Subgroups — #33242705

Order 18688.

674 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989019	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×Dic₅₈₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#22621674	C₂	1	2	yes	maximal, complemented
4	#22612192	C₂	1	1	yes	maximal, characteristic, complemented
5	#22612022	C₂	1	1	yes	maximal, characteristic, complemented
6	#22612017	C₂	1	1	yes	maximal, characteristic, complemented
7	#22611994	C₂	1	2	yes	maximal, complemented
8	#22611873	C₂	1	2	yes	maximal, complemented
9	#22611850	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₅₈₄	C₂	1	1	yes	maximal, characteristic, complemented
11	D₄×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556613	C₂²	1	2	yes	—
13	C₂²×Dic₂₉₂	C₂²	1	2	yes	—
14	#17989246	—	2	2	no	—
15	#17989019	—	2	4	no	—
16	C₂×Dic₅₈₄	—	2	4	no	—
17	C₂×Dic₅₈₄	C₂²	1	2	yes	—
18	C₂×Dic₅₈₄	C₂²	1	1	yes	characteristic
19	C₂×Dic₅₈₄	C₂²	1	1	yes	characteristic
20	C₂×#14556536	C₂²	1	2	yes	—
21	C₂×#14556536	C₂²	1	2	yes	—
22	C₂×D₁₁₆₈	C₂²	1	1	yes	characteristic
23	#17988971	C₂²	1	2	yes	—
24	#17988828	C₂²	1	2	yes	—
25	#17988823	C₂²	1	2	yes	—
26	#17988810	—	2	2	no	—
27	#17988751	—	2	2	no	—
28	#17988728	—	2	2	no	—
29	C₄×Dic₂₉₂	C₂²	1	2	yes	—
30	#17988413	C₂²	1	1	yes	characteristic
31	#17988392	C₂²	1	2	yes	—
32	#17988392	C₂²	1	2	yes	—
33	#17988389	C₂²	1	2	yes	—
34	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
36	C₈×D₄×C₇₃	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₇₃	C₂²	1	1	yes	characteristic
38	C₂²⋊C₁₆×C₇₃	C₂²	1	1	yes	characteristic
39	C₂²⋊C₁₆×C₇₃	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
41	C₂×#11924028	—	2	2	no	—
42	C₂²×Dic₁₄₆	—	2	2	no	—
43	#14556613	—	2	4	no	—
44	#14556613	—	4	2	no	—
45	C₂×Dic₂₉₂	—	2	4	no	—
46	C₂×Dic₂₉₂	—	2	2	no	—
47	C₂×Dic₂₉₂	—	4	2	no	—
48	C₂×Dic₂₉₂	C₂³	1	2	yes	—
49	C₂×Dic₂₉₂	C₂³	1	2	yes	—
50	C₂×#11924017	—	2	2	no	—
51	C₂×#11924017	—	2	2	no	—
52	C₂×D₅₈₄	C₂³	1	2	yes	—
53	#14556579	—	2	2	no	—
54	#14556577	—	4	2	no	—
55	#14556570	—	2	2	no	—
56	#14556569	—	4	2	no	—
57	#14556565	—	2	2	no	—
58	#14556563	—	4	2	no	—
59	C₄×Dic₁₄₆	—	2	2	no	—
60	#14556551	—	4	2	no	—
61	Dic₅₈₄	—	2	4	no	—
62	Dic₅₈₄	—	2	2	no	—
63	Dic₅₈₄	D₄	1	4	yes	complemented
64	#14556536	—	2	4	no	—
65	#14556536	—	4	2	no	—
66	D₁₁₆₈	—	2	2	no	—
67	#14556525	C₂³	1	2	yes	—
68	#14556516	—	2	2	no	—
69	#14556516	—	2	2	no	—
70	#14556513	—	2	2	no	—
71	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
73	C₄×D₄×C₇₃	D₄	1	1	yes	characteristic
74	C₂×C₁₆×C₇₃	—	2	2	no	—
75	C₂×C₁₆×C₇₃	—	2	2	no	—
76	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
77	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
78	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
79	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₇₃	D₄	1	1	yes	characteristic
81	C₂²⋊C₈×C₇₃	D₄	1	1	yes	characteristic
82	C₂²⋊C₈×C₇₃	D₄	1	1	yes	characteristic
83	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
84	#11924028	—	4	4	no	—
85	#11924028	—	8	2	no	—
86	C₂×Dic₁₄₆	—	2	2	no	—
87	C₂×Dic₁₄₆	—	2	2	no	—
88	C₂×Dic₁₄₆	—	4	4	no	—
89	C₂×Dic₁₄₆	—	4	2	no	—
90	C₂×Dic₁₄₆	—	8	2	no	—
91	C₂×D₂₉₂	—	2	2	no	—
92	C₂×C₂²⋊D₇₃	—	4	2	no	—
93	C₂×C₂²⋊D₇₃	—	4	2	no	—
94	C₂²×Dic₇₃	—	4	2	no	—
95	C₂×C₄×D₇₃	—	4	2	no	—
96	Dic₂₉₂	—	2	2	no	—
97	Dic₂₉₂	—	2	2	no	—
98	Dic₂₉₂	—	2	2	no	—
99	Dic₂₉₂	—	4	4	no	—
100	Dic₂₉₂	C₂×D₄	1	4	yes	—
101	#11924017	—	4	4	no	—
102	#11924017	—	8	2	no	—
103	D₅₈₄	—	2	2	no	—
104	D₅₈₄	—	2	2	no	—
105	C₄⋊Dic₇₃	—	2	2	no	—
106	#11924007	—	4	2	no	—
107	#11924007	—	4	2	no	—
108	C₂²⋊Dic₇₃	—	4	2	no	—
109	#11924005	—	4	2	no	—
110	#11924005	—	4	2	no	—
111	#11924005	—	4	2	no	—
112	C₄×Dic₇₃	—	4	2	no	—
113	C₂×D₄×C₇₃	D₈	1	1	yes	characteristic
114	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₇₃	—	2	1	no	—
117	C₂×C₈×C₇₃	—	2	1	no	—
118	C₂×C₈×C₇₃	—	2	1	no	—
119	C₂×C₈×C₇₃	—	2	1	no	—
120	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
124	C₄⋊C₄×C₇₃	D₈	1	1	yes	characteristic
125	C₂²⋊C₄×C₇₃	D₈	1	1	yes	characteristic
126	C₂²⋊C₄×C₇₃	D₈	1	1	yes	characteristic
127	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
128	C₁₆×C₇₃	—	2	1	no	—
129	C₁₆×C₇₃	—	2	1	no	—
130	C₁₆×C₇₃	—	2	1	no	—
131	C₁₆×C₇₃	C₂×D₄	1	2	yes	—
132	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
133	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
134	C₂²×D₇₃	—	4	2	no	—
135	D₄×C₇₃	—	2	2	no	—
136	C₂×C₄×C₇₃	—	2	1	no	—
137	C₂×C₄×C₇₃	—	2	1	no	—
138	C₂×C₄×C₇₃	—	2	1	no	—
139	C₂×C₄×C₇₃	—	2	1	no	—
140	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
145	C₂²⋊D₇₃	—	8	4	no	—
146	C₂²⋊D₇₃	—	8	4	no	—
147	C₂×Dic₇₃	—	4	2	no	—
148	C₂×Dic₇₃	—	4	2	no	—
149	C₂×Dic₇₃	—	4	2	no	—
150	C₂×Dic₇₃	—	8	4	no	—
151	D₂₉₂	—	4	2	no	—
152	D₂₉₂	—	4	2	no	—
153	C₄×D₇₃	—	8	4	no	—
154	Dic₁₄₆	—	2	4	no	—
155	Dic₁₄₆	—	4	2	no	—
156	Dic₁₄₆	—	4	2	no	—
157	Dic₁₄₆	—	4	2	no	—
158	Dic₁₄₆	—	8	4	no	—
159	C₈×C₇₃	—	2	1	no	—
160	C₈×C₇₃	—	2	1	no	—
161	C₈×C₇₃	—	2	1	no	—
162	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
163	C₈×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂²×C₇₃	—	2	1	no	—
165	C₂²×C₇₃	—	2	1	no	—
166	C₂²×C₇₃	—	2	1	no	—
167	C₂²×C₇₃	—	2	1	no	—
168	C₂²×C₇₃	—	2	1	no	—
169	C₂²×C₇₃	#508	1	2	yes	complemented
170	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
171	C₂×D₇₃	—	8	2	no	—
172	C₂×D₇₃	—	8	2	no	—
173	C₂×D₇₃	—	8	2	no	—
174	C₄×C₇₃	—	2	1	no	—
175	C₄×C₇₃	—	2	1	no	—
176	C₄×C₇₃	—	2	1	no	—
177	C₄×C₇₃	—	2	1	no	—
178	C₄×C₇₃	D₄²	1	1	yes	characteristic
179	C₄×C₇₃	#576	1	1	yes	characteristic
180	Dic₇₃	—	4	4	no	—
181	Dic₇₃	—	8	2	no	—
182	Dic₇₃	—	8	2	no	—
183	#33851	—	73	1	no	maximal, Sylow
184	C₂×C₇₃	—	2	2	no	—
185	C₂×C₇₃	—	4	1	no	—
186	C₂×C₇₃	D₄×D₈	1	1	yes	characteristic
187	C₂×C₇₃	C₂×#508	1	1	yes	characteristic
188	C₂×C₇₃	#3417	1	1	yes	characteristic
189	D₇₃	—	16	2	no	—
190	C₂×#508	—	73	1	no	—
191	C₂²×Q₃₂	—	73	1	no	—
192	#3285	—	73	2	no	—
193	#2249	—	73	1	no	—
194	#2216	—	73	1	no	—
195	#2215	—	73	1	no	—
196	#2211	—	73	2	no	—
197	#2189	—	73	2	no	—
198	#2185	—	73	2	no	—
199	C₄×Q₃₂	—	73	1	no	—
200	D₄×C₁₆	D₇₃	1	1	yes	characteristic, complemented
201	C₇₃	#33851	1	1	yes	characteristic, complemented, Sylow
202	C₂×#135	—	73	2	no	—
203	C₂²×Q₁₆	—	73	2	no	—
204	#548	—	146	2	no	—
205	#508	—	146	4	no	—
206	C₂×Q₃₂	—	73	1	no	—
207	C₂×Q₃₂	—	73	2	no	—
208	C₂×Q₃₂	—	73	1	no	—
209	C₂×Q₃₂	—	146	4	no	—
210	C₂×SD₃₂	—	73	2	no	—
211	C₂×SD₃₂	—	73	2	no	—
212	C₂×D₁₆	—	73	1	no	—
213	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
214	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
215	#495	—	73	2	no	—
216	#467	—	73	2	no	—
217	#466	—	73	2	no	—
218	#464	—	146	2	no	—
219	#452	—	146	2	no	—
220	#448	—	146	2	no	—
221	C₄×Q₁₆	—	73	2	no	—
222	C₈×D₄	C₂×D₇₃	1	1	yes	characteristic
223	#366	—	73	1	no	—
224	C₄⋊C₁₆	C₂×D₇₃	1	1	yes	characteristic
225	#358	—	73	2	no	—
226	#358	—	73	2	no	—
227	#357	—	73	2	no	—
228	C₂²⋊C₁₆	C₂×D₇₃	1	1	yes	characteristic
229	C₂²⋊C₁₆	C₂×D₇₃	1	1	yes	characteristic
230	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
231	C₂×#41	—	146	2	no	—
232	C₂²×Q₈	—	146	2	no	—
233	#135	—	292	2	no	—
234	#135	—	146	4	no	—
235	C₂×Q₁₆	—	292	2	no	—
236	C₂×Q₁₆	—	73	2	no	—
237	C₂×Q₁₆	—	73	2	no	—
238	C₂×Q₁₆	—	146	4	no	—
239	C₂×Q₁₆	—	146	2	no	—
240	C₂×SD₁₆	—	146	2	no	—
241	C₂×SD₁₆	—	146	2	no	—
242	C₂×D₈	—	73	2	no	—
243	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
244	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
245	#124	—	292	2	no	—
246	#124	—	146	2	no	—
247	C₂²⋊Q₈	—	292	2	no	—
248	C₂²⋊Q₈	—	146	2	no	—
249	#121	—	292	2	no	—
250	#121	—	146	2	no	—
251	C₄×Q₈	—	146	2	no	—
252	C₄×D₄	—	292	2	no	—
253	C₄×D₄	D₂₉₂	1	1	yes	characteristic
254	Q₃₂	—	73	4	no	—
255	Q₃₂	—	146	4	no	—
256	Q₃₂	—	146	2	no	—
257	SD₃₂	—	292	2	no	—
258	SD₃₂	—	146	4	no	—
259	D₁₆	—	146	2	no	—
260	C₂×C₁₆	—	2	2	no	—
261	C₂×C₁₆	—	2	2	no	—
262	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
263	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
264	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
265	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
266	#107	—	73	2	no	—
267	C₄⋊C₈	D₂₉₂	1	1	yes	characteristic
268	#103	—	146	2	no	—
269	#103	—	146	2	no	—
270	#102	—	146	2	no	—
271	C₂²⋊C₈	D₂₉₂	1	1	yes	characteristic
272	C₂²⋊C₈	D₂₉₂	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
274	#41	—	292	4	no	—
275	#41	—	584	2	no	—
276	C₂×Q₈	—	292	4	no	—
277	C₂×Q₈	—	292	2	no	—
278	C₂×Q₈	—	584	2	no	—
279	C₂×Q₈	—	146	2	no	—
280	C₂×Q₈	—	146	2	no	—
281	C₂×D₄	—	292	2	no	—
282	C₂×D₄	—	292	2	no	—
283	C₂×D₄	—	146	2	no	—
284	C₂×D₄	D₅₈₄	1	1	yes	characteristic
285	C₂²×C₄	—	292	2	no	—
286	C₂²×C₄	—	292	2	no	—
287	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
288	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
289	Q₁₆	—	292	4	no	—
290	Q₁₆	—	73	4	no	—
291	Q₁₆	—	146	2	no	—
292	Q₁₆	—	146	2	no	—
293	Q₁₆	—	146	2	no	—
294	SD₁₆	—	292	4	no	—
295	SD₁₆	—	584	2	no	—
296	D₈	—	146	2	no	—
297	D₈	—	146	2	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
303	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
304	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
305	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
306	C₄⋊C₄	—	292	2	no	—
307	C₄⋊C₄	—	292	2	no	—
308	C₄⋊C₄	—	146	2	no	—
309	C₄⋊C₄	D₅₈₄	1	1	yes	characteristic
310	C₂²⋊C₄	—	292	2	no	—
311	C₂²⋊C₄	—	292	2	no	—
312	C₂²⋊C₄	—	292	2	no	—
313	C₂²⋊C₄	—	292	2	no	—
314	C₂²⋊C₄	D₅₈₄	1	1	yes	characteristic
315	C₂²⋊C₄	D₅₈₄	1	1	yes	characteristic
316	C₄²	—	292	2	no	—
317	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
318	C₁₆	—	2	1	no	—
319	C₁₆	—	2	1	no	—
320	C₁₆	—	2	1	no	—
321	C₁₆	D₄×D₇₃	1	2	yes	—
322	C₂³	—	292	2	no	—
323	C₂³	C₂×D₅₈₄	1	1	yes	characteristic
324	C₂³	C₂×D₅₈₄	1	1	yes	characteristic
325	Q₈	—	292	2	no	—
326	Q₈	—	292	2	no	—
327	Q₈	—	292	2	no	—
328	Q₈	—	584	4	no	—
329	Q₈	—	146	4	no	—
330	D₄	—	2	2	no	—
331	D₄	—	292	2	no	—
332	D₄	—	292	2	no	—
333	D₄	—	584	4	no	—
334	D₄	—	584	4	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	292	2	no	—
340	C₂×C₄	—	292	2	no	—
341	C₂×C₄	—	292	2	no	—
342	C₂×C₄	—	584	4	no	—
343	C₂×C₄	—	584	4	no	—
344	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
345	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
346	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
347	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
348	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	C₂×D₄×D₇₃	1	1	yes	characteristic
353	C₈	#14556637	1	1	yes	characteristic
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	584	2	no	—
360	C₂²	—	584	2	no	—
361	C₂²	—	584	2	no	—
362	C₂²	#17989019	1	2	yes	complemented
363	C₂²	C₂²×D₅₈₄	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	292	4	no	—
369	C₄	—	584	2	no	—
370	C₄	—	584	2	no	—
371	C₄	D₄×D₂₉₂	1	1	yes	characteristic
372	C₄	#17989392	1	1	yes	characteristic
373	C₂	—	2	2	no	—
374	C₂	—	4	1	no	—
375	C₂	—	1168	2	no	—
376	C₂	D₄×D₅₈₄	1	1	yes	characteristic
377	C₂	C₂×#17989019	1	1	yes	characteristic
378	C₂	#22623084	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₈₄
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	D₄×C₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₈
hypercenter	D₄×C₁₆	D₇₃
hypocenter	C₇₃	#33851
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#17989019	C₂	semidirect
2. C₂²×Dic₅₈₄	C₂	semidirect
3. #22621674	C₂	semidirect
4. #22612192	C₂	semidirect
5. #22612022	C₂	semidirect
6. #22612017	C₂	semidirect
7. #22611994	C₂	semidirect
8. #22611873	C₂	semidirect
9. #22611850	C₂	semidirect
10. C₄×Dic₅₈₄	C₂	semidirect
11. D₄×C₁₆×C₇₃	C₂	semidirect
63. Dic₅₈₄	D₄	semidirect
169. C₂²×C₇₃	#508	semidirect
200. D₄×C₁₆	D₇₃	semidirect
201. C₇₃	#33851	semidirect
362. C₂²	#17989019	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989019	2
C₂²×Dic₅₈₄	2
#22621674	2
#22612192	2
#22612022	2
#22612017	2
#22611994	2
#22611873	2
#22611850	2
C₄×Dic₅₈₄	2
D₄×C₁₆×C₇₃	2
#33851	73