Subgroups — C₄×#17989631 #34020489

Order 18688.

1046 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989631	C₂	1	2	yes	maximal
2	C₄×#14556679	C₂	1	2	yes	maximal, complemented
3	C₄×#14556662	C₂	1	1	yes	maximal, characteristic, complemented
4	#22625735	C₂	1	2	yes	maximal, complemented
5	#22625708	C₂	1	4	yes	maximal, complemented
6	C₄×#14556656	C₂	1	2	yes	maximal, complemented
7	C₄×C₈×#64408	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#14556614	C₂	1	1	yes	maximal, characteristic
9	C₂×#14556679	C₂²	1	4	yes	—
10	#17989631	C₄	1	16	yes	direct product factor
11	C₂×#14556662	C₂²	1	2	yes	—
12	C₂×#14556656	C₂²	1	4	yes	—
13	#17989552	—	2	4	no	—
14	#17989552	C₂²	1	4	yes	—
15	C₄×#11924038	—	2	2	no	—
16	C₄×#11924038	C₂²	1	2	yes	—
17	C₄×#11924036	—	2	2	no	—
18	C₄×#11924036	C₂²	1	1	yes	characteristic
19	C₂×C₈×#64408	C₂²	1	2	yes	—
20	C₂×C₈×#64408	C₂²	1	2	yes	—
21	C₄²×#64408	C₂²	1	1	yes	characteristic
22	#17989463	C₂²	1	2	yes	—
23	#17989457	C₂²	1	4	yes	—
24	C₂×#14556614	C₂²	1	1	yes	characteristic
25	C₄×#11924029	C₂²	1	2	yes	—
26	Q₁₆×Dic₇₃	C₄	1	1	yes	characteristic, complemented
27	C₄×#11924015	C₄	1	2	yes	complemented
28	#17988672	C₄	1	2	yes	complemented
29	C₄×D₅₈₄	C₄	1	1	yes	characteristic, complemented
30	#17988669	C₄	1	1	yes	characteristic, complemented
31	#17988624	C₂²	1	1	yes	characteristic
32	#17988582	C₂²	1	2	yes	—
33	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
34	C₄×Q₁₆×C₇₃	C₄	1	1	yes	characteristic, complemented
35	#14556679	—	2	8	no	—
36	#14556679	C₂×C₄	1	16	yes	complemented
37	C₂×#11924038	—	2	4	no	—
38	C₂×#11924038	C₂³	1	4	yes	—
39	C₂×#11924036	—	2	4	no	—
40	C₂×#11924036	C₂³	1	2	yes	—
41	C₂×C₄×#64408	—	2	1	no	—
42	C₂×C₄×#64408	—	2	1	no	—
43	C₂×C₄×#64408	C₂³	1	2	yes	—
44	#14556662	C₂×C₄	1	8	yes	complemented
45	#14556656	C₂×C₄	1	16	yes	complemented
46	#14556646	—	2	4	no	—
47	#14556646	—	2	2	no	—
48	C₄×#10596833	—	2	4	no	—
49	C₄×#10596833	—	2	2	no	—
50	C₄×#10596833	D₄	1	1	yes	characteristic
51	C₄×#10596833	D₄	1	1	yes	characteristic
52	C₄×#10596833	D₄	1	2	yes	—
53	C₈×#64408	—	2	4	no	—
54	C₈×#64408	C₂×C₄	1	8	yes	complemented
55	C₂×#11924029	C₂³	1	2	yes	—
56	#14556614	C₂×C₄	1	8	yes	—
57	C₂×#11924015	C₂×C₄	1	2	yes	—
58	C₂×D₅₈₄	C₂×C₄	1	1	yes	characteristic
59	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
60	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
61	Q₈×Dic₇₃	C₂×C₄	1	2	yes	—
62	C₄×D₂₉₂	—	2	2	no	—
63	C₄×D₂₉₂	C₂×C₄	1	2	yes	—
64	#14556553	—	2	2	no	—
65	#14556553	C₂×C₄	1	2	yes	—
66	#14556549	—	2	2	no	—
67	#14556549	C₂³	1	2	yes	—
68	C₄²×D₇₃	—	2	2	no	—
69	C₄²×D₇₃	C₂³	1	1	yes	characteristic
70	#14556525	C₂×C₄	1	1	yes	characteristic
71	#14556522	C₂×C₄	1	2	yes	—
72	#14556515	C₂×C₄	1	2	yes	—
73	#14556513	C₂×C₄	1	2	yes	—
74	#14556511	C₂×C₄	1	2	yes	—
75	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
76	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
77	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
78	C₂×Q₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
79	C₄×Q₈×C₇₃	C₂×C₄	1	2	yes	—
80	#107×C₇₃	C₂×C₄	1	1	yes	characteristic
81	#103×C₇₃	C₂×C₄	1	2	yes	—
82	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
83	C₂²×#64408	—	2	2	no	—
84	#11924038	—	2	4	no	—
85	#11924038	—	2	16	no	—
86	#11924038	C₂²×C₄	1	8	yes	—
87	#11924036	—	2	16	no	—
88	#11924036	C₂²×C₄	1	4	yes	—
89	#11924036	#41	1	4	yes	—
90	C₂×#10596833	—	2	4	no	—
91	C₂×#10596833	C₂×D₄	1	2	yes	—
92	C₂×#10596833	C₂×D₄	1	2	yes	—
93	C₄×#64408	—	2	4	no	—
94	C₄×#64408	—	2	4	no	—
95	C₄×#64408	—	2	2	no	—
96	C₄×#64408	C₂²×C₄	1	4	yes	—
97	#11924029	—	2	4	no	—
98	#11924029	C₂²×C₄	1	8	yes	—
99	C₂×D₂₉₂	—	2	2	no	—
100	C₂×D₂₉₂	C₂²×C₄	1	2	yes	—
101	C₂×C₄×D₇₃	—	2	2	no	—
102	C₂×C₄×D₇₃	—	2	2	no	—
103	C₂×C₄×D₇₃	—	2	2	no	—
104	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
105	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
106	C₂×C₄×D₇₃	C₂⁴	1	1	yes	characteristic
107	#11924015	C₄²	1	8	yes	complemented
108	D₅₈₄	C₄²	1	4	yes	complemented
109	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
110	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
111	C₈×D₇₃	—	2	2	no	—
112	C₈×D₇₃	C₂²×C₄	1	4	yes	—
113	C₄⋊Dic₇₃	—	2	2	no	—
114	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
115	#11924005	—	2	2	no	—
116	#11924005	—	2	2	no	—
117	#11924005	—	2	2	no	—
118	C₄×Dic₇₃	—	2	2	no	—
119	C₄×Dic₇₃	—	2	2	no	—
120	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
121	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
122	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
123	C₂×Q₈×C₇₃	C₂²×C₄	1	2	yes	—
124	Q₁₆×C₇₃	C₄²	1	4	yes	complemented
125	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
127	C₄⋊C₄×C₇₃	—	2	2	no	—
128	C₄⋊C₄×C₇₃	C₂²×C₄	1	2	yes	—
129	C₄²×C₇₃	—	2	2	no	—
130	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
131	C₂²×D₇₃	—	2	2	no	—
132	C₂²×D₇₃	C₂×#41	1	1	yes	characteristic
133	C₂×#64408	—	2	4	no	—
134	C₂×#64408	—	2	4	no	—
135	C₂×#64408	—	2	4	no	—
136	Q₈×C₇₃	—	2	2	no	—
137	Q₈×C₇₃	C₂×C₄²	1	4	yes	—
138	C₂×C₄×C₇₃	—	2	2	no	—
139	C₂×C₄×C₇₃	—	2	2	no	—
140	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
143	C₂×Dic₇₃	—	2	2	no	—
144	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
145	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
146	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
147	D₂₉₂	—	2	8	no	—
148	D₂₉₂	—	2	2	no	—
149	D₂₉₂	C₂×C₄²	1	4	yes	—
150	C₄×D₇₃	—	2	2	no	—
151	C₄×D₇₃	—	2	8	no	—
152	C₄×D₇₃	—	2	2	no	—
153	C₄×D₇₃	—	4	4	no	—
154	C₄×D₇₃	—	4	4	no	—
155	C₄×D₇₃	C₂³×C₄	1	2	yes	—
156	C₄×D₇₃	C₂×#41	1	2	yes	—
157	#10596833	—	2	8	no	—
158	#10596833	C₄×D₄	1	4	yes	—
159	#10596833	C₄×D₄	1	4	yes	—
160	C₈×C₇₃	—	2	1	no	—
161	C₈×C₇₃	C₂×C₄²	1	2	yes	—
162	#10596831	—	2	1	no	—
163	#10596831	C₂×C₄²	1	2	yes	—
164	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂×D₇₃	—	2	1	no	—
166	C₂×D₇₃	—	2	1	no	—
167	C₂×D₇₃	—	2	4	no	—
168	C₂×D₇₃	—	4	2	no	—
169	C₂×D₇₃	—	4	2	no	—
170	C₂×D₇₃	C₄×#41	1	2	yes	—
171	#64408	—	2	8	no	—
172	C₄×C₇₃	—	2	1	no	—
173	C₄×C₇₃	—	2	4	no	—
174	C₄×C₇₃	—	4	2	no	—
175	C₄×C₇₃	#443	1	2	yes	complemented
176	C₄×C₇₃	C₂²×C₄²	1	1	yes	characteristic
177	C₄×C₇₃	C₄×#41	1	1	yes	characteristic
178	Dic₇₃	—	2	1	no	—
179	Dic₇₃	—	4	2	no	—
180	Dic₇₃	#443	1	2	yes	complemented
181	Dic₇₃	C₂×C₄×D₄	1	2	yes	—
182	C₄×#443	—	73	1	no	maximal, Sylow
183	C₂×C₇₃	C₄²×D₄	1	1	yes	characteristic
184	C₂×C₇₃	C₂×#443	1	1	yes	characteristic
185	C₂×C₇₃	C₂×#443	1	1	yes	characteristic
186	D₇₃	—	2	2	no	—
187	D₇₃	—	4	4	no	—
188	C₂×#443	—	73	2	no	—
189	C₄×#135	—	73	1	no	—
190	C₄×#131	—	73	2	no	—
191	C₄×#108	—	73	1	no	—
192	#1772	—	73	2	no	—
193	#1764	—	73	4	no	—
194	C₄×#104	—	73	2	no	—
195	C₄²×C₈	—	73	1	no	—
196	C₇₃	C₄×#443	1	1	yes	characteristic, complemented, Sylow
197	C₂×#135	—	73	1	no	—
198	C₂×#131	—	73	4	no	—
199	C₄×#41	—	73	2	no	—
200	#443	—	73	16	no	—
201	C₄×Q₁₆	—	73	1	no	—
202	C₄×Q₁₆	#64408	1	1	yes	characteristic, complemented
203	C₄×SD₁₆	—	73	2	no	—
204	C₄×SD₁₆	—	73	2	no	—
205	C₄×D₈	—	73	1	no	—
206	C₄×D₈	—	73	1	no	—
207	C₂×#108	—	73	2	no	—
208	#427	—	73	1	no	—
209	C₂×#104	—	73	4	no	—
210	#416	—	73	2	no	—
211	#405	—	73	4	no	—
212	#405	—	146	4	no	—
213	C₄×M₁₆	—	73	2	no	—
214	C₄×M₁₆	—	146	2	no	—
215	C₂×C₄×C₈	—	73	1	no	—
216	C₂×C₄×C₈	—	73	2	no	—
217	C₂×C₄×C₈	—	73	1	no	—
218	C₂×C₄×C₈	—	73	2	no	—
219	C₂×C₄×C₈	—	146	2	no	—
220	C₄³	—	73	1	no	—
221	#341	—	73	2	no	—
222	#339	—	73	4	no	—
223	C₂×#41	—	73	2	no	—
224	#135	—	73	8	no	—
225	C₂×Q₁₆	C₂×#64408	1	1	yes	characteristic
226	C₂×SD₁₆	—	73	2	no	—
227	C₂×D₈	—	73	1	no	—
228	#131	—	73	16	no	—
229	#131	—	146	8	no	—
230	C₂×M₁₆	—	73	4	no	—
231	C₂×M₁₆	—	146	4	no	—
232	C₂²×C₈	—	73	2	no	—
233	C₂²×C₈	—	73	1	no	—
234	C₂²×C₈	—	73	1	no	—
235	C₂²×C₈	—	146	4	no	—
236	C₄×Q₈	—	73	2	no	—
237	C₄×Q₈	C₂×#64408	1	2	yes	—
238	C₄×D₄	—	73	2	no	—
239	C₄×D₄	—	73	2	no	—
240	C₄×D₄	—	146	2	no	—
241	C₄×D₄	—	146	2	no	—
242	#117	—	73	2	no	—
243	#117	—	146	2	no	—
244	C₂×C₄²	—	73	1	no	—
245	C₂×C₄²	—	73	2	no	—
246	C₂×C₄²	—	146	2	no	—
247	C₂×C₄²	—	146	1	no	—
248	C₂×C₄²	—	146	1	no	—
249	#108	—	73	8	no	—
250	#107	—	73	1	no	—
251	#107	C₂×#64408	1	1	yes	characteristic
252	#106	—	73	2	no	—
253	#104	—	73	16	no	—
254	#103	—	73	2	no	—
255	#103	C₂×#64408	1	2	yes	—
256	#102	—	73	2	no	—
257	#102	—	73	2	no	—
258	#97	—	146	2	no	—
259	#97	—	146	4	no	—
260	C₄×C₈	—	73	1	no	—
261	C₄×C₈	—	73	1	no	—
262	C₄×C₈	—	73	2	no	—
263	C₄×C₈	—	73	8	no	—
264	C₄×C₈	—	73	1	no	—
265	C₄×C₈	—	73	1	no	—
266	C₄×C₈	—	73	1	no	—
267	C₄×C₈	—	146	2	no	—
268	C₄×C₈	—	146	4	no	—
269	C₄×C₈	—	146	4	no	—
270	C₄×C₈	C₂×#64408	1	1	yes	characteristic
271	#41	—	73	8	no	—
272	#41	—	146	4	no	—
273	C₂×Q₈	C₂²×#64408	1	2	yes	—
274	C₂×D₄	—	73	2	no	—
275	C₂×D₄	—	146	2	no	—
276	C₂²×C₄	—	73	1	no	—
277	C₂²×C₄	—	73	1	no	—
278	C₂²×C₄	—	73	1	no	—
279	C₂²×C₄	—	146	2	no	—
280	C₂²×C₄	—	146	2	no	—
281	C₂²×C₄	—	146	2	no	—
282	C₂²×C₄	—	146	2	no	—
283	Q₁₆	C₄×#64408	1	4	yes	complemented
284	SD₁₆	—	73	8	no	—
285	D₈	—	73	4	no	—
286	M₁₆	—	73	8	no	—
287	M₁₆	—	146	16	no	—
288	M₁₆	—	146	4	no	—
289	C₂×C₈	—	73	2	no	—
290	C₂×C₈	—	73	2	no	—
291	C₂×C₈	—	73	4	no	—
292	C₂×C₈	—	73	4	no	—
293	C₂×C₈	—	73	1	no	—
294	C₂×C₈	—	73	4	no	—
295	C₂×C₈	—	73	1	no	—
296	C₂×C₈	—	146	2	no	—
297	C₂×C₈	—	146	16	no	—
298	C₂×C₈	—	146	4	no	—
299	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
300	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	73	2	no	—
303	C₄⋊C₄	—	146	2	no	—
304	C₄⋊C₄	C₂²×#64408	1	2	yes	—
305	C₂²⋊C₄	—	146	2	no	—
306	C₂²⋊C₄	—	146	2	no	—
307	C₂²⋊C₄	—	146	2	no	—
308	C₄²	—	2	2	no	—
309	C₄²	—	73	1	no	—
310	C₄²	—	73	1	no	—
311	C₄²	—	73	1	no	—
312	C₄²	—	73	4	no	—
313	C₄²	—	146	2	no	—
314	C₄²	—	146	2	no	—
315	C₄²	—	146	2	no	—
316	C₄²	—	146	4	no	—
317	C₄²	—	146	4	no	—
318	C₄²	C₂²×#64408	1	1	yes	characteristic
319	C₂³	—	73	1	no	—
320	C₂³	—	146	2	no	—
321	Q₈	—	2	2	no	—
322	Q₈	C₂×C₄×#64408	1	4	yes	—
323	D₄	—	73	4	no	—
324	D₄	—	146	8	no	—
325	D₄	—	146	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	292	4	no	—
329	C₂×C₄	—	292	4	no	—
330	C₂×C₄	—	73	1	no	—
331	C₂×C₄	—	73	1	no	—
332	C₂×C₄	—	73	2	no	—
333	C₂×C₄	—	73	2	no	—
334	C₂×C₄	—	73	1	no	—
335	C₂×C₄	—	146	2	no	—
336	C₂×C₄	—	146	8	no	—
337	C₂×C₄	—	146	2	no	—
338	C₂×C₄	—	146	4	no	—
339	C₂×C₄	—	146	4	no	—
340	C₂×C₄	—	146	4	no	—
341	C₂×C₄	—	146	2	no	—
342	C₂×C₄	D₄×#64408	1	1	yes	characteristic
343	C₂×C₄	D₄×#64408	1	1	yes	characteristic
344	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	73	4	no	—
347	C₈	—	73	4	no	—
348	C₈	—	73	2	no	—
349	C₈	—	146	8	no	—
350	C₈	—	146	1	no	—
351	C₈	C₂×C₄×#64408	1	2	yes	—
352	C₂²	—	292	2	no	—
353	C₂²	—	292	2	no	—
354	C₂²	—	73	2	no	—
355	C₂²	—	146	4	no	—
356	C₂²	—	146	1	no	—
357	C₂²	—	146	1	no	—
358	C₂²	C₂×D₄×#64408	1	1	yes	characteristic
359	C₄	—	2	4	no	—
360	C₄	—	2	1	no	—
361	C₄	—	4	2	no	—
362	C₄	—	292	2	no	—
363	C₄	—	73	2	no	—
364	C₄	—	73	2	no	—
365	C₄	—	146	8	no	—
366	C₄	—	146	1	no	—
367	C₄	#17989631	1	2	yes	direct product factor
368	C₄	C₂²×C₄×#64408	1	1	yes	characteristic
369	C₄	#41×#64408	1	1	yes	characteristic
370	C₂	—	292	4	no	—
371	C₂	—	146	2	no	—
372	C₂	C₄×D₄×#64408	1	1	yes	characteristic
373	C₂	D₇₃⋊#443	1	1	yes	characteristic
374	C₂	C₂×#17989631	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#64408
commutator	C₄×C₇₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₄×Q₁₆×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂×#14556614	C₂²
cosocle	C₂×C₄×D₇₃	C₂⁴
hypercenter	C₄×Q₁₆	#64408
hypocenter	C₇₃	C₄×#443
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×#14556679	C₂	semidirect
3. C₄×#14556662	C₂	semidirect
4. #22625735	C₂	semidirect
5. #22625708	C₂	semidirect
6. C₄×#14556656	C₂	semidirect
7. C₄×C₈×#64408	C₂	semidirect
10. #17989631	C₄	direct (N × H)
26. Q₁₆×Dic₇₃	C₄	semidirect
27. C₄×#11924015	C₄	semidirect
28. #17988672	C₄	semidirect
29. C₄×D₅₈₄	C₄	semidirect
30. #17988669	C₄	semidirect
34. C₄×Q₁₆×C₇₃	C₄	semidirect
36. #14556679	C₂×C₄	semidirect
44. #14556662	C₂×C₄	semidirect
45. #14556656	C₂×C₄	semidirect
54. C₈×#64408	C₂×C₄	semidirect
107. #11924015	C₄²	semidirect
108. D₅₈₄	C₄²	semidirect
124. Q₁₆×C₇₃	C₄²	semidirect
175. C₄×C₇₃	#443	semidirect
180. Dic₇₃	#443	semidirect
196. C₇₃	C₄×#443	semidirect
202. C₄×Q₁₆	#64408	semidirect
283. Q₁₆	C₄×#64408	semidirect
367. C₄	#17989631	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×#17989631	2
C₄×#14556679	2
C₄×#14556662	2
#22625735	2
#22625708	2
C₄×#14556656	2
C₄×C₈×#64408	2
C₄×#14556614	2
C₄×#443	73

Subgroups — C4×#17989631 #34020489

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17989631 #34020489