203. C₇₃	C₂×#1698	semidirect
215. #446	C₂×D₇₃	semidirect
385. C₂	#22609471	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×#17988934	2
C₂×#17988882	2
C₂×#17988787	2
#22609471	2
C₂×#17988340	2
C₂×#17988335	2
C₂×#17988330	2
C₂×#446×C₇₃	2
C₂×#1698	73

Subgroups — C2×#22609471 #33068632

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22609471 #33068632