Subgroups — C₂×#22625005 #34012204

Order 18688.

614 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556666	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14556661	C₂	1	1	yes	maximal, characteristic
3	#22625005	C₂	1	4	yes	maximal, direct product factor
4	C₂×#107×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924038	—	2	1	no	—
6	C₂×#14556666	—	2	1	no	—
7	C₂×#14556666	—	2	2	no	—
8	C₂×#14556666	C₄	1	4	yes	complemented
9	C₂×#14556666	C₂²	1	2	yes	—
10	C₂²×#11924031	—	2	1	no	—
11	C₂×#14556661	—	2	2	no	—
12	C₂×#14556661	—	2	1	no	—
13	C₂×#14556661	C₄	1	4	yes	complemented
14	C₂×#14556661	C₂²	1	2	yes	—
15	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×D₇₃	—	2	1	no	—
17	C₂×#14556525	C₄	1	1	yes	characteristic
18	C₂×#14556524	—	2	1	no	—
19	#107×D₇₃	—	2	1	no	—
20	#107×D₇₃	—	2	1	no	—
21	#107×D₇₃	C₄	1	2	yes	—
22	#107×D₇₃	C₂²	1	2	yes	—
23	C₂×#107×C₇₃	C₄	1	1	yes	characteristic
24	C₂×#11924038	—	2	4	no	—
25	C₂×#11924038	—	2	2	no	—
26	C₂×#11924038	—	4	1	no	—
27	C₂×#11924038	—	4	2	no	—
28	C₂²×#10596833	—	4	1	no	—
29	#14556666	—	2	4	no	—
30	#14556666	—	2	2	no	—
31	#14556666	C₂×C₄	1	4	yes	complemented
32	C₂×#11924031	—	2	4	no	—
33	C₂×#11924031	—	2	2	no	—
34	#14556661	—	2	4	no	—
35	#14556661	—	2	2	no	—
36	#14556661	C₂×C₄	1	4	yes	complemented
37	C₂²×C₄×D₇₃	—	4	1	no	—
38	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
39	C₂²×#10596831	D₄	1	1	yes	characteristic
40	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
44	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
45	C₂×C₈×D₇₃	D₄	1	2	yes	—
46	C₂×C₈×D₇₃	Q₈	1	2	yes	—
47	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
48	C₂×C₄⋊Dic₇₃	—	2	1	no	—
49	C₂×#11924007	—	4	1	no	—
50	C₄⋊C₄×D₇₃	—	2	2	no	—
51	C₄⋊C₄×D₇₃	—	2	2	no	—
52	C₄⋊C₄×D₇₃	—	4	1	no	—
53	C₄⋊C₄×D₇₃	—	4	1	no	—
54	#14556525	—	2	1	no	—
55	#14556525	C₂×C₄	1	2	yes	—
56	#14556524	—	2	2	no	—
57	#14556524	—	2	2	no	—
58	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
60	#107×C₇₃	—	2	1	no	—
61	#107×C₇₃	C₂×C₄	1	2	yes	—
62	#11924038	—	2	4	no	—
63	#11924038	—	4	4	no	—
64	#11924038	—	4	2	no	—
65	C₂×#10596833	—	4	4	no	—
66	C₂×#10596833	—	4	2	no	—
67	#11924031	—	2	4	no	—
68	C₂³×D₇₃	SD₁₆	1	1	yes	characteristic
69	C₂²×Dic₇₃	—	4	1	no	—
70	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
71	C₂×C₄×D₇₃	—	4	2	no	—
72	C₂×C₄×D₇₃	—	4	2	no	—
73	C₂×C₄×D₇₃	—	8	4	no	—
74	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×D₇₃	C₂²⋊C₄	1	2	yes	—
76	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
77	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
78	C₂×C₄×D₇₃	SD₁₆	1	2	yes	—
79	C₂×C₄×D₇₃	SD₁₆	1	1	yes	characteristic
80	C₂×C₄×D₇₃	Q₁₆	1	2	yes	—
81	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
82	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
84	C₂×#10596831	C₄⋊C₄	1	2	yes	—
85	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
86	C₈×D₇₃	C₄²	1	2	yes	—
87	C₈×D₇₃	C₂²⋊C₄	1	2	yes	—
88	C₈×D₇₃	C₂²⋊C₄	1	2	yes	—
89	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
90	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
91	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
92	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
93	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
94	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
95	C₈×D₇₃	C₂×Q₈	1	1	yes	characteristic
96	C₄⋊Dic₇₃	—	2	2	no	—
97	C₄⋊Dic₇₃	—	4	1	no	—
98	#11924007	—	4	2	no	—
99	#11924007	—	4	2	no	—
100	C₂²×C₄×C₇₃	—	4	1	no	—
101	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₈×C₇₃	C₄²	1	2	yes	—
103	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
105	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
106	C₄⋊C₄×C₇₃	—	2	2	no	—
107	C₄⋊C₄×C₇₃	—	4	1	no	—
108	C₂³×C₇₃	#102	1	1	yes	characteristic
109	C₂²×D₇₃	—	2	2	no	—
110	C₂²×D₇₃	—	2	2	no	—
111	C₂²×D₇₃	#102	1	1	yes	characteristic
112	C₂²×D₇₃	#103	1	2	yes	—
113	C₂²×D₇₃	#106	1	2	yes	—
114	C₂²×D₇₃	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₇₃	—	4	2	no	—
116	C₂×C₄×C₇₃	—	8	2	no	—
117	C₂×C₄×C₇₃	#95	1	2	yes	—
118	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
119	C₂×C₄×C₇₃	#103	1	2	yes	—
120	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×Dic₇₃	—	4	2	no	—
122	C₂×Dic₇₃	—	8	2	no	—
123	C₂×Dic₇₃	#102	1	1	yes	characteristic
124	C₂×Dic₇₃	#102	1	2	yes	—
125	C₂×Dic₇₃	#107	1	2	yes	—
126	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
127	C₄×D₇₃	—	8	4	no	—
128	C₄×D₇₃	—	8	4	no	—
129	C₄×D₇₃	#95	1	2	yes	—
130	C₄×D₇₃	#102	1	2	yes	—
131	C₄×D₇₃	#103	1	2	yes	—
132	C₄×D₇₃	#103	1	2	yes	—
133	C₄×D₇₃	#106	1	2	yes	—
134	C₄×D₇₃	#107	1	2	yes	—
135	C₄×D₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₄×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₄×D₇₃	C₂×SD₁₆	1	1	yes	characteristic
138	C₄×D₇₃	C₂×Q₁₆	1	1	yes	characteristic
139	#10596833	—	4	4	no	—
140	C₈×C₇₃	#95	1	2	yes	—
141	C₈×C₇₃	C₂×C₄²	1	1	yes	characteristic
142	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	#10596831	#95	1	2	yes	—
144	#10596831	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	#10596831	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂²×C₇₃	#340	1	2	yes	—
147	C₂²×C₇₃	#361	1	2	yes	—
148	C₂²×C₇₃	#362	1	2	yes	—
149	C₂²×C₇₃	C₂×#102	1	1	yes	characteristic
150	C₂×D₇₃	—	2	1	no	—
151	C₂×D₇₃	—	2	1	no	—
152	C₂×D₇₃	—	2	1	no	—
153	C₂×D₇₃	—	2	1	no	—
154	C₂×D₇₃	—	2	4	no	—
155	C₂×D₇₃	—	2	4	no	—
156	C₂×D₇₃	#340	1	2	yes	—
157	C₂×D₇₃	C₂×#103	1	1	yes	characteristic
158	C₂×D₇₃	C₂×#106	1	1	yes	characteristic
159	C₄×C₇₃	—	8	2	no	—
160	C₄×C₇₃	#340	1	2	yes	—
161	C₄×C₇₃	C₂×#95	1	1	yes	characteristic
162	C₄×C₇₃	C₂×#103	1	1	yes	characteristic
163	Dic₇₃	—	8	2	no	—
164	Dic₇₃	#340	1	2	yes	—
165	Dic₇₃	C₂×#102	1	1	yes	characteristic
166	Dic₇₃	C₂×#107	1	1	yes	characteristic
167	C₂×#1387	—	73	1	no	maximal, Sylow
168	C₂×C₇₃	#1387	1	4	yes	complemented
169	C₂×C₇₃	C₂×#340	1	1	yes	characteristic
170	C₂×C₇₃	C₂×#361	1	1	yes	characteristic
171	C₂×C₇₃	C₂×#362	1	1	yes	characteristic
172	D₇₃	—	2	2	no	—
173	D₇₃	—	2	1	no	—
174	D₇₃	—	2	1	no	—
175	C₂²×M₃₂	—	73	1	no	—
176	C₂²×#108	—	73	1	no	—
177	C₂²×#107	—	73	1	no	—
178	#1387	—	73	4	no	—
179	C₇₃	C₂×#1387	1	1	yes	characteristic, complemented, Sylow
180	C₂²×M₁₆	—	146	1	no	—
181	C₂³×C₈	—	73	1	no	—
182	C₂²×C₄⋊C₄	—	146	1	no	—
183	C₂×M₃₂	—	73	4	no	—
184	C₂×M₃₂	—	73	2	no	—
185	C₂×M₃₂	—	146	1	no	—
186	C₂×M₃₂	—	146	2	no	—
187	C₂²×C₁₆	—	146	1	no	—
188	C₂×#108	—	73	2	no	—
189	C₂×#108	—	73	4	no	—
190	C₂×#108	—	146	1	no	—
191	C₂×#108	—	146	2	no	—
192	C₂×#107	—	73	1	no	—
193	C₂×#107	—	73	2	no	—
194	C₂×#107	—	73	2	no	—
195	C₂×#107	—	146	1	no	—
196	C₂×#107	—	146	1	no	—
197	C₂×#107	—	146	1	no	—
198	C₂×#107	#64408	1	1	yes	characteristic
199	C₂³×C₄	—	292	1	no	—
200	C₂³×C₄	—	73	1	no	—
201	C₂×M₁₆	—	292	1	no	—
202	C₂×M₁₆	—	292	2	no	—
203	C₂×M₁₆	—	146	4	no	—
204	C₂×M₁₆	—	146	2	no	—
205	C₂²×C₈	—	292	1	no	—
206	C₂²×C₈	—	73	1	no	—
207	C₂²×C₈	—	73	2	no	—
208	C₂²×C₈	—	73	2	no	—
209	C₂²×C₈	—	73	1	no	—
210	C₂²×C₈	—	73	1	no	—
211	C₂²×C₈	—	73	2	no	—
212	C₂²×C₈	—	73	2	no	—
213	C₂²×C₈	—	73	1	no	—
214	C₂²×C₈	—	73	1	no	—
215	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	292	1	no	—
218	C₂×C₄⋊C₄	—	292	1	no	—
219	C₂×C₄⋊C₄	—	292	1	no	—
220	C₂×C₄⋊C₄	—	146	2	no	—
221	C₂×C₄⋊C₄	—	146	2	no	—
222	C₂×C₄⋊C₄	—	146	1	no	—
223	M₃₂	—	73	4	no	—
224	M₃₂	—	146	4	no	—
225	M₃₂	—	146	2	no	—
226	C₂×C₁₆	—	146	4	no	—
227	C₂×C₁₆	—	146	2	no	—
228	#108	—	73	4	no	—
229	#108	—	146	4	no	—
230	#108	—	146	2	no	—
231	#107	—	2	1	no	—
232	#107	—	73	2	no	—
233	#107	—	146	2	no	—
234	#107	—	146	2	no	—
235	#107	—	146	1	no	—
236	#107	C₂×#64408	1	2	yes	—
237	C₂⁴	—	73	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	292	2	no	—
240	C₂²×C₄	—	292	2	no	—
241	C₂²×C₄	—	292	1	no	—
242	C₂²×C₄	—	584	4	no	—
243	C₂²×C₄	—	73	1	no	—
244	C₂²×C₄	—	73	2	no	—
245	C₂²×C₄	—	73	2	no	—
246	C₂²×C₄	—	73	1	no	—
247	C₂²×C₄	—	73	1	no	—
248	C₂²×C₄	—	73	2	no	—
249	C₂²×C₄	—	73	2	no	—
250	C₂²×C₄	—	73	1	no	—
251	C₂²×C₄	—	73	1	no	—
252	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
253	M₁₆	—	292	4	no	—
254	M₁₆	—	292	2	no	—
255	M₁₆	—	146	4	no	—
256	C₂×C₈	—	292	4	no	—
257	C₂×C₈	—	292	2	no	—
258	C₂×C₈	—	73	2	no	—
259	C₂×C₈	—	73	2	no	—
260	C₂×C₈	—	73	1	no	—
261	C₂×C₈	—	73	1	no	—
262	C₂×C₈	—	73	2	no	—
263	C₂×C₈	—	73	2	no	—
264	C₂×C₈	—	73	2	no	—
265	C₂×C₈	—	73	2	no	—
266	C₂×C₈	—	73	2	no	—
267	C₂×C₈	—	73	1	no	—
268	C₂×C₈	—	73	1	no	—
269	C₂×C₈	—	73	2	no	—
270	C₂×C₈	—	73	1	no	—
271	C₂×C₈	—	73	1	no	—
272	C₂×C₈	C₄×#64408	1	2	yes	—
273	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
274	C₂×C₈	C₄⋊#64408	1	2	yes	—
275	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	292	2	no	—
279	C₄⋊C₄	—	292	2	no	—
280	C₄⋊C₄	—	292	1	no	—
281	C₄⋊C₄	—	146	2	no	—
282	C₁₆	—	146	4	no	—
283	C₂³	—	73	2	no	—
284	C₂³	—	73	2	no	—
285	C₂³	—	73	1	no	—
286	C₂³	—	73	1	no	—
287	C₂³	—	146	2	no	—
288	C₂³	—	146	2	no	—
289	C₂³	#14556653	1	1	yes	characteristic
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	8	2	no	—
292	C₂×C₄	—	292	2	no	—
293	C₂×C₄	—	584	4	no	—
294	C₂×C₄	—	584	4	no	—
295	C₂×C₄	—	584	2	no	—
296	C₂×C₄	—	73	2	no	—
297	C₂×C₄	—	73	2	no	—
298	C₂×C₄	—	73	1	no	—
299	C₂×C₄	—	73	1	no	—
300	C₂×C₄	—	73	2	no	—
301	C₂×C₄	—	73	2	no	—
302	C₂×C₄	—	73	2	no	—
303	C₂×C₄	—	73	2	no	—
304	C₂×C₄	—	73	1	no	—
305	C₂×C₄	—	73	2	no	—
306	C₂×C₄	—	73	1	no	—
307	C₂×C₄	—	73	1	no	—
308	C₂×C₄	—	73	2	no	—
309	C₂×C₄	—	73	1	no	—
310	C₂×C₄	(C₂×#64408)⋊C₄	1	2	yes	—
311	C₂×C₄	#14556653	1	1	yes	characteristic
312	C₂×C₄	#14556654	1	2	yes	—
313	C₂×C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
314	C₈	—	292	4	no	—
315	C₈	—	73	1	no	—
316	C₈	—	73	2	no	—
317	C₈	—	73	1	no	—
318	C₈	(C₂×#64408)⋊C₄	1	2	yes	—
319	C₈	C₂×C₄×#64408	1	1	yes	characteristic
320	C₈	C₂×C₄⋊#64408	1	1	yes	characteristic
321	C₂²	—	73	2	no	—
322	C₂²	—	73	1	no	—
323	C₂²	—	73	1	no	—
324	C₂²	—	146	1	no	—
325	C₂²	—	146	1	no	—
326	C₂²	—	146	1	no	—
327	C₂²	—	146	1	no	—
328	C₂²	—	146	4	no	—
329	C₂²	—	146	4	no	—
330	C₂²	#17989460	1	2	yes	—
331	C₂²	#17989505	1	2	yes	—
332	C₂²	#17989506	1	2	yes	—
333	C₂²	C₂×#14556653	1	1	yes	characteristic
334	C₄	—	8	2	no	—
335	C₄	—	584	2	no	—
336	C₄	—	73	1	no	—
337	C₄	—	73	2	no	—
338	C₄	—	73	1	no	—
339	C₄	#17989460	1	2	yes	—
340	C₄	C₂×(C₂×#64408)⋊C₄	1	1	yes	characteristic
341	C₄	C₂×#14556654	1	1	yes	characteristic
342	C₂	—	146	2	no	—
343	C₂	—	146	1	no	—
344	C₂	—	146	1	no	—
345	C₂	#22625005	1	4	yes	direct product factor
346	C₂	C₂×#17989460	1	1	yes	characteristic
347	C₂	C₂×#17989505	1	1	yes	characteristic
348	C₂	C₂×#17989506	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556653
commutator	C₈×C₇₃	C₂×C₄²
Frattini	C₂×C₈	C₂²×#64408
Fitting	C₂×#107×C₇₃	C₄
socle	C₂³×C₇₃	#102
prime radical	C₂³×D₇₃	SD₁₆
cosocle	C₂×C₈×D₇₃	C₂³
hypercenter	C₂×#107	#64408
hypocenter	C₇₃	C₂×#1387
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#107

Complements

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

N	H ≅ G/N	type
3. #22625005	C₂	direct (N × H)
8. C₂×#14556666	C₄	semidirect
13. C₂×#14556661	C₄	semidirect
31. #14556666	C₂×C₄	semidirect
36. #14556661	C₂×C₄	semidirect
168. C₂×C₇₃	#1387	semidirect
179. C₇₃	C₂×#1387	semidirect
345. C₂	#22625005	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×#14556666	2
C₂²×#14556661	2
#22625005	2
C₂×#107×D₇₃	2
C₂×#1387	73

Subgroups — C2×#22625005 #34012204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22625005 #34012204