Subgroups — #34030165

Order 18688.

520 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626304	C₂	1	4	yes	maximal, complemented
2	#22626300	C₂	1	4	yes	maximal, complemented
3	#22626274	C₂	1	2	yes	maximal, complemented
4	#22626167	C₂	1	2	yes	maximal
5	C₂×#17989479	C₂	1	2	yes	maximal, complemented
6	#22618359	C₂	1	1	yes	maximal, characteristic
7	C₂×#14556665	C₂²	1	4	yes	—
8	#17989510	C₂²	1	8	yes	—
9	#17989483	C₂²	1	8	yes	—
10	#17989479	C₂²	1	4	yes	—
11	#17989479	C₂²	1	4	yes	—
12	C₂×#14556598	C₄	1	1	yes	characteristic
13	C₂×#14556598	C₂²	1	1	yes	characteristic
14	#17988543	C₄	1	2	yes	—
15	#17988543	C₄	1	1	yes	characteristic
16	#17988543	C₂²	1	2	yes	—
17	#17988543	C₂²	1	1	yes	characteristic
18	C₄×#11924012	C₄	1	1	yes	characteristic
19	C₄×#11924012	C₂²	1	2	yes	—
20	C₂×#14556502	C₄	1	2	yes	—
21	C₂×#14556502	C₂²	1	1	yes	characteristic
22	C₄×#41×C₇₃	C₄	1	1	yes	characteristic
23	#14556665	—	2	4	no	—
24	#14556665	C₂³	1	8	yes	—
25	C₂×#11924031	—	2	4	no	—
26	#14556598	—	2	4	no	—
27	#14556598	—	2	4	no	—
28	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
29	C₂×#11924012	C₂×C₄	1	2	yes	—
30	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
31	C₂×#11924012	C₂³	1	2	yes	—
32	C₂²×#10596831	C₂×C₄	1	2	yes	—
33	C₂²×#10596831	C₂×C₄	1	2	yes	—
34	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
35	C₂²×#10596831	C₂³	1	1	yes	characteristic
36	#14556502	C₂×C₄	1	1	yes	characteristic
37	#14556502	C₂×C₄	1	2	yes	—
38	#14556502	C₂×C₄	1	2	yes	—
39	#14556502	C₂×C₄	1	2	yes	—
40	#14556502	C₂×C₄	1	1	yes	characteristic
41	#14556502	C₂³	1	1	yes	characteristic
42	#14556502	C₂³	1	1	yes	characteristic
43	C₄×#10596831	C₂×C₄	1	2	yes	—
44	C₄×#10596831	C₂×C₄	1	2	yes	—
45	C₄×#10596831	C₂³	1	2	yes	—
46	C₂×#41×C₇₃	C₂×C₄	1	1	yes	characteristic
47	C₄×Q₈×C₇₃	—	2	1	no	—
48	C₄×D₄×C₇₃	—	2	2	no	—
49	C₄×D₄×C₇₃	—	2	1	no	—
50	#117×C₇₃	C₂×C₄	1	2	yes	—
51	#117×C₇₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
54	#11924031	—	2	8	no	—
55	#11924012	—	2	2	no	—
56	#11924012	—	2	4	no	—
57	#11924012	—	2	4	no	—
58	#11924012	M₁₆	1	4	yes	—
59	C₂×#10596831	—	2	2	no	—
60	C₂×#10596831	—	2	4	no	—
61	C₂×#10596831	—	2	4	no	—
62	C₂×#10596831	C₂²×C₄	1	2	yes	—
63	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
64	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
65	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
66	C₂×#10596831	C₂²×C₄	1	2	yes	—
67	C₂×#10596831	#41	1	4	yes	—
68	C₂×#10596831	C₂⁴	1	1	yes	characteristic
69	#41×C₇₃	—	2	2	no	—
70	#41×C₇₃	—	2	2	no	—
71	C₂×Q₈×C₇₃	M₁₆	1	1	yes	characteristic
72	C₂×D₄×C₇₃	M₁₆	1	2	yes	—
73	C₂×D₄×C₇₃	M₁₆	1	1	yes	characteristic
74	C₂²×C₄×C₇₃	—	2	2	no	—
75	C₂²×C₄×C₇₃	—	2	1	no	—
76	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
78	C₄⋊C₄×C₇₃	—	2	2	no	—
79	C₄⋊C₄×C₇₃	—	2	1	no	—
80	C₂²⋊C₄×C₇₃	—	2	2	no	—
81	C₂²⋊C₄×C₇₃	—	2	1	no	—
82	C₄²×C₇₃	—	2	2	no	—
83	C₄²×C₇₃	—	2	1	no	—
84	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
87	C₂³×C₇₃	C₂×M₁₆	1	2	yes	—
88	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
89	Q₈×C₇₃	—	2	2	no	—
90	D₄×C₇₃	—	2	4	no	—
91	D₄×C₇₃	—	2	2	no	—
92	C₂×C₄×C₇₃	—	2	2	no	—
93	C₂×C₄×C₇₃	—	2	2	no	—
94	C₂×C₄×C₇₃	—	2	1	no	—
95	C₂×C₄×C₇₃	—	2	1	no	—
96	C₂×C₄×C₇₃	—	2	2	no	—
97	C₂×C₄×C₇₃	—	8	1	no	—
98	C₂×C₄×C₇₃	—	8	2	no	—
99	C₂×C₄×C₇₃	#117	1	2	yes	—
100	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
101	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
103	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
104	#10596831	—	2	2	no	—
105	#10596831	—	2	1	no	—
106	#10596831	—	2	1	no	—
107	#10596831	#117	1	2	yes	—
108	#10596831	C₂×M₁₆	1	4	yes	—
109	#10596831	C₂×#41	1	2	yes	—
110	C₂²×C₇₃	—	2	2	no	—
111	C₂²×C₇₃	—	4	1	no	—
112	C₂²×C₇₃	—	4	2	no	—
113	C₂²×C₇₃	#432	1	2	yes	—
114	C₂²×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
115	C₄×C₇₃	—	2	2	no	—
116	C₄×C₇₃	—	4	1	no	—
117	C₄×C₇₃	—	4	2	no	—
118	C₄×C₇₃	—	4	1	no	—
119	C₄×C₇₃	#432	1	2	yes	—
120	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
121	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
122	#33426	—	73	1	no	maximal, Sylow
123	C₂×C₇₃	—	2	1	no	—
124	C₂×C₇₃	—	2	2	no	—
125	C₂×C₇₃	—	4	2	no	—
126	C₂×C₇₃	C₂×#432	1	1	yes	characteristic
127	#2875	—	73	1	no	—
128	#2166	—	73	4	no	—
129	#2165	—	73	4	no	—
130	#2153	—	73	2	no	—
131	#2115	—	73	2	no	—
132	C₂×#347	—	73	2	no	—
133	C₇₃	#33426	1	1	yes	characteristic, complemented, Sylow
134	C₂×#131	—	73	1	no	—
135	C₂×#131	—	73	1	no	—
136	C₄×#41	#64408	1	1	yes	characteristic
137	C₂×M₃₂	—	73	4	no	—
138	#405	—	73	1	no	—
139	#405	—	73	2	no	—
140	#405	—	73	1	no	—
141	#405	—	73	2	no	—
142	C₄×M₁₆	—	73	2	no	—
143	C₄×M₁₆	—	73	1	no	—
144	C₂×#97	—	73	2	no	—
145	C₂×#97	—	73	1	no	—
146	#364	—	73	8	no	—
147	#349	—	73	8	no	—
148	#347	—	73	4	no	—
149	#347	—	73	4	no	—
150	C₂×#41	C₂×#64408	1	1	yes	characteristic
151	#131	—	146	4	no	—
152	#131	—	146	4	no	—
153	C₂×M₁₆	—	73	1	no	—
154	C₂×M₁₆	—	73	2	no	—
155	C₂×M₁₆	—	73	2	no	—
156	C₂×M₁₆	—	73	1	no	—
157	C₂²×C₈	—	73	1	no	—
158	C₂²×C₈	—	73	2	no	—
159	C₂²×C₈	—	73	2	no	—
160	C₂²×C₈	—	73	1	no	—
161	C₄×Q₈	—	2	1	no	—
162	C₄×D₄	—	2	2	no	—
163	C₄×D₄	—	2	1	no	—
164	#117	C₂×#64408	1	1	yes	characteristic
165	#117	C₂×#64408	1	2	yes	—
166	C₂×C₄²	C₂×#64408	1	2	yes	—
167	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
168	M₃₂	—	73	8	no	—
169	M₃₂	—	146	4	no	—
170	C₂×C₁₆	—	146	4	no	—
171	#97	—	73	1	no	—
172	#97	—	73	2	no	—
173	#97	—	73	2	no	—
174	#97	—	73	2	no	—
175	#97	—	73	1	no	—
176	#97	—	73	1	no	—
177	#97	—	73	1	no	—
178	C₄×C₈	—	73	2	no	—
179	C₄×C₈	—	73	2	no	—
180	C₄×C₈	—	73	2	no	—
181	#41	—	2	2	no	—
182	#41	—	2	2	no	—
183	C₂×Q₈	#11924037	1	1	yes	characteristic
184	C₂×D₄	#11924037	1	2	yes	—
185	C₂×D₄	#11924037	1	1	yes	characteristic
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	C₂²×#64408	1	2	yes	—
189	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
190	M₁₆	—	73	4	no	—
191	M₁₆	—	146	4	no	—
192	M₁₆	—	146	4	no	—
193	M₁₆	—	146	2	no	—
194	C₂×C₈	—	73	2	no	—
195	C₂×C₈	—	73	1	no	—
196	C₂×C₈	—	73	1	no	—
197	C₂×C₈	—	73	4	no	—
198	C₂×C₈	—	73	2	no	—
199	C₂×C₈	—	73	1	no	—
200	C₂×C₈	—	73	1	no	—
201	C₂×C₈	—	146	4	no	—
202	C₂×C₈	—	146	4	no	—
203	C₂×C₈	—	146	2	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₂²⋊C₄	—	2	1	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	1	no	—
210	C₄²	C₂²×#64408	1	2	yes	—
211	C₄²	C₂²×#64408	1	1	yes	characteristic
212	C₄²	C₂²×#64408	1	1	yes	characteristic
213	C₁₆	—	146	8	no	—
214	C₂³	C₂×#11924037	1	2	yes	—
215	C₂³	C₂×#11924037	1	1	yes	characteristic
216	Q₈	—	2	2	no	—
217	D₄	—	2	4	no	—
218	D₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	8	1	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	#14556670	1	2	yes	—
227	C₂×C₄	C₂×#11924037	1	2	yes	—
228	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
229	C₂×C₄	D₇₃⋊M₁₆	1	2	yes	—
230	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
231	C₈	—	73	2	no	—
232	C₈	—	73	2	no	—
233	C₈	—	73	4	no	—
234	C₈	—	146	2	no	—
235	C₈	—	146	1	no	—
236	C₈	—	146	1	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	2	no	—
240	C₂²	#17989607	1	2	yes	—
241	C₂²	C₂²×#11924037	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	4	1	no	—
244	C₄	—	4	2	no	—
245	C₄	—	4	1	no	—
246	C₄	#17989607	1	2	yes	—
247	C₄	C₂×#14556670	1	1	yes	characteristic
248	C₄	C₂×D₇₃⋊M₁₆	1	1	yes	characteristic
249	C₂	—	2	1	no	—
250	C₂	—	2	2	no	—
251	C₂	—	4	2	no	—
252	C₂	C₂×#17989607	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14556670
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₄×#41×C₇₃	C₄
socle	C₂×C₇₃	C₂×#432
prime radical	C₂×#41×C₇₃	C₂×C₄
cosocle	C₂×#10596831	C₂⁴
hypercenter	C₄×#41	#64408
hypocenter	C₇₃	#33426
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #22626304	C₂	semidirect
2. #22626300	C₂	semidirect
3. #22626274	C₂	semidirect
5. C₂×#17989479	C₂	semidirect
133. C₇₃	#33426	semidirect

Maximal subgroups

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

subgroup	index
#22626304	2
#22626300	2
#22626274	2
#22626167	2
C₂×#17989479	2
#22618359	2
#33426	73