Subgroups — #33104300

Order 18688.

914 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988693	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₉₂	C₂	1	1	yes	maximal, characteristic
3	#22612843	C₂	1	2	yes	maximal
4	#22610667	C₂	1	2	yes	maximal
5	#22610196	C₂	1	4	yes	maximal
6	#22609933	C₂	1	2	yes	maximal
7	#22609798	C₂	1	1	yes	maximal, characteristic
8	C₇₃×#1844	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₉₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₄₆	C₂²	1	2	yes	—
11	C₂×#14556556	C₂²	1	4	yes	—
12	#17988693	—	2	8	no	—
13	C₄×Dic₂₉₂	C₄	1	8	yes	complemented
14	C₂×#14556525	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
16	C₂×#14556516	C₂²	1	2	yes	—
17	C₂×#14556514	C₂²	1	4	yes	—
18	C₂×#14556501	C₂²	1	2	yes	—
19	#17988483	C₂²	1	4	yes	—
20	#17988482	—	2	4	no	—
21	#17988456	—	2	2	no	—
22	C₄⋊C₄×Dic₇₃	—	2	2	no	—
23	C₄×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
24	#17988349	C₂²	1	4	yes	—
25	#17988341	C₂²	1	2	yes	—
26	C₂×#106×C₇₃	C₂²	1	2	yes	—
27	C₂×#106×C₇₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
29	#384×C₇₃	C₂²	1	2	yes	—
30	C₂²×Dic₁₄₆	C₂³	1	2	yes	—
31	C₂×Dic₂₉₂	C₂×C₄	1	4	yes	—
32	C₂×Dic₂₉₂	D₄	1	4	yes	—
33	C₂×Dic₂₉₂	Q₈	1	4	yes	—
34	C₂²×#10596831	C₂³	1	2	yes	—
35	C₄×Dic₁₄₆	—	2	4	no	—
36	C₄×Dic₁₄₆	C₂×C₄	1	8	yes	—
37	#14556556	—	2	8	no	—
38	#14556556	—	4	4	no	—
39	C₂×C₄⋊Dic₇₃	D₄	1	2	yes	—
40	C₂×C₄⋊Dic₇₃	C₂³	1	2	yes	—
41	C₂×#11924007	—	2	4	no	—
42	C₂×#11924007	—	2	4	no	—
43	C₂×#11924007	—	2	2	no	—
44	C₂×C₄×Dic₇₃	—	2	4	no	—
45	C₂×C₄×Dic₇₃	—	2	2	no	—
46	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
47	#14556525	C₂×C₄	1	4	yes	—
48	C₄⋊#10596831	—	2	2	no	—
49	#14556516	C₂×C₄	1	8	yes	—
50	#14556514	—	2	8	no	—
51	#14556501	—	2	4	no	—
52	#14556498	—	2	4	no	—
53	#14556498	D₄	1	2	yes	—
54	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
57	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
58	C₂×C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
59	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
60	#106×C₇₃	—	2	4	no	—
61	#106×C₇₃	—	4	2	no	—
62	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
63	C₂×Dic₁₄₆	—	2	2	no	—
64	C₂×Dic₁₄₆	—	2	2	no	—
65	C₂×Dic₁₄₆	—	2	2	no	—
66	C₂×Dic₁₄₆	C₂²×C₄	1	4	yes	—
67	C₂×Dic₁₄₆	C₂×D₄	1	4	yes	—
68	C₂×Dic₁₄₆	C₂×Q₈	1	4	yes	—
69	C₂²×Dic₇₃	—	2	2	no	—
70	C₂²×Dic₇₃	—	2	2	no	—
71	C₂²×Dic₇₃	C₂×D₄	1	4	yes	—
72	Dic₂₉₂	C₄⋊C₄	1	16	yes	complemented
73	C₂×#10596831	—	2	2	no	—
74	C₂×#10596831	—	2	2	no	—
75	C₂×#10596831	—	2	2	no	—
76	C₄⋊Dic₇₃	—	2	4	no	—
77	C₄⋊Dic₇₃	—	2	2	no	—
78	C₄⋊Dic₇₃	C₂²×C₄	1	4	yes	—
79	#11924007	—	2	8	no	—
80	#11924007	—	4	8	no	—
81	#11924007	—	8	4	no	—
82	C₄×Dic₇₃	—	2	8	no	—
83	C₄×Dic₇₃	—	2	2	no	—
84	C₄×Dic₇₃	—	2	2	no	—
85	C₄×Dic₇₃	—	4	8	no	—
86	C₂²×C₄×C₇₃	—	2	4	no	—
87	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
90	C₂×C₈×C₇₃	—	2	1	no	—
91	C₂×C₈×C₇₃	—	2	1	no	—
92	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
96	C₄⋊C₄×C₇₃	—	2	4	no	—
97	C₄⋊C₄×C₇₃	—	2	4	no	—
98	C₄⋊C₄×C₇₃	—	4	4	no	—
99	C₄⋊C₄×C₇₃	—	8	2	no	—
100	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
101	C₄²×C₇₃	#41	1	2	yes	—
102	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₇₃	—	2	2	no	—
104	C₂×C₄×C₇₃	—	4	4	no	—
105	C₂×C₄×C₇₃	—	4	4	no	—
106	C₂×C₄×C₇₃	—	4	4	no	—
107	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₇₃	#137	1	4	yes	—
110	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂×C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
116	C₂×Dic₇₃	—	2	4	no	—
117	C₂×Dic₇₃	—	2	4	no	—
118	C₂×Dic₇₃	—	2	4	no	—
119	C₂×Dic₇₃	—	2	4	no	—
120	C₂×Dic₇₃	—	2	4	no	—
121	C₂×Dic₇₃	—	2	4	no	—
122	C₂×Dic₇₃	—	2	4	no	—
123	C₂×Dic₇₃	—	4	2	no	—
124	C₂×Dic₇₃	—	4	2	no	—
125	Dic₁₄₆	—	2	4	no	—
126	Dic₁₄₆	—	2	4	no	—
127	Dic₁₄₆	—	2	4	no	—
128	Dic₁₄₆	C₂×C₄⋊C₄	1	8	yes	—
129	C₈×C₇₃	—	2	2	no	—
130	C₈×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
131	#10596831	—	4	2	no	—
132	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
133	C₂²×C₇₃	D₄²	1	1	yes	characteristic
134	C₂²×C₇₃	D₄×Q₈	1	1	yes	characteristic
135	C₂²×C₇₃	C₂×#137	1	2	yes	—
136	C₂²×C₇₃	#577	1	2	yes	—
137	C₄×C₇₃	—	2	2	no	—
138	C₄×C₇₃	—	8	4	no	—
139	C₄×C₇₃	#441	1	4	yes	complemented
140	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
141	C₄×C₇₃	#520	1	1	yes	characteristic
142	C₄×C₇₃	#547	1	1	yes	characteristic
143	C₄×C₇₃	#554	1	1	yes	characteristic
144	Dic₇₃	—	2	8	no	—
145	Dic₇₃	—	4	4	no	—
146	Dic₇₃	—	4	4	no	—
147	#22674	—	73	1	no	maximal, Sylow
148	C₂×C₇₃	D₄×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₇₃	C₂×#441	1	1	yes	characteristic
150	C₂×C₇₃	#2952	1	1	yes	characteristic
151	C₂×C₇₃	#3278	1	2	yes	—
152	C₂×C₇₃	#3387	1	2	yes	—
153	C₂×#441	—	73	2	no	—
154	C₂×C₄×Q₁₆	—	73	1	no	—
155	Q₈×C₄⋊C₄	—	73	2	no	—
156	#1943	—	73	2	no	—
157	#1865	—	73	4	no	—
158	#1844	D₇₃	1	1	yes	characteristic
159	#1803	—	73	2	no	—
160	#1770	—	73	1	no	—
161	C₇₃	#22674	1	1	yes	characteristic, complemented, Sylow
162	C₂²×Q₁₆	—	73	1	no	—
163	C₂×C₄×Q₈	—	73	4	no	—
164	C₂×C₄×Q₈	—	73	2	no	—
165	#441	—	146	8	no	—
166	C₄×Q₁₆	—	73	8	no	—
167	C₂×#107	—	73	1	no	—
168	C₂×#106	C₂×D₇₃	1	2	yes	—
169	C₂×#106	C₂×D₇₃	1	2	yes	—
170	C₂×C₄⋊C₈	—	73	1	no	—
171	C₂×#103	—	73	2	no	—
172	C₂×#103	—	73	4	no	—
173	C₂×#97	—	73	2	no	—
174	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
175	#389	—	73	4	no	—
176	#389	—	146	4	no	—
177	#384	—	146	2	no	—
178	#384	C₂×D₇₃	1	2	yes	—
179	C₄×C₄⋊C₄	—	73	1	no	—
180	C₄×C₄⋊C₄	—	146	2	no	—
181	#340	—	73	4	no	—
182	#336	—	73	2	no	—
183	C₂²×Q₈	—	73	2	no	—
184	C₂×Q₁₆	—	73	4	no	—
185	C₂×Q₁₆	—	73	4	no	—
186	C₂×Q₁₆	—	73	4	no	—
187	C₂²×C₈	—	73	2	no	—
188	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
189	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
190	C₄×Q₈	—	292	4	no	—
191	C₄×Q₈	—	73	8	no	—
192	C₄×Q₈	—	146	8	no	—
193	C₄×Q₈	—	146	4	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	73	2	no	—
197	C₂×C₄⋊C₄	—	73	2	no	—
198	C₂×C₄⋊C₄	—	146	4	no	—
199	C₂×C₄⋊C₄	—	146	4	no	—
200	C₂×C₄⋊C₄	—	146	2	no	—
201	C₂×C₄⋊C₄	C₂²×D₇₃	1	4	yes	—
202	C₂×C₄²	—	73	2	no	—
203	C₂×C₄²	—	146	4	no	—
204	C₂×C₄²	—	146	2	no	—
205	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
206	#107	—	73	4	no	—
207	#106	—	2	4	no	—
208	#106	—	4	2	no	—
209	C₄⋊C₈	—	146	2	no	—
210	#103	—	73	8	no	—
211	#103	—	146	8	no	—
212	#97	—	146	4	no	—
213	C₄×C₈	C₄×D₇₃	1	4	yes	—
214	#95	—	73	2	no	—
215	#95	—	146	4	no	—
216	C₂×Q₈	—	73	4	no	—
217	C₂×Q₈	—	73	4	no	—
218	C₂×Q₈	—	73	4	no	—
219	C₂×Q₈	—	146	2	no	—
220	C₂×Q₈	—	146	2	no	—
221	C₂×Q₈	—	146	2	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	73	4	no	—
224	C₂²×C₄	—	146	2	no	—
225	C₂²×C₄	—	146	2	no	—
226	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
227	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
228	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
229	Q₁₆	—	73	16	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	146	2	no	—
233	C₂×C₈	—	146	2	no	—
234	C₂×C₈	—	146	2	no	—
235	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
236	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
237	C₂×C₈	D₄×D₇₃	1	2	yes	—
238	C₂×C₈	Q₈×D₇₃	1	2	yes	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	4	4	no	—
242	C₄⋊C₄	—	8	2	no	—
243	C₄⋊C₄	—	292	8	no	—
244	C₄⋊C₄	—	584	4	no	—
245	C₄⋊C₄	—	73	4	no	—
246	C₄⋊C₄	—	146	8	no	—
247	C₄⋊C₄	—	146	4	no	—
248	C₄⋊C₄	—	146	2	no	—
249	C₄²	—	292	8	no	—
250	C₄²	—	146	8	no	—
251	C₄²	—	146	2	no	—
252	C₄²	—	146	2	no	—
253	C₄²	C₂×C₄×D₇₃	1	2	yes	—
254	C₄²	#11924029	1	2	yes	—
255	C₂³	C₂×D₄×D₇₃	1	1	yes	characteristic
256	Q₈	—	73	8	no	—
257	Q₈	—	146	4	no	—
258	Q₈	—	146	4	no	—
259	Q₈	—	146	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	292	2	no	—
265	C₂×C₄	—	292	2	no	—
266	C₂×C₄	—	146	4	no	—
267	C₂×C₄	—	146	4	no	—
268	C₂×C₄	—	146	4	no	—
269	C₂×C₄	—	146	4	no	—
270	C₂×C₄	—	146	4	no	—
271	C₂×C₄	—	146	4	no	—
272	C₂×C₄	—	146	4	no	—
273	C₂×C₄	#14556553	1	2	yes	—
274	C₂×C₄	#14556553	1	2	yes	—
275	C₂×C₄	#14556620	1	4	yes	—
276	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
277	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
278	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
279	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
280	C₂×C₄	#14556636	1	1	yes	characteristic
281	C₂×C₄	#14556639	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	292	2	no	—
284	C₈	C₄⋊C₄×D₇₃	1	4	yes	—
285	C₂²	C₂×#14556553	1	1	yes	characteristic
286	C₂²	#17989222	1	1	yes	characteristic
287	C₂²	#17989252	1	1	yes	characteristic
288	C₂²	C₂×#14556620	1	2	yes	—
289	C₂²	#17989393	1	2	yes	—
290	C₄	—	2	2	no	—
291	C₄	—	8	4	no	—
292	C₄	—	292	4	no	—
293	C₄	—	292	4	no	—
294	C₄	—	146	8	no	—
295	C₄	#17988693	1	4	yes	complemented
296	C₄	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
297	C₄	#17989068	1	1	yes	characteristic
298	C₄	#17989240	1	1	yes	characteristic
299	C₄	#17989298	1	1	yes	characteristic
300	C₂	#22612831	1	1	yes	characteristic
301	C₂	C₂×#17988693	1	1	yes	characteristic
302	C₂	#22618814	1	1	yes	characteristic
303	C₂	#22621596	1	2	yes	—
304	C₂	#22622868	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₇₃×#1844	C₂
socle, prime radical	C₂³×C₇₃	C₂²×D₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	#1844	D₇₃
hypocenter	C₇₃	#22674
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
13. C₄×Dic₂₉₂	C₄	semidirect
72. Dic₂₉₂	C₄⋊C₄	semidirect
139. C₄×C₇₃	#441	semidirect
161. C₇₃	#22674	semidirect
295. C₄	#17988693	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988693	2
C₂×C₄×Dic₂₉₂	2
#22612843	2
#22610667	2
#22610196	2
#22609933	2
#22609798	2
C₇₃×#1844	2
#22674	73