Subgroups — #22618477

Order 9344.

738 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17989423	C₂	1	1	yes	maximal, characteristic
2	#17989353	C₂	1	1	yes	maximal, characteristic, complemented
3	#17989055	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988605	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988602	C₂	1	4	yes	maximal, complemented
6	C₂×#14556518	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14556517	C₂	1	1	yes	maximal, characteristic, complemented
8	#104×D₇₃	C₂	1	4	yes	maximal, complemented
9	#421×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#14556639	C₄	1	2	yes	complemented
11	#14556638	—	2	1	no	—
12	#14556638	C₄	1	2	yes	complemented
13	#14556637	—	2	1	no	—
14	#14556637	C₄	1	2	yes	complemented
15	#14556636	—	2	2	no	—
16	#14556636	C₄	1	2	yes	complemented
17	C₂×#11924028	—	2	2	no	—
18	C₂×#11924028	C₂²	1	1	yes	characteristic
19	C₂×#11924028	C₂²	1	1	yes	characteristic
20	#41×D₇₃	—	2	2	no	—
21	#41×D₇₃	C₂²	1	4	yes	—
22	#14556599	—	2	2	no	—
23	#14556597	—	2	2	no	—
24	C₂×#11924012	C₂²	1	1	yes	characteristic, complemented
25	M₁₆×D₇₃	C₂²	1	4	yes	complemented
26	#14556556	—	2	1	no	—
27	#14556553	—	2	1	no	—
28	#14556551	—	2	2	no	—
29	#14556548	C₂²	1	2	yes	complemented
30	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic, complemented
31	C₄²×D₇₃	C₂²	1	2	yes	complemented
32	#14556519	C₂²	1	4	yes	complemented
33	#14556518	C₂²	1	2	yes	complemented
34	#14556518	C₂²	1	2	yes	complemented
35	#14556517	C₂²	1	2	yes	complemented
36	#14556517	C₂²	1	2	yes	complemented
37	C₂×#41×C₇₃	C₂²	1	1	yes	characteristic
38	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic, complemented
39	#117×C₇₃	C₂²	1	1	yes	characteristic, complemented
40	#104×C₇₃	C₂²	1	4	yes	complemented
41	#11924029	—	2	2	no	—
42	#11924029	C₂×C₄	1	4	yes	complemented
43	#11924028	—	2	1	no	—
44	#11924028	—	2	1	no	—
45	#11924028	—	2	4	no	—
46	#11924028	—	2	2	no	—
47	#11924028	—	2	4	no	—
48	#11924028	—	2	2	no	—
49	#11924028	C₂×C₄	1	2	yes	—
50	#11924028	C₂×C₄	1	2	yes	—
51	#11924028	C₂×C₄	1	2	yes	complemented
52	#11924028	D₄	1	2	yes	—
53	#11924028	D₄	1	2	yes	—
54	#11924028	C₂³	1	2	yes	—
55	#11924027	—	2	4	no	—
56	#11924027	—	2	4	no	—
57	#11924027	—	2	2	no	—
58	#11924027	C₂×C₄	1	4	yes	complemented
59	C₂×Dic₁₄₆	—	2	2	no	—
60	C₂×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic, complemented
61	C₂×Dic₁₄₆	D₄	1	1	yes	characteristic
62	Q₈×D₇₃	—	2	2	no	—
63	Q₈×D₇₃	C₂×C₄	1	4	yes	complemented
64	C₂×D₂₉₂	—	2	2	no	—
65	C₂×D₂₉₂	C₂×C₄	1	1	yes	characteristic, complemented
66	C₂×D₂₉₂	D₄	1	1	yes	characteristic
67	C₂×C₂²⋊D₇₃	—	2	1	no	—
68	C₂×C₂²⋊D₇₃	—	2	4	no	—
69	C₂×C₂²⋊D₇₃	D₄	1	2	yes	—
70	D₄×D₇₃	—	2	4	no	—
71	D₄×D₇₃	—	2	4	no	—
72	D₄×D₇₃	—	2	2	no	—
73	D₄×D₇₃	C₂×C₄	1	4	yes	complemented
74	C₂²×Dic₇₃	—	2	1	no	—
75	C₂×C₄×D₇₃	—	2	1	no	—
76	C₂×C₄×D₇₃	—	2	4	no	—
77	C₂×C₄×D₇₃	—	2	2	no	—
78	C₂×C₄×D₇₃	C₂³	1	2	yes	—
79	#11924012	—	2	1	no	—
80	#11924012	C₂³	1	2	yes	—
81	#11924011	—	2	4	no	—
82	C₂×#10596831	—	2	1	no	—
83	C₈×D₇₃	—	2	4	no	—
84	#11924007	—	2	2	no	—
85	#11924005	—	2	2	no	—
86	C₄×Dic₇₃	—	2	1	no	—
87	C₄×Dic₇₃	—	2	2	no	—
88	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
89	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
90	#41×C₇₃	—	2	1	no	—
91	#41×C₇₃	—	2	1	no	—
92	#41×C₇₃	C₂×C₄	1	2	yes	complemented
93	#41×C₇₃	C₂³	1	2	yes	—
94	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic, complemented
95	C₂×D₄×C₇₃	—	2	1	no	—
96	C₂×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic, complemented
97	C₂²×C₄×C₇₃	—	2	1	no	—
98	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
99	M₁₆×C₇₃	—	2	1	no	—
100	M₁₆×C₇₃	C₂³	1	2	yes	—
101	C₂×C₈×C₇₃	—	2	1	no	—
102	C₄⋊C₄×C₇₃	—	2	1	no	—
103	C₂²⋊C₄×C₇₃	—	2	1	no	—
104	C₄²×C₇₃	C₂³	1	2	yes	—
105	C₂³×C₇₃	—	2	1	no	—
106	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₇₃	—	2	1	no	—
108	C₂²×D₇₃	—	2	4	no	—
109	C₂²×D₇₃	C₂×D₄	1	2	yes	—
110	Q₈×C₇₃	—	2	1	no	—
111	Q₈×C₇₃	C₂²×C₄	1	2	yes	—
112	D₄×C₇₃	—	2	2	no	—
113	D₄×C₇₃	—	2	2	no	—
114	D₄×C₇₃	—	2	1	no	—
115	D₄×C₇₃	C₂²×C₄	1	2	yes	—
116	C₂×C₄×C₇₃	—	2	1	no	—
117	C₂×C₄×C₇₃	—	2	2	no	—
118	C₂×C₄×C₇₃	—	2	2	no	—
119	C₂×C₄×C₇₃	—	2	2	no	—
120	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
126	C₂²⋊D₇₃	—	2	2	no	—
127	C₂²⋊D₇₃	—	2	2	no	—
128	C₂²⋊D₇₃	—	2	4	no	—
129	C₂²⋊D₇₃	—	2	4	no	—
130	C₂²⋊D₇₃	C₂²⋊C₄	1	4	yes	—
131	C₂²⋊D₇₃	C₂²⋊C₄	1	4	yes	—
132	C₂×Dic₇₃	—	2	1	no	—
133	C₂×Dic₇₃	—	2	1	no	—
134	C₂×Dic₇₃	—	2	4	no	—
135	C₂×Dic₇₃	—	2	1	no	—
136	C₂×Dic₇₃	—	4	2	no	—
137	C₂×Dic₇₃	C₂×D₄	1	2	yes	—
138	D₂₉₂	—	2	4	no	—
139	D₂₉₂	—	2	1	no	—
140	D₂₉₂	C₂²⋊C₄	1	2	yes	—
141	D₂₉₂	C₂²⋊C₄	1	2	yes	—
142	D₂₉₂	C₂²×C₄	1	2	yes	—
143	C₄×D₇₃	—	2	4	no	—
144	C₄×D₇₃	—	2	2	no	—
145	C₄×D₇₃	—	2	2	no	—
146	C₄×D₇₃	—	2	4	no	—
147	C₄×D₇₃	—	4	4	no	—
148	C₄×D₇₃	C₂²×C₄	1	4	yes	—
149	C₄×D₇₃	C₂×D₄	1	4	yes	—
150	Dic₁₄₆	—	2	1	no	—
151	Dic₁₄₆	—	2	4	no	—
152	Dic₁₄₆	C₂²⋊C₄	1	2	yes	—
153	Dic₁₄₆	C₂²⋊C₄	1	2	yes	—
154	Dic₁₄₆	C₂²×C₄	1	2	yes	—
155	C₈×C₇₃	—	2	2	no	—
156	#10596831	—	2	2	no	—
157	C₂²×C₇₃	—	2	2	no	—
158	C₂²×C₇₃	—	4	1	no	—
159	C₂²×C₇₃	—	8	1	no	—
160	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
163	C₂×D₇₃	—	2	2	no	—
164	C₂×D₇₃	—	4	2	no	—
165	C₂×D₇₃	—	8	4	no	—
166	C₂×D₇₃	—	8	1	no	—
167	C₂×D₇₃	C₂×C₂²⋊C₄	1	4	yes	—
168	C₄×C₇₃	—	2	2	no	—
169	C₄×C₇₃	—	4	2	no	—
170	C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
171	C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
172	C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
173	C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
174	Dic₇₃	—	2	2	no	—
175	Dic₇₃	—	2	2	no	—
176	Dic₇₃	—	4	1	no	—
177	Dic₇₃	C₂×C₂²⋊C₄	1	4	yes	—
178	C₂×C₇₃	—	2	1	no	—
179	C₂×C₇₃	—	2	1	no	—
180	C₂×C₇₃	—	2	1	no	—
181	C₂×C₇₃	—	4	2	no	—
182	C₂×C₇₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
183	D₇₃	—	2	4	no	—
184	D₇₃	—	4	2	no	—
185	#2901	—	73	1	no	maximal, Sylow
186	C₇₃	#2901	1	1	yes	characteristic, complemented, Sylow
187	#585	—	73	1	no	—
188	#568	—	73	1	no	—
189	C₄×#41	—	73	1	no	—
190	#421	—	73	1	no	—
191	#421	—	73	4	no	—
192	#421	D₇₃	1	1	yes	characteristic, complemented
193	C₂×#104	—	73	1	no	—
194	C₂×#104	—	73	1	no	—
195	C₂×#104	—	73	4	no	—
196	2₋¹⁺⁴	—	73	2	no	—
197	2₋¹⁺⁴	—	73	2	no	—
198	2₋¹⁺⁴	—	146	1	no	—
199	H(5,𝔽₂)	—	73	2	no	—
200	H(5,𝔽₂)	—	73	2	no	—
201	H(5,𝔽₂)	—	146	2	no	—
202	H(5,𝔽₂)	—	146	1	no	—
203	C₂×#41	—	73	1	no	—
204	C₂×#41	—	73	4	no	—
205	C₂×#41	—	73	1	no	—
206	C₂×#41	—	146	2	no	—
207	C₂×#41	—	146	2	no	—
208	C₂×#41	C₂×D₇₃	1	1	yes	characteristic
209	#131	—	146	2	no	—
210	#131	—	146	2	no	—
211	C₂×M₁₆	—	73	1	no	—
212	C₂×M₁₆	—	73	4	no	—
213	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic, complemented
214	C₄×Q₈	—	146	1	no	—
215	C₄×D₄	—	146	2	no	—
216	C₄×D₄	—	146	1	no	—
217	#117	—	73	2	no	—
218	#117	C₂×D₇₃	1	1	yes	characteristic, complemented
219	C₂×C₄²	—	73	1	no	—
220	C₂×C₄²	—	73	2	no	—
221	#104	—	73	2	no	—
222	#104	—	73	2	no	—
223	#104	—	73	2	no	—
224	#104	—	73	2	no	—
225	#104	—	73	4	no	—
226	#104	C₂×D₇₃	1	4	yes	complemented
227	#41	—	2	1	no	—
228	#41	—	2	1	no	—
229	#41	—	73	2	no	—
230	#41	—	73	2	no	—
231	#41	—	73	2	no	—
232	#41	—	73	2	no	—
233	#41	—	73	4	no	—
234	#41	—	73	4	no	—
235	#41	—	73	2	no	—
236	#41	—	73	2	no	—
237	#41	—	146	4	no	—
238	#41	—	146	4	no	—
239	#41	—	146	2	no	—
240	#41	—	146	2	no	—
241	#41	—	146	4	no	—
242	#41	—	146	4	no	—
243	#41	—	146	2	no	—
244	#41	—	146	2	no	—
245	#41	—	146	1	no	—
246	#41	—	146	1	no	—
247	#41	C₄×D₇₃	1	2	yes	complemented
248	#41	C₂²×D₇₃	1	2	yes	—
249	C₂×Q₈	—	73	1	no	—
250	C₂×Q₈	—	73	4	no	—
251	C₂×Q₈	—	73	1	no	—
252	C₂×Q₈	—	146	2	no	—
253	C₂×Q₈	—	146	2	no	—
254	C₂×Q₈	C₄×D₇₃	1	1	yes	characteristic, complemented
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	73	2	no	—
257	C₂×D₄	—	73	4	no	—
258	C₂×D₄	—	73	1	no	—
259	C₂×D₄	—	73	1	no	—
260	C₂×D₄	—	146	2	no	—
261	C₂×D₄	—	146	4	no	—
262	C₂×D₄	—	146	2	no	—
263	C₂×D₄	—	146	4	no	—
264	C₂×D₄	—	146	4	no	—
265	C₂×D₄	—	146	1	no	—
266	C₂×D₄	C₄×D₇₃	1	1	yes	characteristic, complemented
267	C₂²×C₄	—	2	1	no	—
268	C₂²×C₄	—	73	2	no	—
269	C₂²×C₄	—	146	2	no	—
270	C₂²×C₄	—	146	4	no	—
271	C₂²×C₄	—	146	1	no	—
272	C₂²×C₄	—	146	1	no	—
273	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
274	M₁₆	—	2	1	no	—
275	M₁₆	—	73	2	no	—
276	M₁₆	—	146	4	no	—
277	M₁₆	—	146	1	no	—
278	M₁₆	C₂²×D₇₃	1	2	yes	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	146	4	no	—
281	C₂×C₈	—	146	1	no	—
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	146	2	no	—
284	C₂²⋊C₄	—	2	1	no	—
285	C₂²⋊C₄	—	146	2	no	—
286	C₄²	—	73	1	no	—
287	C₄²	—	73	1	no	—
288	C₄²	—	146	2	no	—
289	C₄²	—	146	1	no	—
290	C₄²	C₂²×D₇₃	1	2	yes	—
291	C₂³	—	2	1	no	—
292	C₂³	—	73	2	no	—
293	C₂³	—	146	4	no	—
294	C₂³	—	146	1	no	—
295	C₂³	D₄×D₇₃	1	1	yes	characteristic
296	Q₈	—	2	1	no	—
297	Q₈	—	73	2	no	—
298	Q₈	—	73	2	no	—
299	Q₈	—	73	2	no	—
300	Q₈	—	146	4	no	—
301	Q₈	—	146	1	no	—
302	Q₈	C₂×C₄×D₇₃	1	2	yes	—
303	D₄	—	2	2	no	—
304	D₄	—	2	2	no	—
305	D₄	—	2	1	no	—
306	D₄	—	73	4	no	—
307	D₄	—	73	4	no	—
308	D₄	—	73	2	no	—
309	D₄	—	73	2	no	—
310	D₄	—	73	2	no	—
311	D₄	—	146	4	no	—
312	D₄	—	146	4	no	—
313	D₄	—	146	4	no	—
314	D₄	—	146	2	no	—
315	D₄	—	146	2	no	—
316	D₄	—	146	1	no	—
317	D₄	C₂×C₄×D₇₃	1	2	yes	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	292	4	no	—
323	C₂×C₄	—	292	2	no	—
324	C₂×C₄	—	73	2	no	—
325	C₂×C₄	—	73	4	no	—
326	C₂×C₄	—	73	4	no	—
327	C₂×C₄	—	146	4	no	—
328	C₂×C₄	—	146	4	no	—
329	C₂×C₄	—	146	1	no	—
330	C₂×C₄	—	146	4	no	—
331	C₂×C₄	—	146	1	no	—
332	C₂×C₄	—	146	1	no	—
333	C₂×C₄	—	146	2	no	—
334	C₂×C₄	—	146	2	no	—
335	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
336	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
337	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
338	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
339	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
340	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
341	C₈	—	2	2	no	—
342	C₈	—	146	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	4	1	no	—
345	C₂²	—	8	1	no	—
346	C₂²	—	292	2	no	—
347	C₂²	—	584	1	no	—
348	C₂²	—	584	4	no	—
349	C₂²	—	73	4	no	—
350	C₂²	—	146	2	no	—
351	C₂²	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
352	C₂²	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
353	C₂²	C₂×D₄×D₇₃	1	1	yes	characteristic
354	C₄	—	2	2	no	—
355	C₄	—	4	2	no	—
356	C₄	—	292	1	no	—
357	C₄	—	73	4	no	—
358	C₄	—	146	2	no	—
359	C₄	—	146	2	no	—
360	C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
361	C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
362	C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
363	C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
364	C₂	—	2	1	no	—
365	C₂	—	2	1	no	—
366	C₂	—	2	1	no	—
367	C₂	—	4	2	no	—
368	C₂	—	292	2	no	—
369	C₂	—	146	4	no	—
370	C₂	C₂×C₂²⋊C₄×D₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₇₃
commutator	C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#421×C₇₃	C₂
socle	C₂×C₇₃	C₂²×C₂²⋊C₄
hypercenter	#421	D₇₃
hypocenter	C₇₃	#2901
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #421

Complements

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

N	H ≅ G/N	type
2. #17989353	C₂	semidirect
3. #17989055	C₂	semidirect
4. #17988605	C₂	semidirect
5. #17988602	C₂	semidirect
6. C₂×#14556518	C₂	semidirect
7. C₂×#14556517	C₂	semidirect
8. #104×D₇₃	C₂	semidirect
9. #421×C₇₃	C₂	semidirect
10. #14556639	C₄	semidirect
12. #14556638	C₄	semidirect
14. #14556637	C₄	semidirect
16. #14556636	C₄	semidirect
24. C₂×#11924012	C₂²	semidirect
25. M₁₆×D₇₃	C₂²	semidirect
29. #14556548	C₂²	semidirect
30. C₂×C₄×Dic₇₃	C₂²	semidirect
31. C₄²×D₇₃	C₂²	semidirect
32. #14556519	C₂²	semidirect
33. #14556518	C₂²	semidirect
34. #14556518	C₂²	semidirect
35. #14556517	C₂²	semidirect
36. #14556517	C₂²	semidirect
38. C₂×M₁₆×C₇₃	C₂²	semidirect
39. #117×C₇₃	C₂²	semidirect
40. #104×C₇₃	C₂²	semidirect
42. #11924029	C₂×C₄	semidirect
51. #11924028	C₂×C₄	semidirect
58. #11924027	C₂×C₄	semidirect
60. C₂×Dic₁₄₆	C₂×C₄	semidirect
63. Q₈×D₇₃	C₂×C₄	semidirect
65. C₂×D₂₉₂	C₂×C₄	semidirect
73. D₄×D₇₃	C₂×C₄	semidirect
92. #41×C₇₃	C₂×C₄	semidirect
94. C₂×Q₈×C₇₃	C₂×C₄	semidirect
96. C₂×D₄×C₇₃	C₂×C₄	semidirect
186. C₇₃	#2901	semidirect
192. #421	D₇₃	semidirect
213. C₂×M₁₆	C₂×D₇₃	semidirect
218. #117	C₂×D₇₃	semidirect
226. #104	C₂×D₇₃	semidirect
247. #41	C₄×D₇₃	semidirect
254. C₂×Q₈	C₄×D₇₃	semidirect
266. C₂×D₄	C₄×D₇₃	semidirect