Subgroups — C₄×#17988334 #32976935

Order 18688.

540 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556591	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#10596831	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988334	C₂	1	2	yes	maximal
4	#22607815	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556591	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₄₆	—	2	1	no	—
8	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
9	C₂×C₄×#10596831	—	2	1	no	—
10	#17988493	—	2	1	no	—
11	#17988485	—	2	1	no	—
12	#17988485	C₄	1	2	yes	—
13	#17988457	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊Dic₇₃	—	2	1	no	—
15	C₄×C₄⋊Dic₇₃	C₄	1	2	yes	—
16	#17988342	—	2	1	no	—
17	#17988334	C₄	1	8	yes	direct product factor
18	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₇₃	—	2	1	no	—
20	C₄³×C₇₃	C₂²	1	1	yes	characteristic
21	#336×C₇₃	—	2	1	no	—
22	C₂²×Dic₁₄₆	—	2	1	no	—
23	C₂²×#10596831	—	2	2	no	—
24	#14556591	—	2	2	no	—
25	#14556591	C₂×C₄	1	4	yes	—
26	C₄×Dic₁₄₆	—	4	2	no	—
27	C₄×Dic₁₄₆	—	4	2	no	—
28	C₂×C₄⋊Dic₇₃	—	2	1	no	—
29	C₂×C₄⋊Dic₇₃	—	2	1	no	—
30	C₂×C₄⋊Dic₇₃	—	2	1	no	—
31	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
33	C₂×#11924007	—	2	2	no	—
34	C₂×C₄×Dic₇₃	—	2	2	no	—
35	C₄⋊#10596831	—	2	2	no	—
36	C₄⋊#10596831	C₂×C₄	1	4	yes	—
37	C₄×#10596831	—	4	2	no	—
38	#14556498	—	2	2	no	—
39	#14556498	—	2	2	no	—
40	C₂²×C₈×C₇₃	—	2	2	no	—
41	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₇₃	—	2	2	no	—
49	C₄⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
50	C₄×C₈×C₇₃	—	4	2	no	—
51	C₂×Dic₁₄₆	—	2	4	no	—
52	C₂×Dic₁₄₆	—	4	1	no	—
53	C₂×Dic₁₄₆	—	4	1	no	—
54	C₂×Dic₁₄₆	—	4	2	no	—
55	C₂²×Dic₇₃	—	2	2	no	—
56	C₂²×Dic₇₃	—	2	2	no	—
57	C₂×#10596831	—	2	4	no	—
58	C₂×#10596831	—	4	2	no	—
59	C₂×#10596831	—	4	2	no	—
60	C₄⋊Dic₇₃	—	2	2	no	—
61	C₄⋊Dic₇₃	—	2	4	no	—
62	C₄⋊Dic₇₃	—	4	1	no	—
63	C₄⋊Dic₇₃	—	4	1	no	—
64	C₄⋊Dic₇₃	—	4	2	no	—
65	C₄⋊Dic₇₃	—	4	2	no	—
66	C₄⋊Dic₇₃	C₄²	1	4	yes	—
67	#11924007	—	4	4	no	—
68	C₄×Dic₇₃	—	4	4	no	—
69	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₇₃	—	2	4	no	—
77	C₂×C₈×C₇₃	—	4	2	no	—
78	C₂×C₈×C₇₃	—	4	2	no	—
79	C₄²×C₇₃	—	2	2	no	—
80	C₄²×C₇₃	—	2	2	no	—
81	C₄²×C₇₃	—	2	2	no	—
82	C₄²×C₇₃	—	4	1	no	—
83	C₄²×C₇₃	D₈	1	2	yes	—
84	C₄²×C₇₃	SD₁₆	1	2	yes	—
85	C₄²×C₇₃	SD₁₆	1	2	yes	—
86	C₄²×C₇₃	Q₁₆	1	2	yes	—
87	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
88	C₄²×C₇₃	#41	1	2	yes	—
89	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₇₃	—	2	1	no	—
91	C₂×C₄×C₇₃	—	2	1	no	—
92	C₂×C₄×C₇₃	—	2	1	no	—
93	C₂×C₄×C₇₃	—	2	1	no	—
94	C₂×C₄×C₇₃	—	2	1	no	—
95	C₂×C₄×C₇₃	—	2	1	no	—
96	C₂×C₄×C₇₃	—	2	1	no	—
97	C₂×C₄×C₇₃	—	2	1	no	—
98	C₂×C₄×C₇₃	—	2	1	no	—
99	C₂×C₄×C₇₃	#101	1	2	yes	—
100	C₂×C₄×C₇₃	#102	1	2	yes	—
101	C₂×C₄×C₇₃	#102	1	2	yes	—
102	C₂×C₄×C₇₃	#103	1	2	yes	—
103	C₂×C₄×C₇₃	#103	1	2	yes	—
104	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
106	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
112	C₂×C₄×C₇₃	#135	1	2	yes	—
113	C₂×C₄×C₇₃	#135	1	2	yes	—
114	C₂×Dic₇₃	—	2	4	no	—
115	C₂×Dic₇₃	—	4	2	no	—
116	C₂×Dic₇₃	—	4	2	no	—
117	C₂×Dic₇₃	—	4	2	no	—
118	C₂×Dic₇₃	—	4	2	no	—
119	C₂×Dic₇₃	—	4	2	no	—
120	Dic₁₄₆	—	4	4	no	—
121	Dic₁₄₆	—	4	4	no	—
122	C₈×C₇₃	—	4	4	no	—
123	#10596831	—	4	4	no	—
124	C₂²×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
125	C₂²×C₇₃	C₂×#101	1	1	yes	characteristic
126	C₂²×C₇₃	#413	1	1	yes	characteristic
127	C₂²×C₇₃	C₂×#102	1	1	yes	characteristic
128	C₂²×C₇₃	C₂×#103	1	1	yes	characteristic
129	C₂²×C₇₃	#416	1	1	yes	characteristic
130	C₂²×C₇₃	#416	1	1	yes	characteristic
131	C₄×C₇₃	—	2	2	no	—
132	C₄×C₇₃	—	2	2	no	—
133	C₄×C₇₃	—	2	1	no	—
134	C₄×C₇₃	—	2	1	no	—
135	C₄×C₇₃	—	4	1	no	—
136	C₄×C₇₃	#332	1	4	yes	complemented
137	C₄×C₇₃	C₄×D₈	1	2	yes	—
138	C₄×C₇₃	C₄×SD₁₆	1	2	yes	—
139	C₄×C₇₃	C₄×SD₁₆	1	2	yes	—
140	C₄×C₇₃	C₄×Q₁₆	1	2	yes	—
141	Dic₇₃	—	4	4	no	—
142	Dic₇₃	—	8	2	no	—
143	C₄×#332	—	73	1	no	maximal, Sylow
144	C₂×C₇₃	C₂×#332	1	1	yes	characteristic
145	C₂×C₇₃	#1540	1	1	yes	characteristic
146	C₂×C₇₃	#1542	1	1	yes	characteristic
147	C₂×C₇₃	#1544	1	1	yes	characteristic
148	C₂×C₇₃	C₄×#101	1	1	yes	characteristic
149	C₂×C₇₃	C₄×#102	1	1	yes	characteristic
150	C₂×C₇₃	C₄×#103	1	1	yes	characteristic
151	C₄×C₄⋊Q₈	—	73	1	no	—
152	C₄×C₄⋊C₈	—	73	1	no	—
153	C₄×C₄⋊C₈	D₇₃	1	1	yes	characteristic
154	C₂×#332	—	73	2	no	—
155	#1296	—	73	2	no	—
156	C₇₃	C₄×#332	1	1	yes	characteristic, complemented, Sylow
157	C₂×C₄⋊Q₈	—	73	1	no	—
158	C₂×C₄×Q₈	—	146	1	no	—
159	C₂×C₄⋊C₈	—	73	2	no	—
160	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
161	C₂×C₄×C₈	—	2	1	no	—
162	C₂×C₄×C₈	—	146	1	no	—
163	#391	—	146	1	no	—
164	#389	—	73	2	no	—
165	#389	—	146	1	no	—
166	#384	—	73	1	no	—
167	C₄×C₄⋊C₄	—	73	2	no	—
168	C₄×C₄⋊C₄	—	146	1	no	—
169	C₄³	C₂×D₇₃	1	1	yes	characteristic
170	#336	—	2	1	no	—
171	#336	—	146	1	no	—
172	#332	—	73	8	no	—
173	C₂²×Q₈	—	146	1	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	146	2	no	—
176	C₄⋊Q₈	—	73	4	no	—
177	C₄⋊Q₈	—	146	2	no	—
178	C₄×Q₈	—	292	2	no	—
179	C₄×Q₈	—	292	2	no	—
180	C₂×C₄⋊C₄	—	73	2	no	—
181	C₂×C₄⋊C₄	—	73	2	no	—
182	C₂×C₄⋊C₄	—	146	2	no	—
183	C₂×C₄⋊C₄	—	146	1	no	—
184	C₂×C₄⋊C₄	—	146	1	no	—
185	C₂×C₄⋊C₄	—	146	1	no	—
186	C₂×C₄²	—	146	2	no	—
187	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
188	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
189	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
190	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
191	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
192	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
193	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
194	C₄⋊C₈	—	2	2	no	—
195	C₄⋊C₈	—	73	4	no	—
196	C₄⋊C₈	—	146	2	no	—
197	C₄⋊C₈	C₄×D₇₃	1	4	yes	—
198	C₄×C₈	—	4	2	no	—
199	C₄×C₈	—	292	2	no	—
200	#95	—	146	2	no	—
201	#95	—	146	2	no	—
202	C₂×Q₈	—	292	1	no	—
203	C₂×Q₈	—	292	1	no	—
204	C₂×Q₈	—	292	2	no	—
205	C₂×Q₈	—	146	4	no	—
206	C₂²×C₄	—	146	2	no	—
207	C₂²×C₄	—	146	2	no	—
208	C₂²×C₄	#11924005	1	1	yes	characteristic
209	C₂²×C₄	#11924005	1	1	yes	characteristic
210	C₂²×C₄	#11924005	1	1	yes	characteristic
211	C₂²×C₄	#11924005	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	292	2	no	—
219	C₂×C₈	—	292	2	no	—
220	C₂×C₈	—	146	4	no	—
221	C₄⋊C₄	—	292	2	no	—
222	C₄⋊C₄	—	292	4	no	—
223	C₄⋊C₄	—	292	1	no	—
224	C₄⋊C₄	—	292	1	no	—
225	C₄⋊C₄	—	292	2	no	—
226	C₄⋊C₄	—	73	4	no	—
227	C₄⋊C₄	—	146	2	no	—
228	C₄⋊C₄	—	146	4	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	4	1	no	—
233	C₄²	—	292	4	no	—
234	C₄²	#11924013	1	2	yes	—
235	C₄²	#11924015	1	2	yes	—
236	C₄²	#11924017	1	2	yes	—
237	C₄²	Dic₂₉₂	1	2	yes	—
238	C₄²	C₂×C₄×D₇₃	1	2	yes	—
239	C₄²	#11924028	1	2	yes	—
240	C₂³	C₂×#11924005	1	1	yes	characteristic
241	Q₈	—	292	4	no	—
242	Q₈	—	292	4	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	292	2	no	—
253	C₂×C₄	—	292	2	no	—
254	C₂×C₄	—	292	2	no	—
255	C₂×C₄	—	292	2	no	—
256	C₂×C₄	—	292	2	no	—
257	C₂×C₄	—	146	4	no	—
258	C₂×C₄	#14556509	1	2	yes	—
259	C₂×C₄	#14556511	1	2	yes	—
260	C₂×C₄	#14556511	1	2	yes	—
261	C₂×C₄	#14556516	1	2	yes	—
262	C₂×C₄	#14556516	1	2	yes	—
263	C₂×C₄	C₄²×D₇₃	1	1	yes	characteristic
264	C₂×C₄	#14556548	1	1	yes	characteristic
265	C₂×C₄	C₄×C₂²⋊D₇₃	1	2	yes	—
266	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
267	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
268	C₂×C₄	C₂×#11924015	1	1	yes	characteristic
269	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
270	C₂×C₄	C₂×Dic₂₉₂	1	1	yes	characteristic
271	C₂×C₄	#14556611	1	2	yes	—
272	C₂×C₄	#14556613	1	2	yes	—
273	C₈	—	4	4	no	—
274	C₈	—	292	4	no	—
275	C₂²	C₄×#11924005	1	1	yes	characteristic
276	C₂²	C₂×#14556509	1	1	yes	characteristic
277	C₂²	#17988569	1	1	yes	characteristic
278	C₂²	C₂×#14556511	1	1	yes	characteristic
279	C₂²	C₂×#14556516	1	1	yes	characteristic
280	C₂²	#17988581	1	1	yes	characteristic
281	C₂²	#17988584	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	4	1	no	—
287	C₄	—	292	4	no	—
288	C₄	—	584	2	no	—
289	C₄	#17988334	1	4	yes	direct product factor
290	C₄	C₄×#11924013	1	2	yes	—
291	C₄	C₄×#11924015	1	2	yes	—
292	C₄	C₄×#11924017	1	2	yes	—
293	C₄	C₄×Dic₂₉₂	1	2	yes	—
294	C₂	C₂×#17988334	1	1	yes	characteristic
295	C₂	#22608581	1	1	yes	characteristic
296	C₂	#22608586	1	1	yes	characteristic
297	C₂	#22608595	1	1	yes	characteristic
298	C₂	C₄×#14556509	1	1	yes	characteristic
299	C₂	C₄×#14556511	1	1	yes	characteristic
300	C₂	C₄×#14556516	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11924005
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₄×C₄⋊C₈×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₇₃
hypocenter	C₇₃	C₄×#332
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
17. #17988334	C₄	direct (N × H)
136. C₄×C₇₃	#332	semidirect
156. C₇₃	C₄×#332	semidirect
289. C₄	#17988334	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#14556591	2
C₄×C₄⋊#10596831	2
C₂×#17988334	2
#22607815	2
C₄×C₄⋊C₈×C₇₃	2
C₄×#332	73

Subgroups — C4×#17988334 #32976935

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17988334 #32976935