Subgroups — #34030369

Order 18688.

552 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	#22626347	C₂	1	2	yes	maximal, complemented
2	#22626313	C₂	1	4	yes	maximal, complemented
3	D₄×#11924031	C₂	1	2	yes	maximal, complemented
4	#22626295	C₂	1	1	yes	maximal, characteristic, complemented
5	#22626293	C₂	1	2	yes	maximal, complemented
6	C₄×#14556666	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#17989476	C₂	1	2	yes	maximal, complemented
8	C₈×#11924027	C₂	1	1	yes	maximal, characteristic
9	C₂×#14556666	C₂²	1	2	yes	—
10	C₂²×#11924031	C₂²	1	4	yes	—
11	#17989508	C₂²	1	4	yes	—
12	C₄⋊#11924031	C₂²	1	2	yes	—
13	#17989482	C₂²	1	2	yes	—
14	C₂²⋊#11924031	C₂²	1	4	yes	—
15	#17989476	C₂²	1	4	yes	—
16	#17989476	C₂²	1	1	yes	characteristic
17	#17989476	C₂²	1	2	yes	—
18	#17989476	C₂²	1	1	yes	characteristic
19	C₄×#11924031	C₂²	1	2	yes	—
20	C₄×#11924027	C₄	1	1	yes	characteristic
21	C₈×Dic₁₄₆	C₄	1	1	yes	characteristic
22	#17988711	C₂²	1	1	yes	characteristic
23	D₄×#10596831	C₂²	1	1	yes	characteristic
24	C₈×C₂²⋊D₇₃	C₄	1	2	yes	—
25	#17988650	C₂²	1	2	yes	—
26	#17988639	C₄	1	1	yes	characteristic
27	#17988637	C₄	1	2	yes	—
28	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
29	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
30	C₈×D₄×C₇₃	C₄	1	1	yes	characteristic
31	#14556666	—	2	4	no	—
32	C₂×#11924031	—	2	8	no	—
33	C₂×#11924031	C₂³	1	4	yes	—
34	C₂×#11924031	C₂³	1	2	yes	—
35	C₂×#11924031	C₂³	1	2	yes	—
36	C₂×#11924027	C₈	1	1	yes	characteristic
37	C₂²×#10596831	C₂³	1	2	yes	—
38	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
40	C₄×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
41	#14556556	C₈	1	1	yes	characteristic
42	D₄×Dic₇₃	C₈	1	1	yes	characteristic
43	C₄×C₂²⋊D₇₃	C₂×C₄	1	2	yes	—
44	#14556551	C₈	1	2	yes	—
45	#14556549	C₈	1	1	yes	characteristic
46	#14556547	C₈	1	2	yes	—
47	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
48	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
49	C₄⋊#10596831	C₂×C₄	1	1	yes	characteristic
50	#14556520	C₂×C₄	1	2	yes	—
51	#14556520	C₂×C₄	1	2	yes	—
52	C₂²⋊#10596831	C₂×C₄	1	2	yes	—
53	#14556503	C₂×C₄	1	2	yes	—
54	C₄×#10596831	C₂³	1	1	yes	characteristic
55	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
56	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
57	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
58	C₈×Dic₇₃	C₂³	1	2	yes	—
59	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
61	C₄×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
62	C₄⋊C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
63	C₂²⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
65	#11924031	—	2	4	no	—
66	#11924031	—	2	2	no	—
67	#11924031	#41	1	4	yes	complemented
68	#11924027	—	2	4	no	—
69	C₂×Dic₁₄₆	C₂×C₈	1	1	yes	characteristic
70	C₂×C₂²⋊D₇₃	C₂×C₈	1	2	yes	—
71	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
72	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
73	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
74	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
75	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×#10596831	—	2	4	no	—
77	C₂×#10596831	C₂²×C₄	1	2	yes	—
78	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
79	C₂×#10596831	C₂⁴	1	1	yes	characteristic
80	C₈×D₇₃	—	2	2	no	—
81	C₈×D₇₃	—	2	2	no	—
82	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
83	#11924007	C₂×C₈	1	2	yes	—
84	#11924007	C₂×C₈	1	2	yes	—
85	C₂²⋊Dic₇₃	C₂×C₈	1	2	yes	—
86	#11924005	C₂×C₈	1	2	yes	—
87	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
88	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
89	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
90	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
91	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
92	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
93	C₂×D₄×C₇₃	C₂×C₈	1	1	yes	characteristic
94	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₇₃	—	2	4	no	—
96	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
99	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
100	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
101	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
102	C₂³×C₇₃	C₂²×C₈	1	2	yes	—
103	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
104	D₄×C₇₃	M₃₂	1	4	yes	complemented
105	C₂×C₄×C₇₃	—	2	2	no	—
106	C₂×C₄×C₇₃	—	2	2	no	—
107	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
111	C₂²⋊D₇₃	—	2	4	no	—
112	C₂×Dic₇₃	—	2	4	no	—
113	C₂×Dic₇₃	—	4	2	no	—
114	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
116	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
118	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
119	C₄×D₇₃	—	2	1	no	—
120	C₄×D₇₃	—	2	1	no	—
121	C₄×D₇₃	—	2	2	no	—
122	C₄×D₇₃	—	4	1	no	—
123	Dic₁₄₆	—	2	2	no	—
124	C₈×C₇₃	—	2	1	no	—
125	C₈×C₇₃	—	2	2	no	—
126	C₈×C₇₃	#131	1	2	yes	—
127	#10596831	—	2	2	no	—
128	#10596831	—	2	1	no	—
129	#10596831	C₂×#41	1	2	yes	—
130	C₂²×C₇₃	—	2	2	no	—
131	C₂²×C₇₃	—	2	2	no	—
132	C₂²×C₇₃	C₂×M₃₂	1	4	yes	—
133	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
134	C₂×D₇₃	—	2	1	no	—
135	C₂×D₇₃	—	2	1	no	—
136	C₂×D₇₃	—	2	1	no	—
137	C₄×C₇₃	—	2	1	no	—
138	C₄×C₇₃	—	2	2	no	—
139	C₄×C₇₃	C₂×M₃₂	1	2	yes	—
140	C₄×C₇₃	C₄×#41	1	1	yes	characteristic
141	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
142	Dic₇₃	—	2	2	no	—
143	Dic₇₃	—	2	1	no	—
144	Dic₇₃	—	2	1	no	—
145	Dic₇₃	—	2	2	no	—
146	Dic₇₃	—	2	1	no	—
147	#33484	—	73	1	no	maximal, Sylow
148	C₂×C₇₃	—	2	4	no	—
149	C₂×C₇₃	C₈×#41	1	1	yes	characteristic
150	C₂×C₇₃	C₂²×M₃₂	1	1	yes	characteristic
151	C₂×C₇₃	#3407	1	1	yes	characteristic
152	D₇₃	—	4	1	no	—
153	C₈×#41	—	73	1	no	—
154	#2183	—	73	2	no	—
155	#2168	—	73	4	no	—
156	D₄×C₁₆	—	73	2	no	—
157	#2163	—	73	2	no	—
158	#2163	—	73	1	no	—
159	C₄×M₃₂	—	73	1	no	—
160	C₂×#346	—	73	2	no	—
161	C₇₃	#33484	1	1	yes	characteristic, complemented, Sylow
162	C₄×#41	—	73	1	no	—
163	C₂×M₃₂	—	73	2	no	—
164	C₂²×C₁₆	—	73	4	no	—
165	C₈×Q₈	—	73	1	no	—
166	C₈×Q₈	—	73	1	no	—
167	C₈×D₄	—	73	2	no	—
168	C₈×D₄	—	73	1	no	—
169	C₈×D₄	—	73	2	no	—
170	C₈×D₄	#64408	1	1	yes	characteristic
171	#431	—	73	1	no	—
172	#431	—	73	2	no	—
173	C₂×C₄×C₈	—	73	2	no	—
174	C₂×C₄×C₈	—	73	1	no	—
175	C₄⋊C₁₆	—	73	2	no	—
176	C₄⋊C₁₆	—	73	4	no	—
177	C₂²⋊C₁₆	—	73	4	no	—
178	C₂²⋊C₁₆	—	73	2	no	—
179	#346	—	73	1	no	—
180	#346	—	73	2	no	—
181	#346	—	73	4	no	—
182	#346	—	73	1	no	—
183	C₄×C₁₆	—	73	2	no	—
184	C₂×#41	—	73	1	no	—
185	C₂²×C₈	—	73	2	no	—
186	C₂²×C₈	—	73	1	no	—
187	C₂²×C₈	—	73	1	no	—
188	C₂²×C₈	C₂×#64408	1	2	yes	—
189	C₄×Q₈	—	73	1	no	—
190	C₄×Q₈	—	73	1	no	—
191	C₄×D₄	—	73	1	no	—
192	C₄×D₄	—	73	2	no	—
193	C₄×D₄	—	73	2	no	—
194	C₄×D₄	C₂×#64408	1	1	yes	characteristic
195	#117	—	73	1	no	—
196	#117	—	73	2	no	—
197	C₂×C₄²	—	73	2	no	—
198	C₂×C₄²	—	73	1	no	—
199	M₃₂	—	146	4	no	—
200	C₂×C₁₆	—	73	2	no	—
201	C₂×C₁₆	—	73	2	no	—
202	C₂×C₁₆	—	73	4	no	—
203	C₂×C₁₆	—	146	8	no	—
204	C₄⋊C₈	—	73	2	no	—
205	C₄⋊C₈	—	73	2	no	—
206	C₄⋊C₈	—	73	1	no	—
207	C₄⋊C₈	C₂×#64408	1	1	yes	characteristic
208	C₂²⋊C₈	—	73	2	no	—
209	C₂²⋊C₈	—	73	2	no	—
210	C₂²⋊C₈	C₂×#64408	1	2	yes	—
211	C₄×C₈	—	73	2	no	—
212	C₄×C₈	—	73	1	no	—
213	C₄×C₈	—	73	2	no	—
214	C₄×C₈	—	73	1	no	—
215	C₄×C₈	—	73	2	no	—
216	C₄×C₈	—	73	1	no	—
217	C₄×C₈	C₂×#64408	1	1	yes	characteristic
218	#41	—	146	4	no	—
219	C₂×Q₈	—	73	1	no	—
220	C₂×D₄	—	73	2	no	—
221	C₂×D₄	#11924036	1	1	yes	characteristic
222	C₂²×C₄	—	73	2	no	—
223	C₂²×C₄	—	73	1	no	—
224	C₂²×C₄	—	73	2	no	—
225	C₂²×C₄	—	73	1	no	—
226	C₂²×C₄	—	73	1	no	—
227	C₂²×C₄	C₂²×#64408	1	2	yes	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	73	1	no	—
230	C₂×C₈	—	73	2	no	—
231	C₂×C₈	—	73	1	no	—
232	C₂×C₈	—	146	2	no	—
233	C₂×C₈	—	146	4	no	—
234	C₂×C₈	—	146	2	no	—
235	C₂×C₈	C₂²×#64408	1	2	yes	—
236	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
237	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
238	C₄⋊C₄	—	73	2	no	—
239	C₄⋊C₄	—	73	2	no	—
240	C₄⋊C₄	—	73	1	no	—
241	C₄⋊C₄	#11924036	1	1	yes	characteristic
242	C₂²⋊C₄	—	73	2	no	—
243	C₂²⋊C₄	—	73	2	no	—
244	C₂²⋊C₄	#11924036	1	2	yes	—
245	C₄²	—	73	2	no	—
246	C₄²	—	73	1	no	—
247	C₄²	—	73	1	no	—
248	C₄²	—	73	2	no	—
249	C₄²	—	73	1	no	—
250	C₄²	—	73	2	no	—
251	C₄²	C₂²×#64408	1	1	yes	characteristic
252	C₁₆	—	73	4	no	—
253	C₁₆	—	146	4	no	—
254	C₁₆	—	146	2	no	—
255	C₂³	—	73	1	no	—
256	C₂³	C₂×#11924036	1	2	yes	—
257	Q₈	—	146	2	no	—
258	D₄	—	146	4	no	—
259	D₄	#14556666	1	4	yes	complemented
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	292	1	no	—
263	C₂×C₄	—	292	2	no	—
264	C₂×C₄	—	73	1	no	—
265	C₂×C₄	—	73	1	no	—
266	C₂×C₄	—	73	2	no	—
267	C₂×C₄	—	73	1	no	—
268	C₂×C₄	—	73	2	no	—
269	C₂×C₄	—	146	4	no	—
270	C₂×C₄	—	146	2	no	—
271	C₂×C₄	—	146	1	no	—
272	C₂×C₄	—	146	1	no	—
273	C₂×C₄	C₂×#11924036	1	2	yes	—
274	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
275	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
276	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	2	no	—
279	C₈	—	73	2	no	—
280	C₈	—	146	2	no	—
281	C₈	—	146	1	no	—
282	C₈	#14556678	1	2	yes	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	146	1	no	—
286	C₂²	—	146	1	no	—
287	C₂²	—	146	1	no	—
288	C₂²	C₂×#14556666	1	4	yes	—
289	C₂²	C₂²×#11924036	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	2	no	—
292	C₄	—	146	1	no	—
293	C₄	—	146	2	no	—
294	C₄	—	146	1	no	—
295	C₄	—	146	2	no	—
296	C₄	—	146	1	no	—
297	C₄	C₂×#14556666	1	2	yes	—
298	C₄	#41×#64408	1	1	yes	characteristic
299	C₄	C₂×#14556678	1	1	yes	characteristic
300	C₂	—	2	4	no	—
301	C₂	—	292	1	no	—
302	C₂	#22626736	1	1	yes	characteristic
303	C₂	C₂²×#14556666	1	1	yes	characteristic
304	C₂	#22626899	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	C₈×D₄×C₇₃	C₄
prime radical	C₂×#11924027	C₈
cosocle	C₂×#10596831	C₂⁴
hypercenter	C₈×D₄	#64408
hypocenter	C₇₃	#33484
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #22626347	C₂	semidirect
2. #22626313	C₂	semidirect
3. D₄×#11924031	C₂	semidirect
4. #22626295	C₂	semidirect
5. #22626293	C₂	semidirect
6. C₄×#14556666	C₂	semidirect
7. C₂×#17989476	C₂	semidirect
67. #11924031	#41	semidirect
104. D₄×C₇₃	M₃₂	semidirect
161. C₇₃	#33484	semidirect
259. D₄	#14556666	semidirect

Maximal subgroups

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

subgroup	index
#22626347	2
#22626313	2
D₄×#11924031	2
#22626295	2
#22626293	2
C₄×#14556666	2
C₂×#17989476	2
C₈×#11924027	2
#33484	73