Subgroups — C₂²×#17988434 #33244156

Order 18688.

9534 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×C₄×D₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×#11924005	C₂	1	2	yes	maximal
3	C₂³×C₄×Dic₇₃	C₂	1	2	yes	maximal, complemented
4	C₂×#17988434	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#14556498	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#95×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₇₃	C₂²	1	1	yes	characteristic
8	C₂⁴×Dic₇₃	C₂²	1	2	yes	—
9	C₂⁴×Dic₇₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×D₇₃	—	2	16	no	—
11	C₂³×C₄×D₇₃	C₄	1	16	yes	complemented
12	C₂³×C₄×D₇₃	C₂²	1	12	yes	—
13	C₂²×#11924005	C₄	1	32	yes	complemented
14	C₂²×#11924005	C₂²	1	24	yes	—
15	C₂²×C₄×Dic₇₃	—	2	16	no	—
16	C₂²×C₄×Dic₇₃	C₂²	1	24	yes	complemented
17	#17988434	C₂²	1	64	yes	direct product factor
18	C₂×#14556498	C₂²	1	12	yes	complemented
19	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
20	C₂⁴×C₄×C₇₃	C₂²	1	2	yes	—
21	C₂×#95×C₇₃	C₂²	1	12	yes	complemented
22	C₂⁴×D₇₃	—	2	8	no	—
23	C₂⁴×D₇₃	—	2	4	no	—
24	C₂⁴×D₇₃	—	2	4	no	—
25	C₂⁴×D₇₃	C₂×C₄	1	8	yes	—
26	C₂⁴×D₇₃	C₂×C₄	1	16	yes	—
27	C₂⁴×D₇₃	C₂³	1	6	yes	—
28	C₂³×Dic₇₃	—	2	8	no	—
29	C₂³×Dic₇₃	—	2	8	no	—
30	C₂³×Dic₇₃	C₂×C₄	1	8	yes	—
31	C₂³×Dic₇₃	C₂×C₄	1	16	yes	complemented
32	C₂³×Dic₇₃	D₄	1	16	yes	—
33	C₂³×Dic₇₃	C₂³	1	12	yes	—
34	C₂³×Dic₇₃	C₂³	1	6	yes	—
35	C₂²×C₄×D₇₃	—	2	128	no	—
36	C₂²×C₄×D₇₃	—	2	96	no	—
37	C₂²×C₄×D₇₃	C₂×C₄	1	96	yes	complemented
38	C₂²×C₄×D₇₃	C₂³	1	16	yes	—
39	C₂×#11924005	C₂×C₄	1	192	yes	complemented
40	C₂×#11924005	C₂³	1	32	yes	—
41	C₂×C₄×Dic₇₃	—	2	96	no	—
42	C₂×C₄×Dic₇₃	C₂³	1	32	yes	complemented
43	#14556498	C₂³	1	16	yes	complemented
44	C₂⁵×C₇₃	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₇₃	—	2	8	no	—
46	C₂³×C₄×C₇₃	—	2	8	no	—
47	C₂³×C₄×C₇₃	—	2	8	no	—
48	C₂³×C₄×C₇₃	C₂×C₄	1	8	yes	—
49	C₂³×C₄×C₇₃	C₂×C₄	1	16	yes	complemented
50	C₂³×C₄×C₇₃	C₂³	1	6	yes	—
51	C₂³×C₄×C₇₃	C₂³	1	12	yes	—
52	#95×C₇₃	C₂³	1	16	yes	complemented
53	C₂³×D₇₃	—	2	32	no	—
54	C₂³×D₇₃	—	2	32	no	—
55	C₂³×D₇₃	—	2	64	no	—
56	C₂³×D₇₃	—	2	64	no	—
57	C₂³×D₇₃	—	2	24	no	—
58	C₂³×D₇₃	—	2	12	no	—
59	C₂³×D₇₃	—	2	12	no	—
60	C₂³×D₇₃	C₄²	1	64	yes	complemented
61	C₂³×D₇₃	C₂²×C₄	1	48	yes	—
62	C₂³×D₇₃	C₂²×C₄	1	24	yes	—
63	C₂³×D₇₃	C₂⁴	1	4	yes	—
64	C₂²×Dic₇₃	—	2	64	no	—
65	C₂²×Dic₇₃	—	2	24	no	—
66	C₂²×Dic₇₃	—	2	24	no	—
67	C₂²×Dic₇₃	C₂²⋊C₄	1	64	yes	complemented
68	C₂²×Dic₇₃	C₂²×C₄	1	24	yes	—
69	C₂²×Dic₇₃	C₂²×C₄	1	48	yes	complemented
70	C₂²×Dic₇₃	C₂×D₄	1	48	yes	—
71	C₂²×Dic₇₃	C₂⁴	1	8	yes	—
72	C₂²×Dic₇₃	C₂⁴	1	4	yes	—
73	C₂×C₄×D₇₃	—	2	384	no	—
74	C₂×C₄×D₇₃	—	2	64	no	—
75	C₂×C₄×D₇₃	C₂²×C₄	1	64	yes	complemented
76	#11924005	C₂²×C₄	1	128	yes	complemented
77	C₄×Dic₇₃	—	2	64	no	—
78	C₂⁴×C₇₃	C₂²×C₄	1	4	yes	—
79	C₂⁴×C₇₃	C₂²×C₄	1	8	yes	—
80	C₂⁴×C₇₃	C₂×D₄	1	8	yes	—
81	C₂⁴×C₇₃	#41	1	4	yes	—
82	C₂⁴×C₇₃	#41	1	4	yes	—
83	C₂⁴×C₇₃	C₂⁴	1	3	yes	—
84	C₂²×C₄×C₇₃	—	2	32	no	—
85	C₂²×C₄×C₇₃	—	2	64	no	—
86	C₂²×C₄×C₇₃	—	2	24	no	—
87	C₂²×C₄×C₇₃	—	2	24	no	—
88	C₂²×C₄×C₇₃	—	2	24	no	—
89	C₂²×C₄×C₇₃	C₂²×C₄	1	24	yes	—
90	C₂²×C₄×C₇₃	C₂²×C₄	1	48	yes	complemented
91	C₂²×C₄×C₇₃	C₂⁴	1	4	yes	—
92	C₂²×C₄×C₇₃	C₂⁴	1	8	yes	—
93	C₂³×C₇₃	C₂×C₄²	1	16	yes	—
94	C₂³×C₇₃	C₂×C₂²⋊C₄	1	16	yes	—
95	C₂³×C₇₃	#117	1	16	yes	—
96	C₂³×C₇₃	C₄×D₄	1	32	yes	—
97	C₂³×C₇₃	C₄×D₄	1	32	yes	—
98	C₂³×C₇₃	C₂³×C₄	1	6	yes	—
99	C₂³×C₇₃	C₂³×C₄	1	12	yes	—
100	C₂³×C₇₃	C₂²×D₄	1	12	yes	—
101	C₂³×C₇₃	C₂×#41	1	6	yes	—
102	C₂³×C₇₃	C₂×#41	1	6	yes	—
103	C₂³×C₇₃	C₂⁵	1	1	yes	characteristic
104	C₂²×D₇₃	—	2	48	no	—
105	C₂²×D₇₃	—	2	48	no	—
106	C₂²×D₇₃	—	2	96	no	—
107	C₂²×D₇₃	—	2	96	no	—
108	C₂²×D₇₃	—	2	8	no	—
109	C₂²×D₇₃	—	2	4	no	—
110	C₂²×D₇₃	—	2	4	no	—
111	C₂²×D₇₃	—	2	256	no	—
112	C₂²×D₇₃	C₂×C₄²	1	96	yes	complemented
113	C₂²×D₇₃	C₂³×C₄	1	16	yes	—
114	C₂²×D₇₃	C₂³×C₄	1	8	yes	—
115	C₂×C₄×C₇₃	—	2	48	no	—
116	C₂×C₄×C₇₃	—	2	96	no	—
117	C₂×C₄×C₇₃	—	2	8	no	—
118	C₂×C₄×C₇₃	—	2	8	no	—
119	C₂×C₄×C₇₃	—	2	8	no	—
120	C₂×C₄×C₇₃	C₂³×C₄	1	8	yes	—
121	C₂×C₄×C₇₃	C₂³×C₄	1	16	yes	complemented
122	C₂×Dic₇₃	—	2	96	no	—
123	C₂×Dic₇₃	—	2	8	no	—
124	C₂×Dic₇₃	—	2	8	no	—
125	C₂×Dic₇₃	C₂×C₂²⋊C₄	1	96	yes	complemented
126	C₂×Dic₇₃	C₂³×C₄	1	8	yes	—
127	C₂×Dic₇₃	C₂³×C₄	1	16	yes	complemented
128	C₂×Dic₇₃	C₂²×D₄	1	16	yes	—
129	C₄×D₇₃	—	2	128	no	—
130	C₂²×C₇₃	C₄×C₂²⋊C₄	1	64	yes	complemented
131	C₂²×C₇₃	C₂²×C₄²	1	12	yes	—
132	C₂²×C₇₃	C₂²×C₂²⋊C₄	1	12	yes	—
133	C₂²×C₇₃	C₂×#117	1	12	yes	—
134	C₂²×C₇₃	C₂×C₄×D₄	1	24	yes	—
135	C₂²×C₇₃	C₂×C₄×D₄	1	24	yes	—
136	C₂²×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
137	C₂²×C₇₃	C₂⁴×C₄	1	2	yes	—
138	C₂²×C₇₃	C₂³×D₄	1	2	yes	—
139	C₂²×C₇₃	C₂²×#41	1	1	yes	characteristic
140	C₂²×C₇₃	C₂²×#41	1	1	yes	characteristic
141	C₂×D₇₃	—	2	8	no	—
142	C₂×D₇₃	—	2	8	no	—
143	C₂×D₇₃	—	2	16	no	—
144	C₂×D₇₃	—	2	16	no	—
145	C₂×D₇₃	—	2	192	no	—
146	C₂×D₇₃	C₂²×C₄²	1	16	yes	complemented
147	C₄×C₇₃	—	2	8	no	—
148	C₄×C₇₃	—	2	16	no	—
149	Dic₇₃	—	2	16	no	—
150	Dic₇₃	C₂²×C₂²⋊C₄	1	16	yes	complemented
151	C₂²×C₄×C₂²⋊C₄	—	73	1	no	maximal, Sylow
152	C₂×C₇₃	C₂×C₄×C₂²⋊C₄	1	24	yes	complemented
153	C₂×C₇₃	C₂³×C₄²	1	1	yes	characteristic
154	C₂×C₇₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
155	C₂×C₇₃	C₂²×#117	1	1	yes	characteristic
156	C₂×C₇₃	C₂²×C₄×D₄	1	2	yes	—
157	C₂×C₇₃	C₂²×C₄×D₄	1	2	yes	—
158	D₇₃	—	2	16	no	—
159	C₂⁵×C₄	—	73	1	no	—
160	C₂³×C₂²⋊C₄	—	73	2	no	—
161	C₂³×C₄²	—	73	2	no	—
162	C₂×C₄×C₂²⋊C₄	—	73	24	no	—
163	C₂²×#95	—	73	1	no	—
164	C₂²×#95	D₇₃	1	1	yes	characteristic, complemented
165	C₇₃	C₂²×C₄×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
166	C₂⁶	—	73	1	no	—
167	C₂⁴×C₄	—	73	1	no	—
168	C₂⁴×C₄	—	73	2	no	—
169	C₂⁴×C₄	—	73	16	no	—
170	C₂⁴×C₄	—	73	12	no	—
171	C₂⁴×C₄	—	146	16	no	—
172	C₂⁴×C₄	C₂×D₇₃	1	1	yes	characteristic
173	C₂⁴×C₄	C₂×D₇₃	1	2	yes	—
174	C₂²×C₂²⋊C₄	—	73	32	no	—
175	C₂²×C₂²⋊C₄	—	73	24	no	—
176	C₂²×C₄²	—	73	24	no	—
177	C₂²×C₄²	—	146	16	no	—
178	C₄×C₂²⋊C₄	—	73	64	no	—
179	C₂×#95	—	73	12	no	—
180	C₂×#95	C₂×D₇₃	1	12	yes	complemented
181	C₂⁵	—	73	16	no	—
182	C₂⁵	—	73	8	no	—
183	C₂⁵	—	73	6	no	—
184	C₂⁵	—	146	8	no	—
185	C₂⁵	—	146	4	no	—
186	C₂⁵	—	146	4	no	—
187	C₂⁵	C₂²×D₇₃	1	1	yes	characteristic
188	C₂³×C₄	—	2	8	no	—
189	C₂³×C₄	—	2	8	no	—
190	C₂³×C₄	—	2	8	no	—
191	C₂³×C₄	—	73	8	no	—
192	C₂³×C₄	—	73	16	no	—
193	C₂³×C₄	—	73	6	no	—
194	C₂³×C₄	—	73	16	no	—
195	C₂³×C₄	—	73	12	no	—
196	C₂³×C₄	—	73	96	no	—
197	C₂³×C₄	—	73	16	no	—
198	C₂³×C₄	—	146	128	no	—
199	C₂³×C₄	—	146	96	no	—
200	C₂³×C₄	—	146	8	no	—
201	C₂³×C₄	—	146	8	no	—
202	C₂³×C₄	C₄×D₇₃	1	8	yes	—
203	C₂³×C₄	C₄×D₇₃	1	16	yes	complemented
204	C₂³×C₄	C₂²×D₇₃	1	6	yes	—
205	C₂³×C₄	C₂²×D₇₃	1	12	yes	—
206	C₂×C₂²⋊C₄	—	73	192	no	—
207	C₂×C₂²⋊C₄	—	73	32	no	—
208	C₂×C₄²	—	73	32	no	—
209	C₂×C₄²	—	146	96	no	—
210	#95	—	73	16	no	—
211	#95	C₂²×D₇₃	1	16	yes	complemented
212	C₂⁴	—	73	64	no	—
213	C₂⁴	—	73	48	no	—
214	C₂⁴	—	73	24	no	—
215	C₂⁴	—	73	4	no	—
216	C₂⁴	—	146	32	no	—
217	C₂⁴	—	146	32	no	—
218	C₂⁴	—	146	64	no	—
219	C₂⁴	—	146	64	no	—
220	C₂⁴	—	146	24	no	—
221	C₂⁴	—	146	12	no	—
222	C₂⁴	—	146	12	no	—
223	C₂⁴	C₂×C₄×D₇₃	1	4	yes	—
224	C₂⁴	C₂×C₄×D₇₃	1	8	yes	—
225	C₂⁴	D₄×D₇₃	1	8	yes	—
226	C₂⁴	#11924027	1	4	yes	—
227	C₂⁴	#11924029	1	4	yes	—
228	C₂⁴	C₂³×D₇₃	1	3	yes	—
229	C₂²×C₄	—	2	32	no	—
230	C₂²×C₄	—	2	64	no	—
231	C₂²×C₄	—	2	24	no	—
232	C₂²×C₄	—	2	24	no	—
233	C₂²×C₄	—	2	24	no	—
234	C₂²×C₄	—	73	64	no	—
235	C₂²×C₄	—	73	24	no	—
236	C₂²×C₄	—	73	48	no	—
237	C₂²×C₄	—	73	4	no	—
238	C₂²×C₄	—	73	48	no	—
239	C₂²×C₄	—	73	8	no	—
240	C₂²×C₄	—	73	64	no	—
241	C₂²×C₄	—	146	64	no	—
242	C₂²×C₄	—	146	384	no	—
243	C₂²×C₄	—	146	64	no	—
244	C₂²×C₄	—	146	24	no	—
245	C₂²×C₄	—	146	24	no	—
246	C₂²×C₄	C₂×C₄×D₇₃	1	24	yes	—
247	C₂²×C₄	C₂×C₄×D₇₃	1	48	yes	complemented
248	C₂²×C₄	C₂³×D₇₃	1	4	yes	—
249	C₂²×C₄	C₂³×D₇₃	1	8	yes	—
250	C₂²⋊C₄	—	73	128	no	—
251	C₄²	—	146	64	no	—
252	C₂³	—	73	16	no	—
253	C₂³	—	73	8	no	—
254	C₂³	—	73	96	no	—
255	C₂³	—	146	8	no	—
256	C₂³	—	146	4	no	—
257	C₂³	—	146	4	no	—
258	C₂³	—	146	256	no	—
259	C₂³	—	146	48	no	—
260	C₂³	—	146	48	no	—
261	C₂³	—	146	96	no	—
262	C₂³	—	146	96	no	—
263	C₂³	C₄²×D₇₃	1	16	yes	—
264	C₂³	C₂²⋊C₄×D₇₃	1	16	yes	—
265	C₂³	#14556549	1	16	yes	—
266	C₂³	#14556551	1	32	yes	—
267	C₂³	#14556553	1	32	yes	—
268	C₂³	C₂²×C₄×D₇₃	1	6	yes	—
269	C₂³	C₂²×C₄×D₇₃	1	12	yes	—
270	C₂³	C₂×D₄×D₇₃	1	12	yes	—
271	C₂³	C₂×#11924027	1	6	yes	—
272	C₂³	C₂×#11924029	1	6	yes	—
273	C₂³	C₂⁴×D₇₃	1	1	yes	characteristic
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	2	48	no	—
278	C₂×C₄	—	2	96	no	—
279	C₂×C₄	—	73	8	no	—
280	C₂×C₄	—	73	16	no	—
281	C₂×C₄	—	73	96	no	—
282	C₂×C₄	—	73	16	no	—
283	C₂×C₄	—	146	128	no	—
284	C₂×C₄	—	146	8	no	—
285	C₂×C₄	—	146	8	no	—
286	C₂×C₄	—	146	96	no	—
287	C₂×C₄	C₂²×C₄×D₇₃	1	8	yes	—
288	C₂×C₄	C₂²×C₄×D₇₃	1	16	yes	complemented
289	C₂²	—	73	16	no	—
290	C₂²	—	146	192	no	—
291	C₂²	—	146	8	no	—
292	C₂²	—	146	8	no	—
293	C₂²	—	146	16	no	—
294	C₂²	—	146	16	no	—
295	C₂²	#17988434	1	64	yes	direct product factor
296	C₂²	C₂×C₄²×D₇₃	1	12	yes	—
297	C₂²	C₂×C₂²⋊C₄×D₇₃	1	12	yes	—
298	C₂²	C₂×#14556549	1	12	yes	—
299	C₂²	C₂×#14556551	1	24	yes	—
300	C₂²	C₂×#14556553	1	24	yes	—
301	C₂²	C₂³×C₄×D₇₃	1	1	yes	characteristic
302	C₂²	C₂³×C₄×D₇₃	1	2	yes	—
303	C₂²	C₂²×D₄×D₇₃	1	2	yes	—
304	C₂²	C₂²×#11924027	1	1	yes	characteristic
305	C₂²	C₂²×#11924029	1	1	yes	characteristic
306	C₄	—	2	8	no	—
307	C₄	—	2	16	no	—
308	C₄	—	73	16	no	—
309	C₄	—	146	16	no	—
310	C₂	—	146	16	no	—
311	C₂	C₂×#17988434	1	24	yes	direct product factor
312	C₂	C₂²×C₄²×D₇₃	1	1	yes	characteristic
313	C₂	C₂²×C₂²⋊C₄×D₇₃	1	1	yes	characteristic
314	C₂	C₂²×#14556549	1	1	yes	characteristic
315	C₂	C₂²×#14556551	1	2	yes	—
316	C₂	C₂²×#14556553	1	2	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₇₃
commutator	C₂×C₇₃	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₇₃
Fitting	C₂²×#95×C₇₃	C₂
socle	C₂⁵×C₇₃	C₂³
prime radical	C₂⁵×D₇₃	C₂²
cosocle	C₂³×C₇₃	C₂⁵
hypercenter	C₂²×#95	D₇₃
hypocenter	C₇₃	C₂²×C₄×C₂²⋊C₄
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁵ ◁ C₂²×#95

Complements

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

N	H ≅ G/N	type
3. C₂³×C₄×Dic₇₃	C₂	semidirect
4. C₂×#17988434	C₂	direct (N × H)
5. C₂²×#14556498	C₂	semidirect
6. C₂²×#95×C₇₃	C₂	semidirect
11. C₂³×C₄×D₇₃	C₄	semidirect
13. C₂²×#11924005	C₄	semidirect
16. C₂²×C₄×Dic₇₃	C₂²	semidirect
17. #17988434	C₂²	direct (N × H)
18. C₂×#14556498	C₂²	semidirect
21. C₂×#95×C₇₃	C₂²	semidirect
31. C₂³×Dic₇₃	C₂×C₄	semidirect
37. C₂²×C₄×D₇₃	C₂×C₄	semidirect
39. C₂×#11924005	C₂×C₄	semidirect
42. C₂×C₄×Dic₇₃	C₂³	semidirect
43. #14556498	C₂³	semidirect
49. C₂³×C₄×C₇₃	C₂×C₄	semidirect
52. #95×C₇₃	C₂³	semidirect
60. C₂³×D₇₃	C₄²	semidirect
67. C₂²×Dic₇₃	C₂²⋊C₄	semidirect
69. C₂²×Dic₇₃	C₂²×C₄	semidirect
75. C₂×C₄×D₇₃	C₂²×C₄	semidirect
76. #11924005	C₂²×C₄	semidirect
90. C₂²×C₄×C₇₃	C₂²×C₄	semidirect
112. C₂²×D₇₃	C₂×C₄²	semidirect
121. C₂×C₄×C₇₃	C₂³×C₄	semidirect
125. C₂×Dic₇₃	C₂×C₂²⋊C₄	semidirect
127. C₂×Dic₇₃	C₂³×C₄	semidirect
130. C₂²×C₇₃	C₄×C₂²⋊C₄	semidirect
146. C₂×D₇₃	C₂²×C₄²	semidirect
150. Dic₇₃	C₂²×C₂²⋊C₄	semidirect
152. C₂×C₇₃	C₂×C₄×C₂²⋊C₄	semidirect
164. C₂²×#95	D₇₃	semidirect
165. C₇₃	C₂²×C₄×C₂²⋊C₄	semidirect
180. C₂×#95	C₂×D₇₃	semidirect
203. C₂³×C₄	C₄×D₇₃	semidirect
211. #95	C₂²×D₇₃	semidirect
247. C₂²×C₄	C₂×C₄×D₇₃	semidirect
288. C₂×C₄	C₂²×C₄×D₇₃	semidirect
295. C₂²	#17988434	direct (N × H)
311. C₂	C₂×#17988434	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×C₄×D₇₃	2
C₂³×#11924005	2
C₂³×C₄×Dic₇₃	2
C₂×#17988434	2
C₂²×#14556498	2
C₂²×#95×C₇₃	2
C₂²×C₄×C₂²⋊C₄	73

Subgroups — C22×#17988434 #33244156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17988434 #33244156