Subgroups — C₂²×D₄×#10596842 #34035040

Order 18688.

3694 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#10596842	C₂	1	2	yes	maximal, complemented
2	C₂×D₄×#10596842	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊#10596842	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₂²⋊#10596842	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×#10596842	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×#64408	C₂	1	1	yes	maximal, characteristic
7	C₂³×#10596842	—	2	16	no	—
8	C₂³×#10596842	C₂²	1	1	yes	characteristic
9	C₂³×#10596842	C₂²	1	2	yes	—
10	C₂³×#10596842	C₂²	1	1	yes	characteristic
11	C₂³×#10596842	C₂²	1	24	yes	complemented
12	D₄×#10596842	C₂²	1	64	yes	direct product factor
13	C₂×C₄⋊#10596842	C₂²	1	12	yes	complemented
14	C₂×C₂²⋊#10596842	C₂²	1	24	yes	complemented
15	C₂×C₄×#10596842	C₂²	1	12	yes	complemented
16	C₂⁴×#64408	C₂²	1	2	yes	—
17	C₂×D₄×#64408	C₄	1	12	yes	—
18	C₂×D₄×#64408	C₂²	1	12	yes	—
19	C₂²×C₄⋊#64408	C₄	1	1	yes	characteristic
20	C₂²×C₂²⋊#64408	C₄	1	2	yes	—
21	C₂²×C₄×#64408	C₂²	1	1	yes	characteristic
22	C₂²×D₄×D₇₃	C₄	1	1	yes	characteristic
23	C₂²×#10596842	—	2	96	no	—
24	C₂²×#10596842	—	2	8	no	—
25	C₂²×#10596842	—	2	4	no	—
26	C₂²×#10596842	D₄	1	8	yes	direct product factor
27	C₂²×#10596842	C₂³	1	6	yes	—
28	C₂²×#10596842	C₂³	1	12	yes	—
29	C₂²×#10596842	C₂³	1	6	yes	—
30	C₂²×#10596842	C₂³	1	32	yes	complemented
31	C₄⋊#10596842	C₂³	1	16	yes	complemented
32	C₂²⋊#10596842	C₂³	1	32	yes	complemented
33	C₄×#10596842	C₂³	1	16	yes	complemented
34	C₂³×#64408	—	2	8	no	—
35	C₂³×#64408	—	2	8	no	—
36	C₂³×#64408	C₂×C₄	1	1	yes	characteristic
37	C₂³×#64408	C₂×C₄	1	2	yes	—
38	C₂³×#64408	C₂×C₄	1	12	yes	—
39	C₂³×#64408	C₂³	1	12	yes	—
40	C₂³×#64408	C₂³	1	1	yes	characteristic
41	D₄×#64408	C₂×C₄	1	48	yes	—
42	D₄×#64408	C₂³	1	16	yes	—
43	C₂×C₄⋊#64408	C₂×C₄	1	12	yes	—
44	C₂×C₂²⋊#64408	C₂×C₄	1	24	yes	—
45	C₂×C₄×#64408	C₂×C₄	1	6	yes	—
46	C₂×C₄×#64408	C₂³	1	6	yes	—
47	C₂⁴×D₇₃	C₂×C₄	1	2	yes	—
48	C₂²×D₂₉₂	C₈	1	1	yes	characteristic, complemented
49	C₂²×C₂²⋊D₇₃	C₈	1	2	yes	complemented
50	C₂×D₄×D₇₃	C₈	1	12	yes	complemented
51	C₂×D₄×D₇₃	C₂×C₄	1	12	yes	—
52	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×D₄×C₇₃	C₈	1	1	yes	characteristic, complemented
54	C₂×#10596842	—	2	24	no	—
55	C₂×#10596842	—	2	12	no	—
56	C₂×#10596842	—	2	64	no	—
57	C₂×#10596842	C₂×D₄	1	24	yes	direct product factor
58	C₂×#10596842	C₂⁴	1	4	yes	—
59	C₂×#10596842	C₂⁴	1	8	yes	—
60	C₂×#10596842	C₂⁴	1	4	yes	—
61	C₂²×#64408	—	2	8	no	—
62	C₂²×#64408	—	2	4	no	—
63	C₂²×#64408	—	2	48	no	—
64	C₂²×#64408	—	2	24	no	—
65	C₂²×#64408	—	2	24	no	—
66	C₂²×#64408	C₂²×C₄	1	12	yes	—
67	C₂²×#64408	C₂²×C₄	1	24	yes	—
68	C₂²×#64408	C₂²×C₄	1	6	yes	—
69	C₂²×#64408	C₂²×C₄	1	3	yes	—
70	C₂²×#64408	C₂×D₄	1	4	yes	—
71	C₂²×#64408	#41	1	4	yes	—
72	C₂²×#64408	C₂⁴	1	8	yes	—
73	C₂²×#64408	C₂⁴	1	3	yes	—
74	C₄⋊#64408	C₂²×C₄	1	16	yes	—
75	C₂²⋊#64408	C₂²×C₄	1	32	yes	—
76	C₄×#64408	C₂²×C₄	1	12	yes	—
77	C₄×#64408	C₂⁴	1	4	yes	—
78	C₂³×D₇₃	—	2	8	no	—
79	C₂³×D₇₃	—	2	8	no	—
80	C₂³×D₇₃	C₂×C₈	1	12	yes	complemented
81	C₂³×D₇₃	C₂×C₈	1	2	yes	complemented
82	C₂³×D₇₃	C₂²×C₄	1	12	yes	—
83	C₂³×D₇₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×D₂₉₂	C₂×C₈	1	12	yes	complemented
85	C₂×C₂²⋊D₇₃	C₂×C₈	1	24	yes	complemented
86	D₄×D₇₃	C₂×C₈	1	48	yes	complemented
87	D₄×D₇₃	C₂²×C₄	1	16	yes	—
88	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic, complemented
89	C₂×C₄×D₇₃	C₂×C₈	1	6	yes	complemented
90	C₂×C₄×D₇₃	C₂²×C₄	1	6	yes	—
91	C₂⁴×C₇₃	C₂×C₈	1	2	yes	complemented
92	C₂×D₄×C₇₃	C₂×C₈	1	12	yes	complemented
93	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic, complemented
94	C₂³×C₇₃	—	2	8	no	—
95	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
96	C₂³×C₇₃	C₂²×C₈	1	12	yes	complemented
97	C₂²×D₇₃	—	2	8	no	—
98	C₂²×D₇₃	—	2	24	no	—
99	C₂²×D₇₃	—	2	24	no	—
100	C₂²×D₇₃	—	2	48	no	—
101	C₂²×D₇₃	C₄×D₄	1	4	yes	—
102	C₂²×D₇₃	C₂²×C₈	1	3	yes	—
103	C₂²×D₇₃	C₂²×C₈	1	24	yes	complemented
104	C₂²×D₇₃	C₂²×C₈	1	12	yes	complemented
105	C₂²×D₇₃	#131	1	4	yes	—
106	C₂²×D₇₃	C₂³×C₄	1	8	yes	—
107	C₂²×D₇₃	C₂³×C₄	1	3	yes	—
108	C₂×#64408	—	2	24	no	—
109	C₂×#64408	—	2	12	no	—
110	C₂×#64408	—	2	48	no	—
111	C₂×#64408	—	2	8	no	—
112	C₂×#64408	—	2	8	no	—
113	C₂×#64408	C₄×D₄	1	12	yes	—
114	C₂×#64408	C₂³×C₄	1	4	yes	—
115	C₂×#64408	C₂³×C₄	1	8	yes	—
116	C₂×#64408	C₂³×C₄	1	3	yes	—
117	C₂×#64408	C₂²×D₄	1	6	yes	—
118	C₂×#64408	C₂×#41	1	6	yes	—
119	C₂×#64408	C₂⁵	1	1	yes	characteristic
120	#10596842	—	2	8	no	—
121	#10596842	—	2	4	no	—
122	#10596842	C₂²×D₄	1	8	yes	direct product factor
123	D₄×C₇₃	C₂²×C₈	1	16	yes	complemented
124	C₂×C₄×C₇₃	C₂²×C₈	1	6	yes	complemented
125	C₂²⋊D₇₃	C₂²×C₈	1	32	yes	complemented
126	C₂×Dic₇₃	C₂²×C₈	1	6	yes	complemented
127	D₂₉₂	C₂²×C₈	1	16	yes	complemented
128	C₄×D₇₃	C₂²×C₈	1	12	yes	complemented
129	C₄×D₇₃	C₂³×C₄	1	4	yes	—
130	C₂²×C₇₃	—	2	24	no	—
131	C₂²×C₇₃	C₈×D₄	1	4	yes	complemented
132	C₂²×C₇₃	C₂³×C₈	1	3	yes	—
133	C₂²×C₇₃	C₂³×C₈	1	8	yes	complemented
134	C₂×D₇₃	—	2	24	no	—
135	C₂×D₇₃	—	2	8	no	—
136	C₂×D₇₃	—	2	48	no	—
137	C₂×D₇₃	—	2	8	no	—
138	C₂×D₇₃	C₈×D₄	1	12	yes	complemented
139	C₂×D₇₃	C₂×C₄×D₄	1	6	yes	—
140	C₂×D₇₃	C₂³×C₈	1	3	yes	—
141	C₂×D₇₃	C₂³×C₈	1	8	yes	complemented
142	C₂×D₇₃	C₂×#131	1	6	yes	—
143	C₂×D₇₃	C₂⁴×C₄	1	1	yes	characteristic
144	#64408	—	2	8	no	—
145	#64408	—	2	4	no	—
146	#64408	C₂×C₄×D₄	1	6	yes	—
147	#64408	C₂³×D₄	1	1	yes	characteristic
148	#64408	C₂²×#41	1	1	yes	characteristic
149	C₄×C₇₃	C₂³×C₈	1	4	yes	complemented
150	Dic₇₃	C₂³×C₈	1	4	yes	complemented
151	C₂²×C₈×D₄	—	73	1	no	maximal, Sylow
152	C₂×C₇₃	—	2	8	no	—
153	C₂×C₇₃	C₂×C₈×D₄	1	6	yes	complemented
154	C₂×C₇₃	C₂⁴×C₈	1	1	yes	characteristic
155	D₇₃	—	2	8	no	—
156	D₇₃	C₂×C₈×D₄	1	6	yes	complemented
157	D₇₃	C₂²×C₄×D₄	1	1	yes	characteristic
158	D₇₃	C₂²×#131	1	1	yes	characteristic
159	C₂⁴×C₈	—	73	2	no	—
160	C₂²×C₄×D₄	—	73	1	no	—
161	C₂×C₈×D₄	—	73	24	no	—
162	C₂²×C₄⋊C₈	—	73	1	no	—
163	C₂²×C₂²⋊C₈	—	73	2	no	—
164	C₂²×C₄×C₈	—	73	1	no	—
165	C₇₃	C₂²×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
166	C₂³×D₄	—	73	1	no	—
167	C₂⁴×C₄	—	73	2	no	—
168	C₂³×C₈	—	73	1	no	—
169	C₂³×C₈	—	73	1	no	—
170	C₂³×C₈	—	73	2	no	—
171	C₂³×C₈	—	73	24	no	—
172	C₂³×C₈	—	146	16	no	—
173	C₂×C₄×D₄	—	73	12	no	—
174	C₂×C₄×D₄	—	73	12	no	—
175	C₂²×C₄⋊C₄	—	73	1	no	—
176	C₂²×C₂²⋊C₄	—	73	2	no	—
177	C₂²×C₄²	—	73	1	no	—
178	C₈×D₄	—	73	64	no	—
179	C₂×C₄⋊C₈	—	73	12	no	—
180	C₂×C₂²⋊C₈	—	73	24	no	—
181	C₂×C₄×C₈	—	73	12	no	—
182	C₂⁵	—	73	2	no	—
183	C₂²×D₄	—	73	2	no	—
184	C₂²×D₄	—	73	12	no	—
185	C₂²×D₄	—	73	12	no	—
186	C₂²×D₄	—	73	1	no	—
187	C₂²×D₄	#10596842	1	1	yes	characteristic, direct product factor
188	C₂³×C₄	—	73	1	no	—
189	C₂³×C₄	—	73	1	no	—
190	C₂³×C₄	—	73	1	no	—
191	C₂³×C₄	—	73	2	no	—
192	C₂³×C₄	—	73	12	no	—
193	C₂³×C₄	—	73	12	no	—
194	C₂³×C₄	—	146	8	no	—
195	C₂³×C₄	—	146	8	no	—
196	C₂²×C₈	—	73	8	no	—
197	C₂²×C₈	—	73	6	no	—
198	C₂²×C₈	—	73	12	no	—
199	C₂²×C₈	—	73	6	no	—
200	C₂²×C₈	—	73	32	no	—
201	C₂²×C₈	—	146	96	no	—
202	C₂²×C₈	—	146	4	no	—
203	C₂²×C₈	—	146	8	no	—
204	C₄×D₄	—	73	16	no	—
205	C₄×D₄	—	73	48	no	—
206	C₂×C₄⋊C₄	—	73	12	no	—
207	C₂×C₂²⋊C₄	—	73	24	no	—
208	C₂×C₄²	—	73	6	no	—
209	C₂×C₄²	—	73	6	no	—
210	C₄⋊C₈	—	73	16	no	—
211	C₂²⋊C₈	—	73	32	no	—
212	C₄×C₈	—	73	16	no	—
213	C₂⁴	—	73	1	no	—
214	C₂⁴	—	73	2	no	—
215	C₂⁴	—	73	12	no	—
216	C₂⁴	—	73	12	no	—
217	C₂⁴	—	146	8	no	—
218	C₂⁴	—	146	8	no	—
219	C₂⁴	C₂×#10596842	1	2	yes	complemented
220	C₂×D₄	—	73	24	no	—
221	C₂×D₄	—	73	48	no	—
222	C₂×D₄	—	73	16	no	—
223	C₂×D₄	—	73	12	no	—
224	C₂×D₄	C₂×#10596842	1	12	yes	direct product factor
225	C₂²×C₄	—	73	3	no	—
226	C₂²×C₄	—	73	4	no	—
227	C₂²×C₄	—	73	4	no	—
228	C₂²×C₄	—	73	3	no	—
229	C₂²×C₄	—	73	1	no	—
230	C₂²×C₄	—	73	6	no	—
231	C₂²×C₄	—	73	6	no	—
232	C₂²×C₄	—	73	6	no	—
233	C₂²×C₄	—	73	12	no	—
234	C₂²×C₄	—	73	24	no	—
235	C₂²×C₄	—	73	8	no	—
236	C₂²×C₄	—	146	24	no	—
237	C₂²×C₄	—	146	24	no	—
238	C₂²×C₄	—	146	48	no	—
239	C₂²×C₄	—	146	4	no	—
240	C₂²×C₄	—	146	8	no	—
241	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic, complemented
242	C₂×C₈	—	73	24	no	—
243	C₂×C₈	—	73	4	no	—
244	C₂×C₈	—	73	4	no	—
245	C₂×C₈	—	73	8	no	—
246	C₂×C₈	—	146	64	no	—
247	C₂×C₈	—	146	12	no	—
248	C₂×C₈	—	146	24	no	—
249	C₄⋊C₄	—	73	16	no	—
250	C₂²⋊C₄	—	73	32	no	—
251	C₄²	—	73	12	no	—
252	C₄²	—	73	4	no	—
253	C₂³	—	2	8	no	—
254	C₂³	—	73	4	no	—
255	C₂³	—	73	4	no	—
256	C₂³	—	73	3	no	—
257	C₂³	—	73	3	no	—
258	C₂³	—	73	12	no	—
259	C₂³	—	73	8	no	—
260	C₂³	—	73	24	no	—
261	C₂³	—	146	24	no	—
262	C₂³	—	146	24	no	—
263	C₂³	—	146	48	no	—
264	C₂³	—	146	8	no	—
265	C₂³	C₂²×#10596842	1	1	yes	characteristic
266	C₂³	C₂²×#10596842	1	12	yes	complemented
267	D₄	—	73	32	no	—
268	D₄	—	73	16	no	—
269	D₄	C₂²×#10596842	1	16	yes	direct product factor
270	C₂×C₄	—	73	1	no	—
271	C₂×C₄	—	73	3	no	—
272	C₂×C₄	—	73	6	no	—
273	C₂×C₄	—	73	6	no	—
274	C₂×C₄	—	73	12	no	—
275	C₂×C₄	—	73	6	no	—
276	C₂×C₄	—	73	4	no	—
277	C₂×C₄	—	73	12	no	—
278	C₂×C₄	—	73	4	no	—
279	C₂×C₄	—	73	8	no	—
280	C₂×C₄	—	146	8	no	—
281	C₂×C₄	—	146	8	no	—
282	C₂×C₄	—	146	48	no	—
283	C₂×C₄	—	146	24	no	—
284	C₂×C₄	—	146	12	no	—
285	C₂×C₄	C₂²×#10596842	1	6	yes	complemented
286	C₈	—	73	8	no	—
287	C₈	—	146	8	no	—
288	C₈	—	146	4	no	—
289	C₂²	—	2	24	no	—
290	C₂²	—	73	6	no	—
291	C₂²	—	73	6	no	—
292	C₂²	—	73	12	no	—
293	C₂²	—	73	1	no	—
294	C₂²	—	73	3	no	—
295	C₂²	—	73	8	no	—
296	C₂²	—	146	8	no	—
297	C₂²	—	146	8	no	—
298	C₂²	—	146	48	no	—
299	C₂²	—	146	24	no	—
300	C₂²	D₄×#10596842	1	4	yes	direct product factor
301	C₂²	C₂³×#10596842	1	3	yes	—
302	C₂²	C₂³×#10596842	1	8	yes	complemented
303	C₄	—	73	1	no	—
304	C₄	—	73	1	no	—
305	C₄	—	73	6	no	—
306	C₄	—	73	4	no	—
307	C₄	—	146	8	no	—
308	C₄	—	146	4	no	—
309	C₄	C₂³×#10596842	1	4	yes	complemented
310	C₂	—	2	8	no	—
311	C₂	—	73	1	no	—
312	C₂	—	73	1	no	—
313	C₂	—	73	6	no	—
314	C₂	—	146	8	no	—
315	C₂	C₂×D₄×#10596842	1	6	yes	direct product factor
316	C₂	C₂⁴×#10596842	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×#10596842
commutator	C₂×C₇₃	C₂⁴×C₈
Frattini	C₂	C₂⁴×#10596842
Fitting	C₂²×D₄×C₇₃	C₈
socle	C₂³×C₇₃	C₂²×C₈
prime radical	C₂²×D₄×D₇₃	C₄
cosocle	C₂×#64408	C₂⁵
hypercenter	C₂²×D₄	#10596842
hypocenter	C₇₃	C₂²×C₈×D₄
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×D₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×#10596842	C₂	semidirect
2. C₂×D₄×#10596842	C₂	direct (N × H)
3. C₂²×C₄⋊#10596842	C₂	semidirect
4. C₂²×C₂²⋊#10596842	C₂	semidirect
5. C₂²×C₄×#10596842	C₂	semidirect
11. C₂³×#10596842	C₂²	semidirect
12. D₄×#10596842	C₂²	direct (N × H)
13. C₂×C₄⋊#10596842	C₂²	semidirect
14. C₂×C₂²⋊#10596842	C₂²	semidirect
15. C₂×C₄×#10596842	C₂²	semidirect
26. C₂²×#10596842	D₄	direct (N × H)
30. C₂²×#10596842	C₂³	semidirect
31. C₄⋊#10596842	C₂³	semidirect
32. C₂²⋊#10596842	C₂³	semidirect
33. C₄×#10596842	C₂³	semidirect
48. C₂²×D₂₉₂	C₈	semidirect
49. C₂²×C₂²⋊D₇₃	C₈	semidirect
50. C₂×D₄×D₇₃	C₈	semidirect
53. C₂²×D₄×C₇₃	C₈	semidirect
57. C₂×#10596842	C₂×D₄	direct (N × H)
80. C₂³×D₇₃	C₂×C₈	semidirect
81. C₂³×D₇₃	C₂×C₈	semidirect
84. C₂×D₂₉₂	C₂×C₈	semidirect
85. C₂×C₂²⋊D₇₃	C₂×C₈	semidirect
86. D₄×D₇₃	C₂×C₈	semidirect
88. C₂²×Dic₇₃	C₂×C₈	semidirect
89. C₂×C₄×D₇₃	C₂×C₈	semidirect
91. C₂⁴×C₇₃	C₂×C₈	semidirect
92. C₂×D₄×C₇₃	C₂×C₈	semidirect
93. C₂²×C₄×C₇₃	C₂×C₈	semidirect
96. C₂³×C₇₃	C₂²×C₈	semidirect
103. C₂²×D₇₃	C₂²×C₈	semidirect
104. C₂²×D₇₃	C₂²×C₈	semidirect
122. #10596842	C₂²×D₄	direct (N × H)
123. D₄×C₇₃	C₂²×C₈	semidirect
124. C₂×C₄×C₇₃	C₂²×C₈	semidirect
125. C₂²⋊D₇₃	C₂²×C₈	semidirect
126. C₂×Dic₇₃	C₂²×C₈	semidirect
127. D₂₉₂	C₂²×C₈	semidirect
128. C₄×D₇₃	C₂²×C₈	semidirect
131. C₂²×C₇₃	C₈×D₄	semidirect
133. C₂²×C₇₃	C₂³×C₈	semidirect
138. C₂×D₇₃	C₈×D₄	semidirect
141. C₂×D₇₃	C₂³×C₈	semidirect
149. C₄×C₇₃	C₂³×C₈	semidirect
150. Dic₇₃	C₂³×C₈	semidirect
153. C₂×C₇₃	C₂×C₈×D₄	semidirect
156. D₇₃	C₂×C₈×D₄	semidirect
165. C₇₃	C₂²×C₈×D₄	semidirect
187. C₂²×D₄	#10596842	direct (N × H)
219. C₂⁴	C₂×#10596842	semidirect
224. C₂×D₄	C₂×#10596842	direct (N × H)
241. C₂²×C₄	C₂×#10596842	semidirect
266. C₂³	C₂²×#10596842	semidirect
269. D₄	C₂²×#10596842	direct (N × H)
285. C₂×C₄	C₂²×#10596842	semidirect
300. C₂²	D₄×#10596842	direct (N × H)
302. C₂²	C₂³×#10596842	semidirect
309. C₄	C₂³×#10596842	semidirect
315. C₂	C₂×D₄×#10596842	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×#10596842	2
C₂×D₄×#10596842	2
C₂²×C₄⋊#10596842	2
C₂²×C₂²⋊#10596842	2
C₂²×C₄×#10596842	2
C₂²×D₄×#64408	2
C₂²×C₈×D₄	73

Subgroups — C22×D4×#10596842 #34035040

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₄×#10596842 #34035040