Subgroups — #33240256

Order 18688.

770 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989022	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₅₈₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#22611983	C₂	1	4	yes	maximal, complemented
4	#22611973	C₂	1	2	yes	maximal, complemented
5	#22611963	C₂	1	2	yes	maximal, complemented
6	#22611687	C₂	1	1	yes	maximal, characteristic, complemented
7	#22611643	C₂	1	2	yes	maximal, complemented
8	C₇₃×#3147	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₅₈₄	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₂₉₂	—	2	1	no	—
11	#17989022	—	2	8	no	—
12	C₂×#14556534	C₂²	1	4	yes	—
13	C₂×#14556532	C₂²	1	4	yes	—
14	C₂×#14556531	C₂²	1	2	yes	—
15	C₄×D₅₈₄	—	2	1	no	—
16	C₄×D₅₈₄	—	2	1	no	—
17	C₄×D₅₈₄	C₂²	1	2	yes	—
18	C₄×D₅₈₄	C₂²	1	2	yes	—
19	C₂×#14556525	C₂²	1	1	yes	characteristic
20	C₂×#14556513	—	2	1	no	—
21	C₄⋊#11924003	C₂²	1	2	yes	complemented
22	C₄⋊#11924003	C₂²	1	2	yes	complemented
23	#17988391	C₂²	1	4	yes	complemented
24	#17988388	C₂²	1	4	yes	complemented
25	C₂×#135×C₇₃	—	2	1	no	—
26	C₂×#135×C₇₃	C₂²	1	2	yes	—
27	#532×C₇₃	—	2	1	no	—
28	#495×C₇₃	—	2	1	no	—
29	#495×C₇₃	C₂²	1	2	yes	complemented
30	#494×C₇₃	C₂²	1	1	yes	characteristic, complemented
31	#493×C₇₃	C₂²	1	1	yes	characteristic, complemented
32	#492×C₇₃	—	2	1	no	—
33	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
34	C₂²×D₂₉₂	—	2	1	no	—
35	C₂²×C₄×D₇₃	—	4	1	no	—
36	C₂×D₅₈₄	—	2	1	no	—
37	C₂×D₅₈₄	—	2	2	no	—
38	C₂×D₅₈₄	D₄	1	4	yes	complemented
39	C₂×D₅₈₄	C₂³	1	2	yes	—
40	C₄×D₂₉₂	—	2	2	no	—
41	C₄×D₂₉₂	—	2	2	no	—
42	C₄×D₂₉₂	—	4	1	no	—
43	C₄×D₂₉₂	—	4	1	no	—
44	C₂×C₄⋊Dic₇₃	—	2	1	no	—
45	C₂×#11924005	—	4	1	no	—
46	#14556534	—	2	8	no	—
47	#14556532	—	2	8	no	—
48	#14556531	—	2	4	no	—
49	C₂×#11924003	C₂³	1	4	yes	—
50	#14556525	—	2	1	no	—
51	#14556525	C₂³	1	2	yes	—
52	#14556513	—	2	2	no	—
53	#14556513	—	2	2	no	—
54	C₂×#41×C₇₃	—	2	2	no	—
55	#135×C₇₃	—	2	4	no	—
56	#135×C₇₃	—	2	4	no	—
57	#135×C₇₃	—	4	4	no	—
58	C₂×Q₁₆×C₇₃	—	2	1	no	—
59	C₂×Q₁₆×C₇₃	C₂³	1	2	yes	—
60	C₂×SD₁₆×C₇₃	—	2	2	no	—
61	C₂×SD₁₆×C₇₃	—	2	2	no	—
62	C₂×D₈×C₇₃	—	2	1	no	—
63	C₂×D₈×C₇₃	C₂³	1	2	yes	—
64	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
65	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
66	C₄⋊Q₈×C₇₃	—	2	1	no	—
67	#127×C₇₃	—	2	1	no	—
68	#124×C₇₃	—	2	2	no	—
69	#121×C₇₃	—	4	1	no	—
70	#121×C₇₃	—	4	1	no	—
71	C₄×D₄×C₇₃	—	4	1	no	—
72	C₄×D₄×C₇₃	—	4	1	no	—
73	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
74	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
75	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
76	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
78	C₂³×D₇₃	—	4	1	no	—
79	C₂×D₂₉₂	—	2	4	no	—
80	C₂×D₂₉₂	—	2	2	no	—
81	C₂×D₂₉₂	—	4	1	no	—
82	C₂×D₂₉₂	—	4	2	no	—
83	C₂²×Dic₇₃	—	4	1	no	—
84	C₂×C₄×D₇₃	—	4	2	no	—
85	C₂×C₄×D₇₃	—	4	2	no	—
86	C₂×C₄×D₇₃	—	8	4	no	—
87	D₅₈₄	—	2	4	no	—
88	D₅₈₄	—	2	2	no	—
89	D₅₈₄	C₂×D₄	1	4	yes	—
90	C₄⋊Dic₇₃	—	2	2	no	—
91	C₄⋊Dic₇₃	—	4	1	no	—
92	#11924005	—	4	2	no	—
93	#11924005	—	4	2	no	—
94	#11924003	—	2	4	no	—
95	#41×C₇₃	—	4	4	no	—
96	#41×C₇₃	—	4	4	no	—
97	C₂×Q₈×C₇₃	—	2	2	no	—
98	C₂×D₄×C₇₃	—	2	2	no	—
99	C₂×D₄×C₇₃	—	4	2	no	—
100	C₂×D₄×C₇₃	—	4	1	no	—
101	C₂×D₄×C₇₃	—	4	1	no	—
102	C₂²×C₄×C₇₃	—	4	2	no	—
103	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
104	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
105	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
106	Q₁₆×C₇₃	—	2	2	no	—
107	Q₁₆×C₇₃	—	2	2	no	—
108	Q₁₆×C₇₃	—	4	2	no	—
109	SD₁₆×C₇₃	—	4	4	no	—
110	SD₁₆×C₇₃	—	4	4	no	—
111	D₈×C₇₃	—	2	2	no	—
112	D₈×C₇₃	—	2	2	no	—
113	D₈×C₇₃	—	4	2	no	—
114	C₂×C₈×C₇₃	—	2	2	no	—
115	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
116	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₇₃	#41	1	2	yes	—
120	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
121	C₄⋊C₄×C₇₃	—	4	1	no	—
122	C₄⋊C₄×C₇₃	—	4	1	no	—
123	C₂²⋊C₄×C₇₃	—	4	2	no	—
124	C₂²⋊C₄×C₇₃	—	4	2	no	—
125	C₄²×C₇₃	D₈	1	1	yes	characteristic
126	C₄²×C₇₃	D₈	1	1	yes	characteristic
127	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
129	C₂³×C₇₃	—	4	2	no	—
130	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
131	C₂²×D₇₃	—	4	2	no	—
132	C₂²×D₇₃	—	4	4	no	—
133	C₂²×D₇₃	—	8	2	no	—
134	C₂²×D₇₃	—	8	2	no	—
135	Q₈×C₇₃	—	4	2	no	—
136	Q₈×C₇₃	—	4	2	no	—
137	D₄×C₇₃	—	4	2	no	—
138	D₄×C₇₃	—	4	2	no	—
139	D₄×C₇₃	—	8	2	no	—
140	D₄×C₇₃	—	8	4	no	—
141	D₄×C₇₃	—	8	2	no	—
142	C₂×C₄×C₇₃	—	2	2	no	—
143	C₂×C₄×C₇₃	—	4	1	no	—
144	C₂×C₄×C₇₃	—	4	2	no	—
145	C₂×C₄×C₇₃	—	8	4	no	—
146	C₂×C₄×C₇₃	#121	1	2	yes	—
147	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
153	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×Dic₇₃	—	4	2	no	—
155	C₂×Dic₇₃	—	8	2	no	—
156	D₂₉₂	—	2	4	no	—
157	D₂₉₂	—	4	2	no	—
158	D₂₉₂	—	4	4	no	—
159	C₄×D₇₃	—	8	4	no	—
160	C₄×D₇₃	—	8	4	no	—
161	C₈×C₇₃	—	2	1	no	—
162	C₈×C₇₃	—	2	1	no	—
163	C₈×C₇₃	#121	1	2	yes	—
164	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
165	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
166	C₂²×C₇₃	—	2	1	no	—
167	C₂²×C₇₃	—	2	1	no	—
168	C₂²×C₇₃	—	8	2	no	—
169	C₂²×C₇₃	—	8	2	no	—
170	C₂²×C₇₃	—	8	2	no	—
171	C₂²×C₇₃	#459	1	2	yes	—
172	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
173	C₂×D₇₃	—	4	4	no	—
174	C₂×D₇₃	—	8	8	no	—
175	C₂×D₇₃	—	8	2	no	—
176	C₂×D₇₃	—	8	2	no	—
177	C₄×C₇₃	—	2	1	no	—
178	C₄×C₇₃	—	2	1	no	—
179	C₄×C₇₃	—	2	1	no	—
180	C₄×C₇₃	—	8	2	no	—
181	C₄×C₇₃	#459	1	2	yes	—
182	C₄×C₇₃	#509	1	2	yes	complemented
183	C₄×C₇₃	C₂×#121	1	1	yes	characteristic
184	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
185	Dic₇₃	—	8	2	no	—
186	#33580	—	73	1	no	maximal, Sylow
187	C₂×C₇₃	—	2	2	no	—
188	C₂×C₇₃	—	16	2	no	—
189	C₂×C₇₃	C₂×#459	1	1	yes	characteristic
190	C₂×C₇₃	C₂×#509	1	1	yes	characteristic
191	C₂×C₇₃	#3414	1	1	yes	characteristic
192	D₇₃	—	8	4	no	—
193	C₂×#509	—	73	2	no	—
194	#3147	D₇₃	1	1	yes	characteristic, complemented
195	C₂×C₄×D₈	—	73	1	no	—
196	#2210	—	73	4	no	—
197	#2208	—	73	2	no	—
198	#2206	—	73	2	no	—
199	#2150	—	73	1	no	—
200	#2139	—	73	2	no	—
201	C₇₃	#33580	1	1	yes	characteristic, complemented, Sylow
202	C₂×#135	—	2	1	no	—
203	C₂×#135	C₂×D₇₃	1	2	yes	—
204	C₂²×D₈	—	73	1	no	—
205	#532	—	2	1	no	—
206	C₂×C₄×D₄	—	146	1	no	—
207	#509	—	146	8	no	—
208	C₂×SD₃₂	—	73	4	no	—
209	C₂×D₁₆	—	73	4	no	—
210	C₂×M₃₂	—	73	2	no	—
211	#495	—	2	1	no	—
212	#495	C₂×D₇₃	1	2	yes	complemented
213	#494	C₂×D₇₃	1	1	yes	characteristic, complemented
214	#493	C₂×D₇₃	1	1	yes	characteristic, complemented
215	#492	—	2	1	no	—
216	C₄×D₈	—	73	2	no	—
217	C₄×D₈	—	73	2	no	—
218	C₄×D₈	—	146	1	no	—
219	C₄×D₈	—	146	1	no	—
220	C₂×#107	—	73	1	no	—
221	C₂×#102	—	146	1	no	—
222	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
223	C₄⋊C₁₆	—	73	2	no	—
224	C₄⋊C₁₆	—	73	2	no	—
225	#358	—	73	4	no	—
226	#357	—	73	4	no	—
227	C₂×#41	—	2	2	no	—
228	C₂²×D₄	—	146	1	no	—
229	C₂³×C₄	—	292	1	no	—
230	#135	—	2	4	no	—
231	#135	—	2	4	no	—
232	#135	—	4	4	no	—
233	C₂×Q₁₆	—	2	1	no	—
234	C₂×Q₁₆	C₂²×D₇₃	1	2	yes	—
235	C₂×SD₁₆	—	2	2	no	—
236	C₂×SD₁₆	—	2	2	no	—
237	C₂×D₈	—	2	1	no	—
238	C₂×D₈	—	73	4	no	—
239	C₂×D₈	—	73	2	no	—
240	C₂×D₈	—	146	1	no	—
241	C₂×D₈	—	146	2	no	—
242	C₂×D₈	C₂²×D₇₃	1	2	yes	—
243	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
244	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
245	C₄⋊Q₈	—	2	1	no	—
246	#127	—	2	1	no	—
247	#124	—	2	2	no	—
248	#121	—	4	1	no	—
249	#121	—	4	1	no	—
250	C₄×D₄	—	4	1	no	—
251	C₄×D₄	—	4	1	no	—
252	C₄×D₄	—	292	1	no	—
253	C₄×D₄	—	292	1	no	—
254	C₄×D₄	—	146	2	no	—
255	C₄×D₄	—	146	2	no	—
256	C₂×C₄⋊C₄	—	146	1	no	—
257	C₂×C₂²⋊C₄	—	292	1	no	—
258	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
259	SD₃₂	—	146	8	no	—
260	D₁₆	—	146	8	no	—
261	M₃₂	—	146	4	no	—
262	C₂×C₁₆	—	73	4	no	—
263	#107	—	73	2	no	—
264	#107	—	146	1	no	—
265	#102	—	146	2	no	—
266	#102	—	146	2	no	—
267	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
268	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
271	C₂⁴	—	292	1	no	—
272	#41	—	4	4	no	—
273	#41	—	4	4	no	—
274	C₂×Q₈	—	2	2	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	4	2	no	—
277	C₂×D₄	—	4	1	no	—
278	C₂×D₄	—	4	1	no	—
279	C₂×D₄	—	292	1	no	—
280	C₂×D₄	—	292	2	no	—
281	C₂×D₄	—	146	4	no	—
282	C₂×D₄	—	146	2	no	—
283	C₂²×C₄	—	4	2	no	—
284	C₂²×C₄	—	292	1	no	—
285	C₂²×C₄	—	292	2	no	—
286	C₂²×C₄	—	292	2	no	—
287	C₂²×C₄	—	584	4	no	—
288	C₂²×C₄	#11924013	1	1	yes	characteristic
289	C₂²×C₄	#11924013	1	1	yes	characteristic
290	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
291	Q₁₆	—	2	2	no	—
292	Q₁₆	—	2	2	no	—
293	Q₁₆	—	4	2	no	—
294	SD₁₆	—	4	4	no	—
295	SD₁₆	—	4	4	no	—
296	D₈	—	2	2	no	—
297	D₈	—	2	2	no	—
298	D₈	—	4	2	no	—
299	D₈	—	73	4	no	—
300	D₈	—	146	4	no	—
301	D₈	—	146	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	D₄×D₇₃	1	2	yes	—
304	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
305	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
306	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
307	C₂×C₈	#11924027	1	2	yes	—
308	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
309	C₄⋊C₄	—	4	1	no	—
310	C₄⋊C₄	—	4	1	no	—
311	C₄⋊C₄	—	292	1	no	—
312	C₄⋊C₄	—	146	2	no	—
313	C₂²⋊C₄	—	4	2	no	—
314	C₂²⋊C₄	—	4	2	no	—
315	C₂²⋊C₄	—	292	2	no	—
316	C₂²⋊C₄	—	292	2	no	—
317	C₄²	#11924013	1	1	yes	characteristic
318	C₄²	#11924013	1	1	yes	characteristic
319	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
320	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
321	C₁₆	—	146	4	no	—
322	C₂³	—	4	2	no	—
323	C₂³	—	292	4	no	—
324	C₂³	—	292	2	no	—
325	C₂³	—	584	2	no	—
326	C₂³	—	584	2	no	—
327	C₂³	C₂×#11924013	1	1	yes	characteristic
328	Q₈	—	4	2	no	—
329	Q₈	—	4	2	no	—
330	D₄	—	4	2	no	—
331	D₄	—	4	2	no	—
332	D₄	—	8	2	no	—
333	D₄	—	8	4	no	—
334	D₄	—	8	2	no	—
335	D₄	—	292	4	no	—
336	D₄	—	292	2	no	—
337	D₄	—	146	4	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	8	4	no	—
342	C₂×C₄	—	292	2	no	—
343	C₂×C₄	—	584	4	no	—
344	C₂×C₄	—	584	2	no	—
345	C₂×C₄	—	584	4	no	—
346	C₂×C₄	#14556562	1	2	yes	—
347	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
348	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
349	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
350	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
351	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
352	C₂×C₄	#14556617	1	2	yes	—
353	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	#14556562	1	2	yes	—
357	C₈	C₂×D₄×D₇₃	1	1	yes	characteristic
358	C₈	C₂×#11924027	1	1	yes	characteristic
359	C₂²	—	2	1	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	8	2	no	—
362	C₂²	—	8	2	no	—
363	C₂²	—	8	2	no	—
364	C₂²	—	292	4	no	—
365	C₂²	—	584	8	no	—
366	C₂²	—	584	2	no	—
367	C₂²	—	584	2	no	—
368	C₂²	#17988779	1	2	yes	—
369	C₂²	C₂²×#11924013	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	8	2	no	—
374	C₄	—	584	2	no	—
375	C₄	#17988779	1	2	yes	—
376	C₄	#17989022	1	2	yes	complemented
377	C₄	C₂×#14556562	1	1	yes	characteristic
378	C₄	C₂×#14556617	1	1	yes	characteristic
379	C₂	—	2	2	no	—
380	C₂	—	16	2	no	—
381	C₂	—	584	4	no	—
382	C₂	C₂×#17988779	1	1	yes	characteristic
383	C₂	C₂×#17989022	1	1	yes	characteristic
384	C₂	#22623065	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11924013
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#3147	C₂
socle	C₂²×C₇₃	C₂²×D₈
hypercenter	#3147	D₇₃
hypocenter	C₇₃	#33580
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×#17989022	C₂	semidirect
2. C₂×C₄×D₅₈₄	C₂	semidirect
3. #22611983	C₂	semidirect
4. #22611973	C₂	semidirect
5. #22611963	C₂	semidirect
6. #22611687	C₂	semidirect
7. #22611643	C₂	semidirect
8. C₇₃×#3147	C₂	semidirect
21. C₄⋊#11924003	C₂²	semidirect
22. C₄⋊#11924003	C₂²	semidirect
23. #17988391	C₂²	semidirect
24. #17988388	C₂²	semidirect
29. #495×C₇₃	C₂²	semidirect
30. #494×C₇₃	C₂²	semidirect
31. #493×C₇₃	C₂²	semidirect
38. C₂×D₅₈₄	D₄	semidirect
182. C₄×C₇₃	#509	semidirect
194. #3147	D₇₃	semidirect
201. C₇₃	#33580	semidirect
212. #495	C₂×D₇₃	semidirect
213. #494	C₂×D₇₃	semidirect
214. #493	C₂×D₇₃	semidirect
376. C₄	#17989022	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989022	2
C₂×C₄×D₅₈₄	2
#22611983	2
#22611973	2
#22611963	2
#22611687	2
#22611643	2
C₇₃×#3147	2
#33580	73