Subgroups — C₄×#17989688 #34034096

Order 18688.

490 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989688	C₂	1	2	yes	maximal
2	C₄×#14556688	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#14556686	C₂	1	1	yes	maximal, characteristic, complemented
4	#22627011	C₂	1	2	yes	maximal, complemented
5	C₄×#14556679	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556688	C₂²	1	2	yes	—
7	C₂×#14556686	C₂²	1	2	yes	—
8	#17989688	C₄	1	8	yes	direct product factor
9	#17989681	—	2	1	no	—
10	C₄×#11924040	—	2	1	no	—
11	C₄×#11924040	—	2	1	no	—
12	C₄×#11924040	—	2	1	no	—
13	C₂×#14556679	C₄	1	1	yes	characteristic
14	C₂×#14556679	C₂²	1	1	yes	characteristic
15	#17989552	—	2	1	no	—
16	#17989552	—	2	1	no	—
17	#17989552	C₄	1	1	yes	characteristic
18	#17989552	C₂²	1	1	yes	characteristic
19	C₄×#11924038	—	2	1	no	—
20	C₄×#11924038	C₄	1	1	yes	characteristic
21	C₄×#11924036	—	2	1	no	—
22	C₄×#11924036	C₂²	1	1	yes	characteristic
23	C₄×#11924029	C₄	1	1	yes	characteristic
24	#14556688	—	2	2	no	—
25	#14556688	C₂×C₄	1	4	yes	complemented
26	#14556686	—	2	2	no	—
27	#14556686	C₂×C₄	1	4	yes	complemented
28	C₂×#11924040	—	2	2	no	—
29	C₂×#11924040	—	2	2	no	—
30	#14556679	—	2	2	no	—
31	#14556679	—	2	2	no	—
32	#14556679	C₈	1	4	yes	—
33	#14556679	C₂×C₄	1	4	yes	—
34	C₂×#11924038	—	2	1	no	—
35	C₂×#11924038	—	2	1	no	—
36	C₂×#11924038	C₂×C₄	1	1	yes	characteristic
37	C₂×#11924038	C₂×C₄	1	1	yes	characteristic
38	C₂×#11924036	—	2	1	no	—
39	C₂×#11924036	—	2	1	no	—
40	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
41	C₂×#11924036	C₂³	1	1	yes	characteristic
42	#14556646	—	2	1	no	—
43	#14556646	—	2	1	no	—
44	#14556646	—	2	1	no	—
45	C₄×#10596833	—	2	1	no	—
46	C₄×#10596833	—	2	1	no	—
47	C₄×#10596833	—	2	1	no	—
48	C₄×#10596833	D₄	1	1	yes	characteristic
49	C₄×#10596833	D₄	1	1	yes	characteristic
50	C₄×#10596833	D₄	1	1	yes	characteristic
51	C₄×#10596833	D₄	1	1	yes	characteristic
52	C₂×#11924029	C₂×C₄	1	1	yes	characteristic
53	Q₈×Dic₇₃	C₈	1	1	yes	characteristic
54	C₄×D₂₉₂	—	2	1	no	—
55	C₄×D₂₉₂	C₈	1	1	yes	characteristic
56	#14556553	—	2	1	no	—
57	#14556553	C₈	1	1	yes	characteristic
58	#14556549	—	2	1	no	—
59	#14556549	C₂×C₄	1	1	yes	characteristic
60	C₄²×D₇₃	—	2	1	no	—
61	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
62	C₄×Q₈×C₇₃	C₈	1	1	yes	characteristic
63	#11924040	—	2	4	no	—
64	#11924040	—	2	4	no	—
65	#11924038	—	2	1	no	—
66	#11924038	—	2	1	no	—
67	#11924038	—	2	4	no	—
68	#11924038	—	2	4	no	—
69	#11924038	C₄²	1	2	yes	—
70	#11924038	C₂×C₈	1	2	yes	—
71	#11924036	—	2	4	no	—
72	#11924036	—	2	4	no	—
73	#11924036	C₂×C₈	1	2	yes	—
74	#11924036	M₁₆	1	2	yes	—
75	#11924036	C₂²×C₄	1	2	yes	—
76	#11924036	#41	1	2	yes	—
77	C₂×#10596833	—	2	1	no	—
78	C₂×#10596833	—	2	1	no	—
79	C₂×#10596833	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×#10596833	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×#10596833	C₂×D₄	1	1	yes	characteristic
82	C₂×#10596833	C₂×D₄	1	1	yes	characteristic
83	#11924029	—	2	1	no	—
84	#11924029	—	2	1	no	—
85	#11924029	C₄²	1	2	yes	—
86	#11924029	C₂×C₈	1	2	yes	—
87	C₂×D₂₉₂	—	2	1	no	—
88	C₂×D₂₉₂	C₂×C₈	1	1	yes	characteristic
89	C₂×C₄×D₇₃	—	2	1	no	—
90	C₂×C₄×D₇₃	—	2	1	no	—
91	C₂×C₄×D₇₃	—	2	1	no	—
92	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
93	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
94	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
95	C₄⋊Dic₇₃	—	2	1	no	—
96	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
97	#11924005	—	2	1	no	—
98	#11924005	—	2	1	no	—
99	#11924005	—	2	1	no	—
100	C₄×Dic₇₃	—	2	1	no	—
101	C₄×Dic₇₃	—	2	1	no	—
102	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
103	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
104	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
105	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
106	C₄⋊C₄×C₇₃	—	2	1	no	—
107	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
108	C₄²×C₇₃	—	2	1	no	—
109	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
110	C₂²×D₇₃	—	2	1	no	—
111	C₂²×D₇₃	C₂×M₁₆	1	1	yes	characteristic
112	Q₈×C₇₃	—	2	1	no	—
113	Q₈×C₇₃	C₄×C₈	1	2	yes	—
114	C₂×C₄×C₇₃	—	2	1	no	—
115	C₂×C₄×C₇₃	—	2	1	no	—
116	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
119	C₂×Dic₇₃	—	2	1	no	—
120	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
121	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
122	C₂×Dic₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	D₂₉₂	—	2	2	no	—
124	D₂₉₂	—	2	2	no	—
125	D₂₉₂	—	2	1	no	—
126	D₂₉₂	C₄×C₈	1	2	yes	—
127	C₄×D₇₃	—	2	2	no	—
128	C₄×D₇₃	—	2	2	no	—
129	C₄×D₇₃	—	2	2	no	—
130	C₄×D₇₃	—	2	2	no	—
131	C₄×D₇₃	—	4	2	no	—
132	C₄×D₇₃	—	4	2	no	—
133	C₄×D₇₃	C₂×C₄²	1	1	yes	characteristic
134	C₄×D₇₃	#117	1	1	yes	characteristic
135	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
136	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
137	#10596833	—	2	2	no	—
138	#10596833	—	2	2	no	—
139	#10596833	C₂²⋊C₈	1	2	yes	—
140	#10596833	C₂²⋊C₈	1	2	yes	—
141	#10596833	C₄×D₄	1	2	yes	—
142	#10596833	C₄×D₄	1	2	yes	—
143	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
144	C₂×D₇₃	—	2	1	no	—
145	C₂×D₇₃	—	2	1	no	—
146	C₂×D₇₃	—	2	2	no	—
147	C₂×D₇₃	—	4	1	no	—
148	C₂×D₇₃	—	4	1	no	—
149	C₂×D₇₃	C₄×M₁₆	1	1	yes	characteristic
150	C₂×D₇₃	#431	1	1	yes	characteristic
151	C₄×C₇₃	—	2	1	no	—
152	C₄×C₇₃	—	2	2	no	—
153	C₄×C₇₃	—	4	1	no	—
154	C₄×C₇₃	#350	1	2	yes	complemented
155	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
156	C₄×C₇₃	#431	1	1	yes	characteristic
157	Dic₇₃	—	2	1	no	—
158	Dic₇₃	—	4	1	no	—
159	Dic₇₃	#350	1	2	yes	complemented
160	Dic₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
161	Dic₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
162	C₄×#350	—	73	1	no	maximal, Sylow
163	C₂×C₇₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
164	C₂×C₇₃	C₂×#350	1	1	yes	characteristic
165	C₂×C₇₃	C₂×#350	1	1	yes	characteristic
166	D₇₃	—	2	1	no	—
167	D₇₃	—	2	1	no	—
168	D₇₃	—	4	2	no	—
169	C₄×#131	—	73	1	no	—
170	C₂×#350	—	73	2	no	—
171	C₄×M₃₂	—	73	1	no	—
172	C₂×C₄×C₁₆	—	73	1	no	—
173	#1379	—	73	2	no	—
174	C₇₃	C₄×#350	1	1	yes	characteristic, complemented, Sylow
175	C₂×#131	—	73	1	no	—
176	C₂×#131	—	73	1	no	—
177	C₄×#41	—	73	1	no	—
178	C₂×M₃₂	—	73	2	no	—
179	C₂²×C₁₆	—	73	2	no	—
180	#405	—	73	1	no	—
181	#405	—	73	1	no	—
182	#405	—	146	1	no	—
183	#405	—	146	1	no	—
184	C₄×M₁₆	—	73	1	no	—
185	C₄×M₁₆	—	146	1	no	—
186	C₂×C₄×C₈	—	73	1	no	—
187	C₂×C₄×C₈	—	146	1	no	—
188	#350	—	73	8	no	—
189	#346	—	146	1	no	—
190	C₄×C₁₆	—	146	1	no	—
191	C₄×C₁₆	—	146	1	no	—
192	C₄×C₁₆	—	146	1	no	—
193	C₂×#41	—	73	1	no	—
194	#131	—	73	4	no	—
195	#131	—	73	4	no	—
196	#131	—	146	2	no	—
197	#131	—	146	2	no	—
198	C₂×M₁₆	—	73	1	no	—
199	C₂×M₁₆	—	73	1	no	—
200	C₂×M₁₆	—	146	1	no	—
201	C₂×M₁₆	—	146	1	no	—
202	C₂²×C₈	—	73	1	no	—
203	C₂²×C₈	—	73	1	no	—
204	C₂²×C₈	—	146	1	no	—
205	C₂²×C₈	—	146	1	no	—
206	C₄×Q₈	—	73	1	no	—
207	C₄×Q₈	#10596842	1	1	yes	characteristic
208	C₄×D₄	—	73	1	no	—
209	C₄×D₄	—	73	1	no	—
210	C₄×D₄	—	146	1	no	—
211	C₄×D₄	—	146	1	no	—
212	#117	—	73	1	no	—
213	#117	—	146	1	no	—
214	C₂×C₄²	—	73	1	no	—
215	C₂×C₄²	—	146	1	no	—
216	M₃₂	—	73	4	no	—
217	M₃₂	—	146	2	no	—
218	C₂×C₁₆	—	73	4	no	—
219	C₂×C₁₆	—	146	2	no	—
220	C₂×C₁₆	—	146	2	no	—
221	C₂×C₁₆	—	146	2	no	—
222	#97	—	146	1	no	—
223	#97	—	146	1	no	—
224	#97	—	146	1	no	—
225	C₄×C₈	—	73	1	no	—
226	C₄×C₈	—	73	1	no	—
227	C₄×C₈	—	73	1	no	—
228	C₄×C₈	—	73	1	no	—
229	C₄×C₈	—	146	1	no	—
230	C₄×C₈	—	146	1	no	—
231	C₄×C₈	—	146	1	no	—
232	#41	—	73	2	no	—
233	#41	—	73	2	no	—
234	#41	—	146	1	no	—
235	#41	—	146	1	no	—
236	C₂×Q₈	C₂×#10596842	1	1	yes	characteristic
237	C₂×D₄	—	73	1	no	—
238	C₂×D₄	—	146	1	no	—
239	C₂²×C₄	—	73	1	no	—
240	C₂²×C₄	—	73	1	no	—
241	C₂²×C₄	—	73	1	no	—
242	C₂²×C₄	—	146	1	no	—
243	C₂²×C₄	—	146	1	no	—
244	C₂²×C₄	—	146	1	no	—
245	M₁₆	—	73	2	no	—
246	M₁₆	—	73	2	no	—
247	M₁₆	—	146	4	no	—
248	M₁₆	—	146	4	no	—
249	M₁₆	—	146	1	no	—
250	M₁₆	—	146	1	no	—
251	C₂×C₈	—	73	1	no	—
252	C₂×C₈	—	73	1	no	—
253	C₂×C₈	—	73	1	no	—
254	C₂×C₈	—	73	1	no	—
255	C₂×C₈	—	73	2	no	—
256	C₂×C₈	—	73	2	no	—
257	C₂×C₈	—	73	2	no	—
258	C₂×C₈	—	73	2	no	—
259	C₂×C₈	—	146	4	no	—
260	C₂×C₈	—	146	4	no	—
261	C₂×C₈	—	146	1	no	—
262	C₂×C₈	—	146	1	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	73	1	no	—
265	C₄⋊C₄	—	146	1	no	—
266	C₄⋊C₄	C₂×#10596842	1	1	yes	characteristic
267	C₂²⋊C₄	—	146	1	no	—
268	C₂²⋊C₄	—	146	1	no	—
269	C₂²⋊C₄	—	146	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	73	1	no	—
272	C₄²	—	73	1	no	—
273	C₄²	—	73	1	no	—
274	C₄²	—	146	1	no	—
275	C₄²	—	146	1	no	—
276	C₄²	C₂×#10596842	1	1	yes	characteristic
277	C₁₆	—	146	4	no	—
278	C₁₆	—	146	4	no	—
279	C₂³	—	73	1	no	—
280	C₂³	—	146	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	C₄×#10596842	1	2	yes	—
283	D₄	—	73	2	no	—
284	D₄	—	146	2	no	—
285	D₄	—	146	2	no	—
286	D₄	—	146	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	292	2	no	—
290	C₂×C₄	—	292	2	no	—
291	C₂×C₄	—	73	1	no	—
292	C₂×C₄	—	73	1	no	—
293	C₂×C₄	—	73	1	no	—
294	C₂×C₄	—	73	1	no	—
295	C₂×C₄	—	73	1	no	—
296	C₂×C₄	—	73	1	no	—
297	C₂×C₄	—	73	1	no	—
298	C₂×C₄	—	146	2	no	—
299	C₂×C₄	—	146	2	no	—
300	C₂×C₄	—	146	2	no	—
301	C₂×C₄	—	146	1	no	—
302	C₂×C₄	—	146	2	no	—
303	C₂×C₄	C₂²⋊#10596842	1	1	yes	characteristic
304	C₂×C₄	C₂²⋊#10596842	1	1	yes	characteristic
305	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
306	C₈	—	73	2	no	—
307	C₈	—	73	2	no	—
308	C₈	—	73	2	no	—
309	C₈	—	73	2	no	—
310	C₈	—	146	2	no	—
311	C₈	—	146	2	no	—
312	C₂²	—	292	1	no	—
313	C₂²	—	292	1	no	—
314	C₂²	—	73	1	no	—
315	C₂²	—	73	1	no	—
316	C₂²	—	146	2	no	—
317	C₂²	—	146	1	no	—
318	C₂²	—	146	1	no	—
319	C₂²	C₂×C₂²⋊#10596842	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	2	no	—
322	C₄	—	4	1	no	—
323	C₄	—	292	1	no	—
324	C₄	—	73	2	no	—
325	C₄	—	73	1	no	—
326	C₄	—	73	1	no	—
327	C₄	—	146	1	no	—
328	C₄	#17989688	1	2	yes	direct product factor
329	C₄	C₂×C₄×#10596842	1	1	yes	characteristic
330	C₄	#17989700	1	1	yes	characteristic
331	C₂	—	292	2	no	—
332	C₂	—	146	1	no	—
333	C₂	—	146	1	no	—
334	C₂	C₄×C₂²⋊#10596842	1	1	yes	characteristic
335	C₂	#22627111	1	1	yes	characteristic
336	C₂	C₂×#17989688	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#10596842
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×#10596842
Fitting	C₄×Q₈×C₇₃	C₈
socle	C₂²×C₇₃	C₂×C₂²⋊C₈
prime radical	C₂×#11924029	C₂×C₄
cosocle	C₂×#11924036	C₂³
hypercenter	C₄×Q₈	#10596842
hypocenter	C₇₃	C₄×#350
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₄×#14556688	C₂	semidirect
3. C₄×#14556686	C₂	semidirect
4. #22627011	C₂	semidirect
8. #17989688	C₄	direct (N × H)
25. #14556688	C₂×C₄	semidirect
27. #14556686	C₂×C₄	semidirect
154. C₄×C₇₃	#350	semidirect
159. Dic₇₃	#350	semidirect
174. C₇₃	C₄×#350	semidirect
328. C₄	#17989688	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×#17989688	2
C₄×#14556688	2
C₄×#14556686	2
#22627011	2
C₄×#14556679	2
C₄×#350	73

Subgroups — C4×#17989688 #34034096

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17989688 #34034096