Subgroups — C₄×D₇₃⋊#108 #34020275

Order 18688.

1086 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₇₃⋊#108	C₂	1	2	yes	maximal
2	C₄×D₇₃⋊M₁₆	C₂	1	2	yes	maximal
3	C₄×#14556662	C₂	1	4	yes	maximal, complemented
4	C₄×#14556661	C₂	1	4	yes	maximal, complemented
5	#22625623	C₂	1	2	yes	maximal
6	C₂×C₄×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂×D₇₃⋊M₁₆	C₂²	1	4	yes	—
8	C₂×#14556662	C₂²	1	4	yes	—
9	D₇₃⋊#108	C₄	1	16	yes	direct product factor
10	C₂×#14556661	C₂²	1	4	yes	—
11	C₄×#11924038	C₂²	1	8	yes	—
12	C₄×#11924037	—	2	4	no	—
13	#17989546	—	2	2	no	—
14	C₄×#11924036	—	2	2	no	—
15	#17989463	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
20	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
21	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
22	C₄×C₈×D₇₃	C₄	1	2	yes	—
23	C₄×C₈×D₇₃	C₄	1	2	yes	—
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
27	C₂×#11924038	C₂³	1	8	yes	—
28	D₇₃⋊M₁₆	—	2	8	no	—
29	D₇₃⋊M₁₆	C₂×C₄	1	16	yes	—
30	C₂×#11924037	—	2	4	no	—
31	C₂×#11924036	—	2	4	no	—
32	#14556662	C₂×C₄	1	16	yes	complemented
33	#14556661	C₂×C₄	1	16	yes	complemented
34	#14556646	—	2	8	no	—
35	C₄×#10596833	—	2	8	no	—
36	C₂²×C₄×D₇₃	Q₈	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	D₄	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	#11924038	—	2	8	no	—
68	#11924038	C₂²×C₄	1	16	yes	—
69	#11924037	—	2	16	no	—
70	#11924036	—	2	16	no	—
71	C₂×#10596833	—	2	8	no	—
72	C₂³×D₇₃	C₂×Q₈	1	1	yes	characteristic
73	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
74	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×D₇₃	—	2	8	no	—
76	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
77	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
78	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×D₇₃	C₂²×C₄	1	4	yes	—
80	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×D₇₃	C₂×Q₈	1	2	yes	—
82	C₂×C₄×D₇₃	C₂×Q₈	1	2	yes	—
83	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
84	C₂×C₄×D₇₃	#41	1	4	yes	—
85	C₂×C₄×D₇₃	C₂⁴	1	1	yes	characteristic
86	C₂×#10596831	—	2	2	no	—
87	C₂×#10596831	C₄²	1	4	yes	—
88	C₂×#10596831	C₂²×C₄	1	2	yes	—
89	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
90	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
91	C₈×D₇₃	—	2	4	no	—
92	C₈×D₇₃	—	2	4	no	—
93	C₈×D₇₃	C₄²	1	4	yes	—
94	C₈×D₇₃	C₄²	1	4	yes	—
95	C₈×D₇₃	C₂²×C₄	1	4	yes	—
96	C₈×D₇₃	C₂²×C₄	1	4	yes	—
97	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
98	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
99	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
100	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
102	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
103	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
105	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₇₃	—	2	2	no	—
107	C₂×C₈×C₇₃	C₄²	1	4	yes	—
108	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
109	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
112	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
113	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
114	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×D₇₃	—	2	4	no	—
116	C₂²×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂²×D₇₃	C₄×Q₈	1	4	yes	—
118	C₂²×D₇₃	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₇₃	—	2	4	no	—
120	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₇₃	#117	1	2	yes	—
125	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×Dic₇₃	—	2	4	no	—
127	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×Dic₇₃	C₄×D₄	1	4	yes	—
131	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
132	C₄×D₇₃	—	2	4	no	—
133	C₄×D₇₃	—	2	4	no	—
134	C₄×D₇₃	—	2	4	no	—
135	C₄×D₇₃	#108	1	8	yes	complemented
136	C₄×D₇₃	C₂×C₄²	1	2	yes	—
137	C₄×D₇₃	#117	1	2	yes	—
138	C₄×D₇₃	C₄×D₄	1	4	yes	—
139	C₄×D₇₃	C₄×Q₈	1	4	yes	—
140	C₄×D₇₃	C₂³×C₄	1	2	yes	—
141	C₄×D₇₃	C₂×#41	1	2	yes	—
142	#10596833	—	2	16	no	—
143	C₈×C₇₃	—	2	2	no	—
144	C₈×C₇₃	C₂×C₄²	1	2	yes	—
145	C₈×C₇₃	C₂×C₄²	1	2	yes	—
146	#10596831	—	2	2	no	—
147	#10596831	C₂×C₄²	1	2	yes	—
148	#10596831	C₂×C₄²	1	2	yes	—
149	C₂²×C₇₃	—	2	2	no	—
150	C₂²×C₇₃	C₄×C₄⋊C₄	1	2	yes	—
151	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₂×D₇₃	—	2	2	no	—
153	C₂×D₇₃	—	2	8	no	—
154	C₂×D₇₃	C₄×C₄⋊C₄	1	2	yes	—
155	C₂×D₇₃	C₂×#108	1	4	yes	—
156	C₂×D₇₃	C₂×C₄×Q₈	1	2	yes	—
157	C₄×C₇₃	—	2	2	no	—
158	C₄×C₇₃	C₄×C₄⋊C₄	1	2	yes	—
159	C₄×C₇₃	C₂×#108	1	4	yes	complemented
160	C₄×C₇₃	C₂²×C₄²	1	1	yes	characteristic
161	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
162	Dic₇₃	—	2	2	no	—
163	Dic₇₃	C₄×C₄⋊C₄	1	2	yes	—
164	Dic₇₃	C₂×#108	1	4	yes	complemented
165	Dic₇₃	C₂×C₄×D₄	1	2	yes	—
166	C₂×C₄×#108	—	73	1	no	maximal, Sylow
167	C₂×C₇₃	—	2	2	no	—
168	C₂×C₇₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₇₃	C₂²×#108	1	2	yes	—
170	D₇₃	—	2	2	no	—
171	D₇₃	C₄×#108	1	4	yes	complemented
172	C₂²×#108	—	73	2	no	—
173	C₂×C₄×M₁₆	—	73	2	no	—
174	C₂²×C₄×C₈	—	73	1	no	—
175	C₄×#108	—	73	4	no	—
176	C₄×#108	—	73	4	no	—
177	C₂×#341	—	73	2	no	—
178	C₇₃	C₂×C₄×#108	1	1	yes	characteristic, complemented, Sylow
179	C₂²×M₁₆	—	73	4	no	—
180	C₂³×C₈	—	73	1	no	—
181	C₂³×C₈	—	73	1	no	—
182	C₂²×C₄²	—	73	1	no	—
183	C₂×#108	—	73	4	no	—
184	C₂×#108	—	73	16	no	—
185	C₂×#108	—	73	4	no	—
186	C₄×M₁₆	—	73	8	no	—
187	C₄×M₁₆	—	146	4	no	—
188	C₂×#97	—	146	2	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	—	146	2	no	—
197	C₂×C₄×C₈	#64408	1	1	yes	characteristic
198	#341	—	73	8	no	—
199	C₂³×C₄	—	73	1	no	—
200	C₂³×C₄	—	73	2	no	—
201	C₂×M₁₆	—	73	16	no	—
202	C₂×M₁₆	—	73	8	no	—
203	C₂×M₁₆	—	146	4	no	—
204	C₂×M₁₆	—	146	8	no	—
205	C₂²×C₈	—	73	1	no	—
206	C₂²×C₈	—	73	8	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	1	no	—
212	C₂²×C₈	—	73	2	no	—
213	C₂²×C₈	—	73	2	no	—
214	C₂²×C₈	—	146	4	no	—
215	C₂²×C₈	—	146	4	no	—
216	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
217	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
218	C₂×C₄²	—	73	2	no	—
219	C₂×C₄²	—	73	1	no	—
220	C₂×C₄²	—	73	1	no	—
221	C₂×C₄²	—	73	1	no	—
222	C₂×C₄²	—	73	1	no	—
223	C₂×C₄²	—	73	1	no	—
224	C₂×C₄²	—	73	2	no	—
225	C₂×C₄²	—	73	2	no	—
226	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
227	#108	—	73	16	no	—
228	#108	—	73	16	no	—
229	#97	—	146	8	no	—
230	C₄×C₈	—	73	2	no	—
231	C₄×C₈	—	73	2	no	—
232	C₄×C₈	—	73	2	no	—
233	C₄×C₈	—	73	2	no	—
234	C₄×C₈	—	73	2	no	—
235	C₄×C₈	—	73	2	no	—
236	C₄×C₈	—	146	8	no	—
237	C₄×C₈	C₂×#64408	1	2	yes	—
238	C₄×C₈	C₂×#64408	1	2	yes	—
239	C₂⁴	—	73	1	no	—
240	C₂²×C₄	—	73	1	no	—
241	C₂²×C₄	—	73	1	no	—
242	C₂²×C₄	—	73	1	no	—
243	C₂²×C₄	—	73	4	no	—
244	C₂²×C₄	—	73	2	no	—
245	C₂²×C₄	—	73	1	no	—
246	C₂²×C₄	—	73	1	no	—
247	C₂²×C₄	—	73	4	no	—
248	C₂²×C₄	—	73	2	no	—
249	C₂²×C₄	—	73	2	no	—
250	C₂²×C₄	—	73	2	no	—
251	C₂²×C₄	—	73	2	no	—
252	C₂²×C₄	—	146	8	no	—
253	C₂²×C₄	C₄⋊#64408	1	2	yes	—
254	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
255	M₁₆	—	73	16	no	—
256	M₁₆	—	146	16	no	—
257	M₁₆	—	146	8	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	73	1	no	—
260	C₂×C₈	—	73	1	no	—
261	C₂×C₈	—	73	4	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	2	no	—
267	C₂×C₈	—	146	4	no	—
268	C₂×C₈	—	146	4	no	—
269	C₂×C₈	—	146	8	no	—
270	C₂×C₈	—	146	16	no	—
271	C₂×C₈	—	146	2	no	—
272	C₂×C₈	C₄×#64408	1	4	yes	—
273	C₂×C₈	C₂²×#64408	1	2	yes	—
274	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
275	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
276	C₄²	—	73	2	no	—
277	C₄²	—	73	2	no	—
278	C₄²	—	73	1	no	—
279	C₄²	—	73	2	no	—
280	C₄²	—	73	1	no	—
281	C₄²	—	73	2	no	—
282	C₄²	—	73	2	no	—
283	C₄²	C₄⋊#64408	1	1	yes	characteristic
284	C₄²	C₄⋊#64408	1	1	yes	characteristic
285	C₄²	C₂²×#64408	1	2	yes	—
286	C₂³	—	73	1	no	—
287	C₂³	—	73	1	no	—
288	C₂³	—	73	4	no	—
289	C₂³	—	146	4	no	—
290	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	73	1	no	—
293	C₂×C₄	—	73	2	no	—
294	C₂×C₄	—	73	1	no	—
295	C₂×C₄	—	73	4	no	—
296	C₂×C₄	—	73	2	no	—
297	C₂×C₄	—	73	2	no	—
298	C₂×C₄	—	73	2	no	—
299	C₂×C₄	—	73	2	no	—
300	C₂×C₄	—	73	4	no	—
301	C₂×C₄	—	73	4	no	—
302	C₂×C₄	—	73	8	no	—
303	C₂×C₄	—	73	2	no	—
304	C₂×C₄	—	146	4	no	—
305	C₂×C₄	—	146	4	no	—
306	C₂×C₄	—	146	4	no	—
307	C₂×C₄	—	146	4	no	—
308	C₂×C₄	C₂×C₄×#64408	1	2	yes	—
309	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
310	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
311	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
312	C₂×C₄	#14556670	1	2	yes	—
313	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
314	C₈	—	2	2	no	—
315	C₈	—	73	2	no	—
316	C₈	—	73	2	no	—
317	C₈	—	146	16	no	—
318	C₈	—	146	2	no	—
319	C₈	C₂×C₄×#64408	1	2	yes	—
320	C₈	C₂×C₄×#64408	1	2	yes	—
321	C₂²	—	2	2	no	—
322	C₂²	—	73	4	no	—
323	C₂²	—	73	2	no	—
324	C₂²	—	73	2	no	—
325	C₂²	—	146	8	no	—
326	C₂²	—	146	2	no	—
327	C₂²	C₄×C₄⋊#64408	1	2	yes	—
328	C₂²	C₂²×C₄⋊#64408	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	73	4	no	—
331	C₄	—	73	2	no	—
332	C₄	—	73	2	no	—
333	C₄	—	146	2	no	—
334	C₄	C₄×C₄⋊#64408	1	2	yes	—
335	C₄	D₇₃⋊#108	1	4	yes	direct product factor
336	C₄	C₂²×C₄×#64408	1	1	yes	characteristic
337	C₄	C₂×#14556670	1	1	yes	characteristic
338	C₂	—	2	2	no	—
339	C₂	—	73	4	no	—
340	C₂	—	146	2	no	—
341	C₂	C₂×C₄×C₄⋊#64408	1	1	yes	characteristic
342	C₂	C₂×D₇₃⋊#108	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×Q₈
cosocle	C₂×C₄×D₇₃	C₂⁴
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	C₂×C₄×#108
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×#14556662	C₂	semidirect
4. C₄×#14556661	C₂	semidirect
9. D₇₃⋊#108	C₄	direct (N × H)
32. #14556662	C₂×C₄	semidirect
33. #14556661	C₂×C₄	semidirect
135. C₄×D₇₃	#108	semidirect
159. C₄×C₇₃	C₂×#108	semidirect
164. Dic₇₃	C₂×#108	semidirect
171. D₇₃	C₄×#108	semidirect
178. C₇₃	C₂×C₄×#108	semidirect
335. C₄	D₇₃⋊#108	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×D₇₃⋊#108	2
C₄×D₇₃⋊M₁₆	2
C₄×#14556662	2
C₄×#14556661	2
#22625623	2
C₂×C₄×C₈×D₇₃	2
C₂×C₄×#108	73

Subgroups — C4×D73⋊#108 #34020275

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₇₃⋊#108 #34020275