Subgroups — C₄×#17989597 #34020257

Order 18688.

1182 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	C₂×#17989597	C₂	1	2	yes	maximal
2	C₄×#14556670	C₂	1	2	yes	maximal
3	C₄×#14556660	C₂	1	4	yes	maximal, complemented
4	C₄×#14556659	C₂	1	4	yes	maximal, complemented
5	#22625617	C₂	1	2	yes	maximal
6	C₂×C₄×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#14556670	C₂²	1	4	yes	—
8	#17989597	C₄	1	16	yes	direct product factor
9	C₂×#14556660	C₂²	1	4	yes	—
10	C₂×#14556659	C₂²	1	4	yes	—
11	C₄×C₄⋊#64408	C₂²	1	8	yes	—
12	C₄×C₂²⋊#64408	—	2	4	no	—
13	#17989523	—	2	2	no	—
14	C₄²×#64408	—	2	2	no	—
15	#17989461	C₂²	1	8	yes	—
16	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
18	C₂×C₄²×D₇₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×#10596831	C₄	1	1	yes	characteristic, complemented
20	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic, complemented
21	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic, complemented
22	C₄×C₈×D₇₃	C₄	1	2	yes	complemented
23	C₄×C₈×D₇₃	C₄	1	2	yes	complemented
24	C₄×C₈×D₇₃	C₂²	1	2	yes	—
25	C₄×C₈×D₇₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic, complemented
27	#14556670	—	2	8	no	—
28	#14556670	C₂×C₄	1	16	yes	—
29	C₂×C₄⋊#64408	C₂³	1	8	yes	—
30	C₂×C₂²⋊#64408	—	2	4	no	—
31	C₂×C₄×#64408	—	2	4	no	—
32	C₂×C₄×#64408	—	2	8	no	—
33	#14556660	C₂×C₄	1	16	yes	complemented
34	#14556659	C₂×C₄	1	16	yes	complemented
35	(C₂×#64408)⋊C₄	—	2	8	no	—
36	C₂²×C₄×D₇₃	D₄	1	2	yes	—
37	C₂²×C₄×D₇₃	C₂³	1	1	yes	characteristic
38	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
39	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₇₃	—	2	4	no	—
41	C₂×C₈×D₇₃	C₂×C₄	1	8	yes	—
42	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₈×D₇₃	C₂³	1	2	yes	—
46	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
47	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
48	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×Dic₇₃	Q₈	1	2	yes	—
50	C₄²×D₇₃	C₂×C₄	1	2	yes	—
51	C₄²×D₇₃	D₄	1	1	yes	characteristic
52	C₄²×D₇₃	D₄	1	1	yes	characteristic
53	C₄²×D₇₃	Q₈	1	1	yes	characteristic
54	C₄²×D₇₃	Q₈	1	1	yes	characteristic
55	C₄²×D₇₃	C₂³	1	2	yes	—
56	C₄×#10596831	C₂×C₄	1	2	yes	—
57	C₄×#10596831	C₂×C₄	1	2	yes	—
58	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
59	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
60	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
61	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
62	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
67	C₂²×#64408	—	2	8	no	—
68	C₄⋊#64408	—	2	8	no	—
69	C₄⋊#64408	C₂²×C₄	1	16	yes	—
70	C₂²⋊#64408	—	2	16	no	—
71	C₄×#64408	—	2	16	no	—
72	C₄×#64408	—	4	16	no	—
73	C₂³×D₇₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
75	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₄×D₇₃	—	2	8	no	—
77	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
78	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
79	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×D₇₃	C₂²×C₄	1	4	yes	—
81	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
82	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
84	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₄×D₇₃	#41	1	4	yes	—
86	C₂×C₄×D₇₃	C₂⁴	1	1	yes	characteristic
87	C₂×#10596831	—	2	2	no	—
88	C₂×#10596831	C₄²	1	4	yes	complemented
89	C₂×#10596831	C₂²×C₄	1	2	yes	—
90	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
91	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
92	C₈×D₇₃	—	2	4	no	—
93	C₈×D₇₃	—	2	4	no	—
94	C₈×D₇₃	C₄²	1	4	yes	complemented
95	C₈×D₇₃	C₄²	1	4	yes	complemented
96	C₈×D₇₃	C₂²×C₄	1	4	yes	—
97	C₈×D₇₃	C₂²×C₄	1	4	yes	—
98	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
99	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
100	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
102	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
103	C₄×Dic₇₃	C₂×Q₈	1	2	yes	—
104	C₄×Dic₇₃	C₂×Q₈	1	2	yes	—
105	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
106	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₇₃	—	2	2	no	—
108	C₂×C₈×C₇₃	C₄²	1	4	yes	complemented
109	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
113	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
114	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
115	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×D₇₃	—	2	4	no	—
117	C₂²×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×D₇₃	C₄×D₄	1	4	yes	—
119	C₂²×D₇₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×#64408	—	2	16	no	—
121	C₂×#64408	—	4	16	no	—
122	C₂×C₄×C₇₃	—	2	4	no	—
123	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
124	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₇₃	#117	1	2	yes	—
128	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×Dic₇₃	—	2	4	no	—
130	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×Dic₇₃	C₄×Q₈	1	4	yes	—
134	C₂×Dic₇₃	C₂²×Q₈	1	1	yes	characteristic
135	C₄×D₇₃	—	2	4	no	—
136	C₄×D₇₃	—	2	4	no	—
137	C₄×D₇₃	—	2	4	no	—
138	C₄×D₇₃	C₂×C₄²	1	2	yes	—
139	C₄×D₇₃	#117	1	2	yes	—
140	C₄×D₇₃	C₄×D₄	1	4	yes	—
141	C₄×D₇₃	C₄×Q₈	1	4	yes	—
142	C₄×D₇₃	#135	1	8	yes	—
143	C₄×D₇₃	C₂³×C₄	1	2	yes	—
144	C₄×D₇₃	C₂×#41	1	2	yes	—
145	C₈×C₇₃	—	2	2	no	—
146	C₈×C₇₃	C₂×C₄²	1	2	yes	—
147	C₈×C₇₃	C₂×C₄²	1	2	yes	—
148	#10596831	—	2	2	no	—
149	#10596831	C₂×C₄²	1	2	yes	—
150	#10596831	C₂×C₄²	1	2	yes	—
151	C₂²×C₇₃	—	2	2	no	—
152	C₂²×C₇₃	C₄×C₄⋊C₄	1	2	yes	—
153	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
154	C₂×D₇₃	—	2	2	no	—
155	C₂×D₇₃	—	2	8	no	—
156	C₂×D₇₃	C₄×C₄⋊C₄	1	2	yes	—
157	C₂×D₇₃	C₂×C₄×D₄	1	2	yes	—
158	C₂×D₇₃	C₂×#135	1	4	yes	—
159	#64408	—	4	16	no	—
160	C₄×C₇₃	—	2	2	no	—
161	C₄×C₇₃	C₄×C₄⋊C₄	1	2	yes	—
162	C₄×C₇₃	#427	1	4	yes	complemented
163	C₄×C₇₃	C₂²×C₄²	1	1	yes	characteristic
164	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
165	Dic₇₃	—	2	2	no	—
166	Dic₇₃	C₄×C₄⋊C₄	1	2	yes	—
167	Dic₇₃	#427	1	4	yes	complemented
168	Dic₇₃	C₂×C₄×Q₈	1	2	yes	—
169	C₄×#427	—	73	1	no	maximal, Sylow
170	C₂×C₇₃	—	2	2	no	—
171	C₂×C₇₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₇₃	C₂×#427	1	2	yes	—
173	D₇₃	—	2	2	no	—
174	D₇₃	C₄×#135	1	4	yes	—
175	C₂×#427	—	73	2	no	—
176	C₂²×C₄×C₈	—	73	1	no	—
177	C₄×#117	—	73	2	no	—
178	C₄×#107	—	73	4	no	—
179	C₄×#106	—	73	4	no	—
180	#1735	—	73	2	no	—
181	C₇₃	C₄×#427	1	1	yes	characteristic, complemented, Sylow
182	C₂³×C₈	—	73	1	no	—
183	C₂³×C₈	—	73	1	no	—
184	C₂×#117	—	73	4	no	—
185	C₂²×C₄²	—	73	1	no	—
186	#427	—	73	16	no	—
187	C₂×#107	—	73	4	no	—
188	C₂×#106	—	73	4	no	—
189	C₂×C₄×C₈	—	73	1	no	—
190	C₂×C₄×C₈	—	73	1	no	—
191	C₂×C₄×C₈	—	73	1	no	—
192	C₂×C₄×C₈	—	73	2	no	—
193	C₂×C₄×C₈	—	73	2	no	—
194	C₂×C₄×C₈	—	73	2	no	—
195	C₂×C₄×C₈	—	73	2	no	—
196	C₂×C₄×C₈	#64408	1	1	yes	characteristic, complemented
197	C₄×C₄⋊C₄	—	73	8	no	—
198	C₄×C₂²⋊C₄	—	146	4	no	—
199	#376	—	146	2	no	—
200	C₄³	—	146	2	no	—
201	#340	—	73	8	no	—
202	C₂³×C₄	—	73	1	no	—
203	C₂³×C₄	—	73	2	no	—
204	C₂²×C₈	—	73	1	no	—
205	C₂²×C₈	—	73	8	no	—
206	C₂²×C₈	—	73	1	no	—
207	C₂²×C₈	—	73	1	no	—
208	C₂²×C₈	—	73	1	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	—	146	4	no	—
214	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
215	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
216	#117	—	73	16	no	—
217	#117	—	146	8	no	—
218	C₂×C₄⋊C₄	—	73	8	no	—
219	C₂×C₂²⋊C₄	—	146	4	no	—
220	C₂×C₄²	—	73	2	no	—
221	C₂×C₄²	—	73	1	no	—
222	C₂×C₄²	—	73	1	no	—
223	C₂×C₄²	—	73	1	no	—
224	C₂×C₄²	—	73	1	no	—
225	C₂×C₄²	—	73	1	no	—
226	C₂×C₄²	—	73	2	no	—
227	C₂×C₄²	—	73	2	no	—
228	C₂×C₄²	—	146	8	no	—
229	C₂×C₄²	—	146	4	no	—
230	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
231	#107	—	73	16	no	—
232	#106	—	73	16	no	—
233	C₄×C₈	—	73	2	no	—
234	C₄×C₈	—	73	2	no	—
235	C₄×C₈	—	73	2	no	—
236	C₄×C₈	—	73	2	no	—
237	C₄×C₈	—	73	2	no	—
238	C₄×C₈	—	73	2	no	—
239	C₄×C₈	C₂×#64408	1	2	yes	—
240	C₄×C₈	C₂×#64408	1	2	yes	—
241	#95	—	146	8	no	—
242	C₂⁴	—	73	1	no	—
243	C₂²×C₄	—	73	1	no	—
244	C₂²×C₄	—	73	1	no	—
245	C₂²×C₄	—	73	1	no	—
246	C₂²×C₄	—	73	4	no	—
247	C₂²×C₄	—	73	1	no	—
248	C₂²×C₄	—	73	2	no	—
249	C₂²×C₄	—	73	1	no	—
250	C₂²×C₄	—	73	4	no	—
251	C₂²×C₄	—	73	2	no	—
252	C₂²×C₄	—	73	2	no	—
253	C₂²×C₄	—	73	2	no	—
254	C₂²×C₄	—	73	2	no	—
255	C₂²×C₄	—	146	8	no	—
256	C₂²×C₄	—	146	8	no	—
257	C₂²×C₄	C₄⋊#64408	1	2	yes	—
258	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	73	1	no	—
261	C₂×C₈	—	73	1	no	—
262	C₂×C₈	—	73	4	no	—
263	C₂×C₈	—	73	4	no	—
264	C₂×C₈	—	73	4	no	—
265	C₂×C₈	—	73	4	no	—
266	C₂×C₈	—	73	4	no	—
267	C₂×C₈	—	73	2	no	—
268	C₂×C₈	—	146	4	no	—
269	C₂×C₈	—	146	4	no	—
270	C₂×C₈	—	146	2	no	—
271	C₂×C₈	C₄×#64408	1	4	yes	complemented
272	C₂×C₈	C₂²×#64408	1	2	yes	—
273	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
274	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
275	C₄⋊C₄	—	73	16	no	—
276	C₄⋊C₄	—	146	8	no	—
277	C₂²⋊C₄	—	146	16	no	—
278	C₄²	—	292	16	no	—
279	C₄²	—	73	2	no	—
280	C₄²	—	73	1	no	—
281	C₄²	—	73	2	no	—
282	C₄²	—	73	2	no	—
283	C₄²	—	73	1	no	—
284	C₄²	—	73	2	no	—
285	C₄²	—	73	2	no	—
286	C₄²	—	146	16	no	—
287	C₄²	C₄⋊#64408	1	1	yes	characteristic
288	C₄²	C₄⋊#64408	1	1	yes	characteristic
289	C₄²	C₂²×#64408	1	2	yes	—
290	C₂³	—	73	1	no	—
291	C₂³	—	73	1	no	—
292	C₂³	—	73	4	no	—
293	C₂³	—	146	4	no	—
294	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	292	16	no	—
297	C₂×C₄	—	73	2	no	—
298	C₂×C₄	—	73	1	no	—
299	C₂×C₄	—	73	1	no	—
300	C₂×C₄	—	73	4	no	—
301	C₂×C₄	—	73	2	no	—
302	C₂×C₄	—	73	2	no	—
303	C₂×C₄	—	73	2	no	—
304	C₂×C₄	—	73	2	no	—
305	C₂×C₄	—	73	4	no	—
306	C₂×C₄	—	73	4	no	—
307	C₂×C₄	—	73	8	no	—
308	C₂×C₄	—	73	2	no	—
309	C₂×C₄	—	146	4	no	—
310	C₂×C₄	—	146	4	no	—
311	C₂×C₄	—	146	16	no	—
312	C₂×C₄	—	146	4	no	—
313	C₂×C₄	—	146	4	no	—
314	C₂×C₄	C₂×C₄×#64408	1	2	yes	—
315	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
316	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
317	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
318	C₂×C₄	#14556670	1	2	yes	—
319	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
320	C₈	—	2	2	no	—
321	C₈	—	73	2	no	—
322	C₈	—	73	2	no	—
323	C₈	—	146	2	no	—
324	C₈	C₂×C₄×#64408	1	2	yes	—
325	C₈	C₂×C₄×#64408	1	2	yes	—
326	C₂²	—	2	2	no	—
327	C₂²	—	73	4	no	—
328	C₂²	—	73	2	no	—
329	C₂²	—	73	2	no	—
330	C₂²	—	146	8	no	—
331	C₂²	—	146	2	no	—
332	C₂²	C₄×C₄⋊#64408	1	2	yes	—
333	C₂²	C₂²×C₄⋊#64408	1	1	yes	characteristic
334	C₄	—	2	2	no	—
335	C₄	—	292	16	no	—
336	C₄	—	73	4	no	—
337	C₄	—	73	2	no	—
338	C₄	—	73	2	no	—
339	C₄	—	146	2	no	—
340	C₄	C₄×C₄⋊#64408	1	2	yes	—
341	C₄	#17989597	1	4	yes	direct product factor
342	C₄	C₂²×C₄×#64408	1	1	yes	characteristic
343	C₄	C₂×#14556670	1	1	yes	characteristic
344	C₂	—	2	2	no	—
345	C₂	—	73	4	no	—
346	C₂	—	146	2	no	—
347	C₂	C₂×C₄×C₄⋊#64408	1	1	yes	characteristic
348	C₂	C₂×#17989597	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄⋊#64408
commutator	C₄×C₇₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₇₃	C₂×D₄
cosocle	C₂×C₄×D₇₃	C₂⁴
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	C₄×#427
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₄×#14556660	C₂	semidirect
4. C₄×#14556659	C₂	semidirect
8. #17989597	C₄	direct (N × H)
19. C₂×C₄×#10596831	C₄	semidirect
20. C₂×C₈×Dic₇₃	C₄	semidirect
21. C₂×C₈×Dic₇₃	C₄	semidirect
22. C₄×C₈×D₇₃	C₄	semidirect
23. C₄×C₈×D₇₃	C₄	semidirect
26. C₂×C₄×C₈×C₇₃	C₄	semidirect
33. #14556660	C₂×C₄	semidirect
34. #14556659	C₂×C₄	semidirect
88. C₂×#10596831	C₄²	semidirect
94. C₈×D₇₃	C₄²	semidirect
95. C₈×D₇₃	C₄²	semidirect
108. C₂×C₈×C₇₃	C₄²	semidirect
162. C₄×C₇₃	#427	semidirect
167. Dic₇₃	#427	semidirect
181. C₇₃	C₄×#427	semidirect
196. C₂×C₄×C₈	#64408	semidirect
271. C₂×C₈	C₄×#64408	semidirect
341. C₄	#17989597	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×#17989597	2
C₄×#14556670	2
C₄×#14556660	2
C₄×#14556659	2
#22625617	2
C₂×C₄×C₈×D₇₃	2
C₄×#427	73

Subgroups — C4×#17989597 #34020257

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17989597 #34020257