Subgroups — C₂²×#17988499 #33252164

Order 18688.

7118 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×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₂²⋊Dic₇₃	C₂	1	2	yes	maximal
3	C₂×#17988499	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#14556498	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#14556498	C₂	1	2	yes	maximal, complemented
6	C₂³×C₂²⋊C₄×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₄₆	—	2	16	no	—
8	C₂³×Dic₁₄₆	—	2	8	no	—
9	C₂³×Dic₁₄₆	C₂²	1	12	yes	complemented
10	C₂⁴×Dic₇₃	C₂²	1	2	yes	—
11	C₂⁴×Dic₇₃	C₂²	1	2	yes	—
12	C₂²×C₂²⋊Dic₇₃	—	2	16	no	—
13	C₂²×C₂²⋊Dic₇₃	C₂²	1	24	yes	—
14	#17988499	C₂²	1	64	yes	direct product factor
15	C₂×#14556498	C₂²	1	24	yes	complemented
16	C₂×#14556498	C₂²	1	12	yes	complemented
17	C₂⁶×C₇₃	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
19	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
20	C₂²×C₂²⋊C₄×C₇₃	—	2	8	no	—
21	C₂²×C₂²⋊C₄×C₇₃	C₂²	1	12	yes	—
22	C₂²×Dic₁₄₆	—	2	128	no	—
23	C₂²×Dic₁₄₆	—	2	96	no	—
24	C₂²×Dic₁₄₆	—	2	48	no	—
25	C₂²×Dic₁₄₆	C₂³	1	16	yes	complemented
26	C₂³×Dic₇₃	—	2	8	no	—
27	C₂³×Dic₇₃	—	2	8	no	—
28	C₂³×Dic₇₃	—	2	8	no	—
29	C₂³×Dic₇₃	—	2	8	no	—
30	C₂³×Dic₇₃	—	2	8	no	—
31	C₂³×Dic₇₃	D₄	1	16	yes	—
32	C₂³×Dic₇₃	C₂³	1	12	yes	—
33	C₂³×Dic₇₃	C₂³	1	12	yes	—
34	C₂×C₂²⋊Dic₇₃	—	2	96	no	—
35	C₂×C₂²⋊Dic₇₃	C₂³	1	32	yes	—
36	#14556498	C₂³	1	32	yes	complemented
37	#14556498	C₂³	1	16	yes	complemented
38	C₂⁵×C₇₃	—	2	4	no	—
39	C₂⁵×C₇₃	—	2	8	no	—
40	C₂⁵×C₇₃	—	2	8	no	—
41	C₂⁵×C₇₃	—	2	4	no	—
42	C₂⁵×C₇₃	Q₈	1	8	yes	complemented
43	C₂⁵×C₇₃	C₂³	1	6	yes	—
44	C₂⁵×C₇₃	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₇₃	—	2	8	no	—
46	C₂³×C₄×C₇₃	—	2	4	no	—
47	C₂³×C₄×C₇₃	—	2	8	no	—
48	C₂³×C₄×C₇₃	D₄	1	8	yes	—
49	C₂³×C₄×C₇₃	C₂³	1	6	yes	—
50	C₂³×C₄×C₇₃	C₂³	1	6	yes	—
51	C₂×C₂²⋊C₄×C₇₃	—	2	48	no	—
52	C₂×C₂²⋊C₄×C₇₃	C₂³	1	16	yes	—
53	C₂×Dic₁₄₆	—	2	384	no	—
54	C₂×Dic₁₄₆	—	2	32	no	—
55	C₂×Dic₁₄₆	—	2	64	no	—
56	C₂²×Dic₇₃	—	2	64	no	—
57	C₂²×Dic₇₃	—	2	24	no	—
58	C₂²×Dic₇₃	—	2	24	no	—
59	C₂²×Dic₇₃	—	2	24	no	—
60	C₂²×Dic₇₃	—	2	24	no	—
61	C₂²×Dic₇₃	—	2	24	no	—
62	C₂²×Dic₇₃	—	4	32	no	—
63	C₂²×Dic₇₃	C₂×D₄	1	48	yes	—
64	C₂²×Dic₇₃	C₂⁴	1	8	yes	—
65	C₂²×Dic₇₃	C₂⁴	1	8	yes	—
66	C₂²⋊Dic₇₃	—	2	64	no	—
67	C₂⁴×C₇₃	—	2	64	no	—
68	C₂⁴×C₇₃	—	2	32	no	—
69	C₂⁴×C₇₃	—	2	12	no	—
70	C₂⁴×C₇₃	—	2	24	no	—
71	C₂⁴×C₇₃	—	2	24	no	—
72	C₂⁴×C₇₃	—	2	12	no	—
73	C₂⁴×C₇₃	—	4	16	no	—
74	C₂⁴×C₇₃	—	4	32	no	—
75	C₂⁴×C₇₃	—	4	16	no	—
76	C₂⁴×C₇₃	C₂×D₄	1	8	yes	—
77	C₂⁴×C₇₃	C₂×D₄	1	4	yes	—
78	C₂⁴×C₇₃	C₂×Q₈	1	4	yes	—
79	C₂⁴×C₇₃	C₂×Q₈	1	24	yes	complemented
80	C₂⁴×C₇₃	#41	1	8	yes	—
81	C₂⁴×C₇₃	#41	1	4	yes	—
82	C₂⁴×C₇₃	C₂⁴	1	4	yes	—
83	C₂⁴×C₇₃	C₂⁴	1	3	yes	—
84	C₂²×C₄×C₇₃	—	2	32	no	—
85	C₂²×C₄×C₇₃	—	2	24	no	—
86	C₂²×C₄×C₇₃	—	2	12	no	—
87	C₂²×C₄×C₇₃	—	2	24	no	—
88	C₂²×C₄×C₇₃	—	4	16	no	—
89	C₂²×C₄×C₇₃	C₂×D₄	1	24	yes	—
90	C₂²×C₄×C₇₃	C₂⁴	1	4	yes	—
91	C₂²×C₄×C₇₃	C₂⁴	1	4	yes	—
92	C₂²⋊C₄×C₇₃	—	2	32	no	—
93	C₂³×C₇₃	—	2	96	no	—
94	C₂³×C₇₃	—	2	48	no	—
95	C₂³×C₇₃	—	2	4	no	—
96	C₂³×C₇₃	—	2	8	no	—
97	C₂³×C₇₃	—	2	8	no	—
98	C₂³×C₇₃	—	2	4	no	—
99	C₂³×C₇₃	—	4	24	no	—
100	C₂³×C₇₃	—	4	48	no	—
101	C₂³×C₇₃	—	4	24	no	—
102	C₂³×C₇₃	—	4	128	no	—
103	C₂³×C₇₃	#120	1	16	yes	—
104	C₂³×C₇₃	C₂²⋊Q₈	1	32	yes	—
105	C₂³×C₇₃	C₂²⋊Q₈	1	16	yes	—
106	C₂³×C₇₃	#124	1	32	yes	—
107	C₂³×C₇₃	#124	1	16	yes	—
108	C₂³×C₇₃	C₂²×D₄	1	12	yes	—
109	C₂³×C₇₃	C₂²×D₄	1	6	yes	—
110	C₂³×C₇₃	C₂²×Q₈	1	6	yes	—
111	C₂³×C₇₃	C₂²×Q₈	1	8	yes	complemented
112	C₂³×C₇₃	C₂×#41	1	12	yes	—
113	C₂³×C₇₃	C₂×#41	1	6	yes	—
114	C₂³×C₇₃	C₂⁵	1	1	yes	characteristic
115	C₂×C₄×C₇₃	—	2	48	no	—
116	C₂×C₄×C₇₃	—	2	8	no	—
117	C₂×C₄×C₇₃	—	2	4	no	—
118	C₂×C₄×C₇₃	—	2	8	no	—
119	C₂×C₄×C₇₃	—	4	24	no	—
120	C₂×C₄×C₇₃	C₂²×D₄	1	8	yes	—
121	C₂×Dic₇₃	—	2	96	no	—
122	C₂×Dic₇₃	—	2	8	no	—
123	C₂×Dic₇₃	—	2	8	no	—
124	C₂×Dic₇₃	—	2	8	no	—
125	C₂×Dic₇₃	—	2	8	no	—
126	C₂×Dic₇₃	—	2	8	no	—
127	C₂×Dic₇₃	—	4	48	no	—
128	C₂×Dic₇₃	C₂²×D₄	1	16	yes	—
129	Dic₁₄₆	—	2	128	no	—
130	C₂²×C₇₃	—	2	16	no	—
131	C₂²×C₇₃	—	2	8	no	—
132	C₂²×C₇₃	—	4	96	no	—
133	C₂²×C₇₃	—	4	4	no	—
134	C₂²×C₇₃	—	4	8	no	—
135	C₂²×C₇₃	—	4	4	no	—
136	C₂²×C₇₃	#393	1	64	yes	complemented
137	C₂²×C₇₃	C₂×#120	1	12	yes	—
138	C₂²×C₇₃	C₂×C₂²⋊Q₈	1	24	yes	—
139	C₂²×C₇₃	C₂×C₂²⋊Q₈	1	12	yes	—
140	C₂²×C₇₃	C₂×#124	1	24	yes	—
141	C₂²×C₇₃	C₂×#124	1	12	yes	—
142	C₂²×C₇₃	C₂³×D₄	1	2	yes	—
143	C₂²×C₇₃	C₂³×D₄	1	1	yes	characteristic
144	C₂²×C₇₃	C₂³×Q₈	1	1	yes	characteristic
145	C₂²×C₇₃	C₂²×#41	1	2	yes	—
146	C₂²×C₇₃	C₂²×#41	1	1	yes	characteristic
147	C₄×C₇₃	—	2	8	no	—
148	C₄×C₇₃	—	4	4	no	—
149	Dic₇₃	—	2	16	no	—
150	Dic₇₃	—	4	8	no	—
151	C₂²×#393	—	73	1	no	maximal, Sylow
152	C₂×C₇₃	—	4	8	no	—
153	C₂×C₇₃	C₂×#393	1	24	yes	complemented
154	C₂×C₇₃	C₂²×#120	1	1	yes	characteristic
155	C₂×C₇₃	C₂²×C₂²⋊Q₈	1	2	yes	—
156	C₂×C₇₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
157	C₂×C₇₃	C₂²×#124	1	2	yes	—
158	C₂×C₇₃	C₂²×#124	1	1	yes	characteristic
159	C₂⁴×Q₈	—	73	1	no	—
160	C₂³×C₂²⋊C₄	—	73	2	no	—
161	C₂³×C₂²⋊C₄	D₇₃	1	1	yes	characteristic
162	C₂×#393	—	73	24	no	—
163	C₂²×#95	—	73	2	no	—
164	C₂²×#95	—	73	1	no	—
165	C₇₃	C₂²×#393	1	1	yes	characteristic, complemented, Sylow
166	C₂⁶	C₂×D₇₃	1	1	yes	characteristic
167	C₂³×Q₈	—	73	12	no	—
168	C₂³×Q₈	—	146	16	no	—
169	C₂³×Q₈	—	146	8	no	—
170	C₂⁴×C₄	—	73	2	no	—
171	C₂⁴×C₄	—	73	2	no	—
172	C₂⁴×C₄	C₂×D₇₃	1	1	yes	characteristic
173	C₂⁴×C₄	C₂×D₇₃	1	1	yes	characteristic
174	C₂²×C₂²⋊C₄	—	2	8	no	—
175	C₂²×C₂²⋊C₄	—	73	24	no	—
176	C₂²×C₂²⋊C₄	—	146	16	no	—
177	C₂²×C₂²⋊C₄	C₂×D₇₃	1	12	yes	—
178	#393	—	73	64	no	—
179	C₂×#95	—	73	24	no	—
180	C₂×#95	—	73	12	no	—
181	C₂⁵	—	2	8	no	—
182	C₂⁵	—	2	4	no	—
183	C₂⁵	—	2	8	no	—
184	C₂⁵	—	2	4	no	—
185	C₂⁵	Dic₁₄₆	1	8	yes	complemented
186	C₂⁵	C₂²×D₇₃	1	6	yes	—
187	C₂⁵	C₂²×D₇₃	1	1	yes	characteristic
188	C₂²×Q₈	—	73	16	no	—
189	C₂²×Q₈	—	146	96	no	—
190	C₂²×Q₈	—	146	128	no	—
191	C₂²×Q₈	—	146	48	no	—
192	C₂³×C₄	—	2	8	no	—
193	C₂³×C₄	—	2	4	no	—
194	C₂³×C₄	—	2	8	no	—
195	C₂³×C₄	—	73	16	no	—
196	C₂³×C₄	—	73	12	no	—
197	C₂³×C₄	—	73	12	no	—
198	C₂³×C₄	—	146	8	no	—
199	C₂³×C₄	—	146	8	no	—
200	C₂³×C₄	—	146	8	no	—
201	C₂³×C₄	—	146	8	no	—
202	C₂³×C₄	—	146	8	no	—
203	C₂³×C₄	C₂²⋊D₇₃	1	8	yes	—
204	C₂³×C₄	C₂²×D₇₃	1	6	yes	—
205	C₂³×C₄	C₂²×D₇₃	1	6	yes	—
206	C₂×C₂²⋊C₄	—	2	48	no	—
207	C₂×C₂²⋊C₄	—	73	32	no	—
208	C₂×C₂²⋊C₄	—	146	96	no	—
209	C₂×C₂²⋊C₄	C₂²×D₇₃	1	16	yes	—
210	#95	—	73	32	no	—
211	#95	—	73	16	no	—
212	C₂⁴	—	2	64	no	—
213	C₂⁴	—	2	32	no	—
214	C₂⁴	—	2	24	no	—
215	C₂⁴	—	2	12	no	—
216	C₂⁴	—	2	24	no	—
217	C₂⁴	—	2	12	no	—
218	C₂⁴	—	4	16	no	—
219	C₂⁴	—	4	32	no	—
220	C₂⁴	—	4	16	no	—
221	C₂⁴	D₄×D₇₃	1	8	yes	—
222	C₂⁴	C₂×C₂²⋊D₇₃	1	4	yes	—
223	C₂⁴	C₂×Dic₁₄₆	1	4	yes	—
224	C₂⁴	C₂×Dic₁₄₆	1	24	yes	complemented
225	C₂⁴	#11924027	1	8	yes	—
226	C₂⁴	#11924028	1	4	yes	—
227	C₂⁴	C₂³×D₇₃	1	4	yes	—
228	C₂⁴	C₂³×D₇₃	1	3	yes	—
229	C₂×Q₈	—	146	64	no	—
230	C₂×Q₈	—	146	384	no	—
231	C₂×Q₈	—	146	32	no	—
232	C₂²×C₄	—	2	32	no	—
233	C₂²×C₄	—	2	24	no	—
234	C₂²×C₄	—	2	12	no	—
235	C₂²×C₄	—	2	24	no	—
236	C₂²×C₄	—	4	16	no	—
237	C₂²×C₄	—	292	32	no	—
238	C₂²×C₄	—	73	48	no	—
239	C₂²×C₄	—	73	8	no	—
240	C₂²×C₄	—	73	8	no	—
241	C₂²×C₄	—	146	24	no	—
242	C₂²×C₄	—	146	24	no	—
243	C₂²×C₄	—	146	24	no	—
244	C₂²×C₄	—	146	64	no	—
245	C₂²×C₄	—	146	24	no	—
246	C₂²×C₄	—	146	24	no	—
247	C₂²×C₄	C₂×C₂²⋊D₇₃	1	24	yes	—
248	C₂²×C₄	C₂³×D₇₃	1	4	yes	—
249	C₂²×C₄	C₂³×D₇₃	1	4	yes	—
250	C₂²⋊C₄	—	2	32	no	—
251	C₂²⋊C₄	—	146	64	no	—
252	C₂³	—	2	8	no	—
253	C₂³	—	2	4	no	—
254	C₂³	—	2	8	no	—
255	C₂³	—	2	4	no	—
256	C₂³	—	2	96	no	—
257	C₂³	—	2	48	no	—
258	C₂³	—	4	128	no	—
259	C₂³	—	4	24	no	—
260	C₂³	—	4	48	no	—
261	C₂³	—	4	24	no	—
262	C₂³	#14556559	1	16	yes	—
263	C₂³	#14556568	1	32	yes	—
264	C₂³	#14556570	1	16	yes	—
265	C₂³	#14556577	1	32	yes	—
266	C₂³	#14556578	1	16	yes	—
267	C₂³	C₂×D₄×D₇₃	1	12	yes	—
268	C₂³	C₂²×C₂²⋊D₇₃	1	6	yes	—
269	C₂³	C₂²×Dic₁₄₆	1	6	yes	—
270	C₂³	C₂²×Dic₁₄₆	1	8	yes	complemented
271	C₂³	C₂×#11924027	1	12	yes	—
272	C₂³	C₂×#11924028	1	6	yes	—
273	C₂³	C₂⁴×D₇₃	1	1	yes	characteristic
274	Q₈	—	146	128	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	48	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	4	24	no	—
280	C₂×C₄	—	292	48	no	—
281	C₂×C₄	—	73	16	no	—
282	C₂×C₄	—	146	8	no	—
283	C₂×C₄	—	146	8	no	—
284	C₂×C₄	—	146	8	no	—
285	C₂×C₄	—	146	8	no	—
286	C₂×C₄	—	146	8	no	—
287	C₂×C₄	—	146	96	no	—
288	C₂×C₄	C₂²×C₂²⋊D₇₃	1	8	yes	—
289	C₂²	—	2	16	no	—
290	C₂²	—	2	8	no	—
291	C₂²	—	4	96	no	—
292	C₂²	—	4	4	no	—
293	C₂²	—	4	8	no	—
294	C₂²	—	4	4	no	—
295	C₂²	#17988499	1	64	yes	direct product factor
296	C₂²	C₂×#14556559	1	12	yes	—
297	C₂²	C₂×#14556568	1	24	yes	—
298	C₂²	C₂×#14556570	1	12	yes	—
299	C₂²	C₂×#14556577	1	24	yes	—
300	C₂²	C₂×#14556578	1	12	yes	—
301	C₂²	C₂²×D₄×D₇₃	1	2	yes	—
302	C₂²	C₂³×C₂²⋊D₇₃	1	1	yes	characteristic
303	C₂²	C₂³×Dic₁₄₆	1	1	yes	characteristic
304	C₂²	C₂²×#11924027	1	2	yes	—
305	C₂²	C₂²×#11924028	1	1	yes	characteristic
306	C₄	—	2	8	no	—
307	C₄	—	4	4	no	—
308	C₄	—	292	8	no	—
309	C₄	—	146	16	no	—
310	C₂	—	4	8	no	—
311	C₂	C₂×#17988499	1	24	yes	direct product factor
312	C₂	C₂²×#14556559	1	1	yes	characteristic
313	C₂	C₂²×#14556568	1	2	yes	—
314	C₂	C₂²×#14556570	1	1	yes	characteristic
315	C₂	C₂²×#14556577	1	2	yes	—
316	C₂	C₂²×#14556578	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₇₃
commutator, cosocle	C₂³×C₇₃	C₂⁵
Frattini	C₂³	C₂⁴×D₇₃
Fitting	C₂³×C₂²⋊C₄×C₇₃	C₂
socle	C₂⁵×C₇₃	C₂³
prime radical	C₂⁶×C₇₃	C₂²
hypercenter	C₂³×C₂²⋊C₄	D₇₃
hypocenter	C₇₃	C₂²×#393
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×Dic₁₄₆	C₂	semidirect
3. C₂×#17988499	C₂	direct (N × H)
4. C₂²×#14556498	C₂	semidirect
5. C₂²×#14556498	C₂	semidirect
9. C₂³×Dic₁₄₆	C₂²	semidirect
14. #17988499	C₂²	direct (N × H)
15. C₂×#14556498	C₂²	semidirect
16. C₂×#14556498	C₂²	semidirect
25. C₂²×Dic₁₄₆	C₂³	semidirect
36. #14556498	C₂³	semidirect
37. #14556498	C₂³	semidirect
42. C₂⁵×C₇₃	Q₈	semidirect
79. C₂⁴×C₇₃	C₂×Q₈	semidirect
111. C₂³×C₇₃	C₂²×Q₈	semidirect
136. C₂²×C₇₃	#393	semidirect
153. C₂×C₇₃	C₂×#393	semidirect
165. C₇₃	C₂²×#393	semidirect
185. C₂⁵	Dic₁₄₆	semidirect
224. C₂⁴	C₂×Dic₁₄₆	semidirect
270. C₂³	C₂²×Dic₁₄₆	semidirect
295. C₂²	#17988499	direct (N × H)
311. C₂	C₂×#17988499	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×Dic₁₄₆	2
C₂³×C₂²⋊Dic₇₃	2
C₂×#17988499	2
C₂²×#14556498	2
C₂³×C₂²⋊C₄×C₇₃	2
C₂²×#393	73

Subgroups — C22×#17988499 #33252164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17988499 #33252164