Subgroups — #34013877

Order 18688.

318 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989510	C₂	1	2	yes	maximal
2	#22626263	C₂	1	1	yes	maximal, characteristic
3	C₄×#14556665	C₂	1	1	yes	maximal, characteristic
4	#22624975	C₂	1	2	yes	maximal
5	C₄⋊C₄×#10596831	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556665	C₂²	1	2	yes	—
7	C₂×#14556665	C₂²	1	1	yes	characteristic
8	C₂×#14556665	C₂²	1	1	yes	characteristic
9	#17989510	—	2	4	no	—
10	C₄⋊#11924031	—	2	1	no	—
11	C₄⋊#11924031	—	2	1	no	—
12	#17989475	—	2	1	no	—
13	C₄×#11924031	—	2	1	no	—
14	C₂×C₄⋊#10596831	C₄	1	1	yes	characteristic
15	C₂×C₄×#10596831	C₂²	1	2	yes	—
16	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
17	#17988342	C₄	1	2	yes	—
18	C₄×C₄⋊C₄×C₇₃	C₄	1	1	yes	characteristic
19	#14556665	—	2	2	no	—
20	#14556665	—	2	1	no	—
21	#14556665	—	2	1	no	—
22	#14556665	—	2	2	no	—
23	#14556665	D₄	1	2	yes	—
24	#14556665	Q₈	1	2	yes	—
25	C₂×#11924031	—	2	1	no	—
26	C₂×#11924031	—	2	1	no	—
27	C₂×#11924031	—	2	2	no	—
28	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
29	C₂²×#10596831	C₂×C₄	1	2	yes	—
30	C₂²×#10596831	C₂³	1	1	yes	characteristic
31	C₄⋊#10596831	C₈	1	2	yes	—
32	C₄⋊#10596831	C₈	1	2	yes	—
33	C₄×#10596831	—	2	2	no	—
34	C₄×#10596831	—	2	2	no	—
35	C₄×#10596831	D₄	1	1	yes	characteristic
36	C₄×#10596831	D₄	1	1	yes	characteristic
37	C₄×#10596831	Q₈	1	1	yes	characteristic
38	C₄×#10596831	Q₈	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
40	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
42	#95×C₇₃	—	2	1	no	—
43	#11924031	—	2	2	no	—
44	#11924031	—	2	2	no	—
45	#11924031	—	4	2	no	—
46	C₂×#10596831	—	2	2	no	—
47	C₂×#10596831	—	2	1	no	—
48	C₂×#10596831	—	2	1	no	—
49	C₂×#10596831	—	2	2	no	—
50	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
51	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
52	C₂×#10596831	C₂×C₈	1	4	yes	—
53	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
54	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
55	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
56	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
57	C₂×#10596831	#41	1	1	yes	characteristic
58	C₂×#10596831	#41	1	1	yes	characteristic
59	C₂²×C₄×C₇₃	—	2	2	no	—
60	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
63	C₄⋊C₄×C₇₃	—	2	4	no	—
64	C₄²×C₇₃	—	2	2	no	—
65	C₄²×C₇₃	—	2	2	no	—
66	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
67	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
68	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
69	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
70	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₇₃	—	2	2	no	—
72	C₂×C₄×C₇₃	—	2	4	no	—
73	C₂×C₄×C₇₃	—	4	1	no	—
74	C₂×C₄×C₇₃	—	4	2	no	—
75	C₂×C₄×C₇₃	—	4	2	no	—
76	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
85	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
86	#10596831	—	2	1	no	—
87	#10596831	—	2	1	no	—
88	#10596831	—	2	4	no	—
89	#10596831	#121	1	1	yes	characteristic
90	#10596831	C₂²⋊Q₈	1	1	yes	characteristic
91	#10596831	#125	1	1	yes	characteristic
92	#10596831	C₄⋊Q₈	1	1	yes	characteristic
93	C₂²×C₇₃	—	2	1	no	—
94	C₂²×C₇₃	—	2	1	no	—
95	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
96	C₂²×C₇₃	#436	1	1	yes	characteristic
97	C₂²×C₇₃	#446	1	1	yes	characteristic
98	C₄×C₇₃	—	2	1	no	—
99	C₄×C₇₃	—	2	1	no	—
100	C₄×C₇₃	—	2	1	no	—
101	C₄×C₇₃	—	4	4	no	—
102	C₄×C₇₃	#364	1	2	yes	complemented
103	C₄×C₇₃	#389	1	1	yes	characteristic
104	C₄×C₇₃	#424	1	1	yes	characteristic
105	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
106	C₄×C₇₃	C₈×Q₈	1	1	yes	characteristic
107	#13163	—	73	1	no	maximal, Sylow
108	C₂×C₇₃	—	2	1	no	—
109	C₂×C₇₃	—	2	1	no	—
110	C₂×C₇₃	#1939	1	1	yes	characteristic
111	C₂×C₇₃	C₂×#364	1	1	yes	characteristic
112	C₂×C₇₃	#2153	1	1	yes	characteristic
113	C₂×#364	—	73	2	no	—
114	#2150	—	73	1	no	—
115	C₄×M₃₂	—	73	1	no	—
116	C₈×C₄⋊C₄	—	73	1	no	—
117	#1376	—	73	2	no	—
118	C₇₃	#13163	1	1	yes	characteristic, complemented, Sylow
119	C₂×M₃₂	—	73	1	no	—
120	C₂×M₃₂	—	73	1	no	—
121	C₂×M₃₂	—	73	2	no	—
122	C₂×C₄⋊C₈	—	73	1	no	—
123	C₂×C₄×C₈	—	73	1	no	—
124	C₂×C₄×C₈	—	73	2	no	—
125	C₄×C₄⋊C₄	#64408	1	1	yes	characteristic
126	#364	—	146	4	no	—
127	C₄⋊C₁₆	—	146	1	no	—
128	C₄⋊C₁₆	—	146	1	no	—
129	#346	—	146	1	no	—
130	C₄×C₁₆	—	146	1	no	—
131	#336	—	73	2	no	—
132	C₂²×C₈	—	73	1	no	—
133	C₂²×C₈	—	73	1	no	—
134	C₂²×C₈	—	73	2	no	—
135	C₂×C₄⋊C₄	—	2	1	no	—
136	C₂×C₄²	C₂×#64408	1	2	yes	—
137	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
138	M₃₂	—	73	2	no	—
139	M₃₂	—	73	2	no	—
140	M₃₂	—	146	2	no	—
141	M₃₂	—	146	2	no	—
142	M₃₂	—	146	1	no	—
143	M₃₂	—	146	1	no	—
144	C₂×C₁₆	—	146	2	no	—
145	C₂×C₁₆	—	146	1	no	—
146	C₂×C₁₆	—	146	1	no	—
147	C₄⋊C₈	—	73	2	no	—
148	C₄⋊C₈	—	73	2	no	—
149	C₄×C₈	—	73	1	no	—
150	C₄×C₈	—	73	1	no	—
151	C₄×C₈	—	73	1	no	—
152	C₄×C₈	—	73	1	no	—
153	C₄×C₈	—	146	2	no	—
154	C₄×C₈	—	146	2	no	—
155	#95	—	2	1	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	#11924037	1	1	yes	characteristic
158	C₂²×C₄	#11924037	1	1	yes	characteristic
159	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
160	C₂×C₈	—	73	1	no	—
161	C₂×C₈	—	73	1	no	—
162	C₂×C₈	—	73	1	no	—
163	C₂×C₈	—	73	1	no	—
164	C₂×C₈	—	73	1	no	—
165	C₂×C₈	—	73	1	no	—
166	C₂×C₈	—	73	4	no	—
167	C₂×C₈	—	73	1	no	—
168	C₂×C₈	—	73	1	no	—
169	C₂×C₈	—	146	1	no	—
170	C₂×C₈	—	146	1	no	—
171	C₂×C₈	—	146	2	no	—
172	C₂×C₈	—	146	2	no	—
173	C₄⋊C₄	—	2	4	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	C₄⋊#64408	1	1	yes	characteristic
177	C₄²	C₄⋊#64408	1	1	yes	characteristic
178	C₄²	#11924036	1	1	yes	characteristic
179	C₄²	#11924036	1	1	yes	characteristic
180	C₁₆	—	292	2	no	—
181	C₁₆	—	146	2	no	—
182	C₁₆	—	146	2	no	—
183	C₂³	C₂×#11924037	1	1	yes	characteristic
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	#14556658	1	1	yes	characteristic
190	C₂×C₄	#14556658	1	1	yes	characteristic
191	C₂×C₄	#14556658	1	1	yes	characteristic
192	C₂×C₄	#14556658	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
194	C₂×C₄	D₄×#64408	1	1	yes	characteristic
195	C₂×C₄	Q₈×#64408	1	1	yes	characteristic
196	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
197	C₂×C₄	#14556678	1	1	yes	characteristic
198	C₂×C₄	#14556679	1	1	yes	characteristic
199	C₈	—	73	1	no	—
200	C₈	—	73	1	no	—
201	C₈	—	73	1	no	—
202	C₈	—	73	1	no	—
203	C₈	—	146	1	no	—
204	C₈	—	146	1	no	—
205	C₈	—	146	4	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	C₂×#14556658	1	1	yes	characteristic
209	C₂²	#17989622	1	1	yes	characteristic
210	C₂²	#17989638	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	4	no	—
215	C₄	#17989510	1	2	yes	complemented
216	C₄	#17989538	1	1	yes	characteristic
217	C₄	#17989594	1	1	yes	characteristic
218	C₄	#17989615	1	1	yes	characteristic
219	C₄	#17989636	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#22626029	1	1	yes	characteristic
223	C₂	C₂×#17989510	1	1	yes	characteristic
224	C₂	#22626274	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₄×C₄⋊C₄×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄⋊C₈
prime radical	C₂³×C₇₃	C₂×M₁₆
cosocle	C₂²×#10596831	C₂³
hypercenter	C₄×C₄⋊C₄	#64408
hypocenter	C₇₃	#13163
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
102. C₄×C₇₃	#364	semidirect
118. C₇₃	#13163	semidirect
215. C₄	#17989510	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989510	2
#22626263	2
C₄×#14556665	2
#22624975	2
C₄⋊C₄×#10596831	2
#13163	73