Subgroups — #34030206

Order 18688.

690 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989645	C₂	1	2	yes	maximal, complemented
2	#22626315	C₂	1	4	yes	maximal, complemented
3	D₄×#11924031	C₂	1	4	yes	maximal, complemented
4	#22626263	C₂	1	1	yes	maximal, characteristic, complemented
5	#22626161	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×#11924031	C₂	1	1	yes	maximal, characteristic, complemented
7	#22618643	C₂	1	1	yes	maximal, characteristic
8	#17989645	—	2	8	no	—
9	C₂×#14556665	C₂²	1	4	yes	—
10	C₂×#14556665	C₂²	1	2	yes	—
11	C₂²×#11924031	C₂²	1	4	yes	—
12	C₂²×#11924031	C₂²	1	1	yes	characteristic
13	C₂²×#11924031	C₂²	1	1	yes	characteristic
14	C₄⋊#11924031	C₂²	1	4	yes	—
15	C₂²⋊#11924031	C₂²	1	8	yes	—
16	C₄×#11924031	C₂²	1	2	yes	—
17	C₄×#11924031	C₂²	1	2	yes	—
18	C₂×#14556598	C₂²	1	2	yes	—
19	#17988665	C₄	1	4	yes	—
20	D₄×#10596831	C₂²	1	4	yes	—
21	#17988611	C₄	1	1	yes	characteristic
22	#17988554	C₄	1	2	yes	—
23	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
24	#532×C₇₃	C₄	1	1	yes	characteristic
25	#14556665	—	2	8	no	—
26	#14556665	—	2	4	no	—
27	C₂×#11924031	—	2	8	no	—
28	C₂×#11924031	—	2	2	no	—
29	C₂×#11924031	D₄	1	4	yes	complemented
30	C₂×#11924031	C₂³	1	4	yes	—
31	C₂×#11924031	C₂³	1	2	yes	—
32	C₂×#11924031	C₂³	1	1	yes	characteristic
33	C₂×#11924031	C₂³	1	1	yes	characteristic
34	#14556598	—	2	4	no	—
35	#14556598	—	2	4	no	—
36	C₂×#11924012	C₂×C₄	1	2	yes	—
37	C₂×#11924012	C₂×C₄	1	4	yes	—
38	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
39	C₂²×#10596831	C₂³	1	4	yes	—
40	C₂²×#10596831	C₂³	1	1	yes	characteristic
41	C₄⋊#10596831	C₂×C₄	1	4	yes	—
42	C₂²⋊#10596831	C₂×C₄	1	8	yes	—
43	C₄×#10596831	C₂×C₄	1	2	yes	—
44	C₄×#10596831	C₂³	1	2	yes	—
45	C₂×#41×C₇₃	C₂×C₄	1	2	yes	—
46	C₄⋊Q₈×C₇₃	C₈	1	1	yes	characteristic
47	#127×C₇₃	C₈	1	1	yes	characteristic
48	#124×C₇₃	C₈	1	2	yes	—
49	#121×C₇₃	C₈	1	4	yes	—
50	C₄×D₄×C₇₃	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
52	#11924031	—	2	4	no	—
53	#11924031	—	2	2	no	—
54	#11924031	C₂×D₄	1	2	yes	—
55	#11924031	C₂×D₄	1	2	yes	—
56	#11924012	—	2	4	no	—
57	#11924012	—	2	8	no	—
58	C₂×#10596831	—	2	2	no	—
59	C₂×#10596831	—	2	4	no	—
60	C₂×#10596831	—	2	4	no	—
61	C₂×#10596831	C₂²×C₄	1	2	yes	—
62	C₂×#10596831	C₂²×C₄	1	4	yes	—
63	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
64	C₂×#10596831	C₂×D₄	1	2	yes	—
65	C₂×#10596831	#41	1	2	yes	—
66	C₂×#10596831	C₂⁴	1	1	yes	characteristic
67	#41×C₇₃	—	2	4	no	—
68	#41×C₇₃	—	2	4	no	—
69	C₂×Q₈×C₇₃	C₂×C₈	1	2	yes	—
70	C₂×D₄×C₇₃	C₂×C₈	1	4	yes	—
71	C₂×D₄×C₇₃	C₂×C₈	1	4	yes	—
72	C₂×D₄×C₇₃	C₂×C₈	1	2	yes	—
73	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
74	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
76	C₄⋊C₄×C₇₃	C₂×C₈	1	4	yes	—
77	C₂²⋊C₄×C₇₃	C₂×C₈	1	8	yes	—
78	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
79	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
80	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
81	C₂³×C₇₃	C₂²×C₈	1	4	yes	—
82	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
83	Q₈×C₇₃	—	2	4	no	—
84	D₄×C₇₃	—	2	8	no	—
85	D₄×C₇₃	—	2	4	no	—
86	D₄×C₇₃	—	2	8	no	—
87	C₂×C₄×C₇₃	—	2	4	no	—
88	C₂×C₄×C₇₃	—	2	4	no	—
89	C₂×C₄×C₇₃	—	2	4	no	—
90	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
91	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
92	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
93	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₇₃	#131	1	2	yes	—
96	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
97	#10596831	—	2	4	no	—
98	#10596831	—	2	2	no	—
99	#10596831	C₄×D₄	1	2	yes	—
100	#10596831	C₂²×D₄	1	1	yes	characteristic
101	#10596831	C₂×#41	1	1	yes	characteristic
102	C₂²×C₇₃	—	2	2	no	—
103	C₂²×C₇₃	—	2	8	no	—
104	C₂²×C₇₃	—	2	4	no	—
105	C₂²×C₇₃	C₈×D₄	1	2	yes	—
106	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
107	C₄×C₇₃	—	2	2	no	—
108	C₄×C₇₃	—	2	4	no	—
109	C₄×C₇₃	C₈×D₄	1	2	yes	—
110	C₄×C₇₃	#504	1	4	yes	complemented
111	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
112	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
113	#33434	—	73	1	no	maximal, Sylow
114	C₂×C₇₃	—	2	2	no	—
115	C₂×C₇₃	—	4	4	no	—
116	C₂×C₇₃	C₂×C₈×D₄	1	1	yes	characteristic
117	C₂×C₇₃	C₂×#504	1	2	yes	—
118	C₂×#504	—	73	2	no	—
119	#2931	—	73	1	no	—
120	#2169	—	73	4	no	—
121	D₄×C₁₆	—	73	4	no	—
122	#2150	—	73	1	no	—
123	#2113	—	73	2	no	—
124	C₂×C₄×C₁₆	—	73	1	no	—
125	C₇₃	#33434	1	1	yes	characteristic, complemented, Sylow
126	C₂×#131	—	73	2	no	—
127	#532	#64408	1	1	yes	characteristic
128	#504	—	146	8	no	—
129	C₂×M₃₂	—	73	2	no	—
130	C₂×M₃₂	—	73	4	no	—
131	C₂²×C₁₆	—	73	1	no	—
132	C₂²×C₁₆	—	73	1	no	—
133	C₂²×C₁₆	—	73	4	no	—
134	#436	—	73	4	no	—
135	C₈×D₄	—	73	4	no	—
136	#423	—	73	1	no	—
137	#408	—	73	2	no	—
138	C₂×C₄×C₈	—	73	1	no	—
139	C₄⋊C₁₆	—	73	4	no	—
140	C₂²⋊C₁₆	—	73	8	no	—
141	C₄×C₁₆	—	73	2	no	—
142	C₄×C₁₆	—	73	2	no	—
143	C₂×#41	C₂×#64408	1	2	yes	—
144	#131	—	146	4	no	—
145	#131	—	146	4	no	—
146	C₂×M₁₆	—	73	2	no	—
147	C₂×M₁₆	—	73	4	no	—
148	C₂²×C₈	—	73	1	no	—
149	C₂²×C₈	—	73	4	no	—
150	C₂²×C₈	—	73	1	no	—
151	C₄⋊Q₈	#10596833	1	1	yes	characteristic
152	#127	#10596833	1	1	yes	characteristic
153	#124	#10596833	1	2	yes	—
154	#121	#10596833	1	4	yes	—
155	C₄×D₄	C₂×#64408	1	4	yes	—
156	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
157	M₃₂	—	146	8	no	—
158	M₃₂	—	146	4	no	—
159	C₂×C₁₆	—	73	1	no	—
160	C₂×C₁₆	—	73	1	no	—
161	C₂×C₁₆	—	73	4	no	—
162	C₂×C₁₆	—	73	2	no	—
163	C₂×C₁₆	—	73	4	no	—
164	C₂×C₁₆	—	146	8	no	—
165	C₂×C₁₆	—	146	2	no	—
166	C₄⋊C₈	—	73	4	no	—
167	C₂²⋊C₈	—	73	8	no	—
168	C₄×C₈	—	73	2	no	—
169	C₄×C₈	—	73	2	no	—
170	#41	—	2	4	no	—
171	#41	—	2	4	no	—
172	C₂×Q₈	C₂×#10596833	1	2	yes	—
173	C₂×D₄	C₂×#10596833	1	4	yes	—
174	C₂×D₄	C₂×#10596833	1	4	yes	—
175	C₂×D₄	C₂×#10596833	1	2	yes	—
176	C₂²×C₄	C₂×#10596833	1	2	yes	—
177	C₂²×C₄	C₂²×#64408	1	4	yes	—
178	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
179	M₁₆	—	146	8	no	—
180	M₁₆	—	146	4	no	—
181	C₂×C₈	—	73	1	no	—
182	C₂×C₈	—	73	1	no	—
183	C₂×C₈	—	73	2	no	—
184	C₂×C₈	—	73	2	no	—
185	C₂×C₈	—	73	2	no	—
186	C₂×C₈	—	73	4	no	—
187	C₂×C₈	—	146	4	no	—
188	C₂×C₈	—	146	4	no	—
189	C₂×C₈	—	146	2	no	—
190	C₄⋊C₄	C₂×#10596833	1	4	yes	—
191	C₂²⋊C₄	C₂×#10596833	1	8	yes	—
192	C₄²	C₂×#10596833	1	1	yes	characteristic
193	C₄²	C₂×#10596833	1	1	yes	characteristic
194	C₄²	C₂²×#64408	1	2	yes	—
195	C₁₆	—	73	2	no	—
196	C₁₆	—	73	2	no	—
197	C₁₆	—	146	2	no	—
198	C₁₆	—	146	4	no	—
199	C₂³	C₂²×#10596833	1	4	yes	—
200	C₂³	C₂²×#10596833	1	1	yes	characteristic
201	Q₈	—	2	4	no	—
202	D₄	—	2	8	no	—
203	D₄	—	2	8	no	—
204	D₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	D₄×#64408	1	2	yes	—
209	C₂×C₄	C₂²×#10596833	1	2	yes	—
210	C₂×C₄	C₂²×#10596833	1	4	yes	—
211	C₂×C₄	C₂²×#10596833	1	2	yes	—
212	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
213	C₂×C₄	#14556678	1	2	yes	—
214	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
215	C₈	—	73	1	no	—
216	C₈	—	73	1	no	—
217	C₈	—	73	2	no	—
218	C₈	—	146	4	no	—
219	C₈	—	146	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	8	no	—
223	C₂²	D₄×#10596833	1	2	yes	—
224	C₂²	C₂³×#10596833	1	1	yes	characteristic
225	C₄	—	2	4	no	—
226	C₄	—	2	2	no	—
227	C₄	D₄×#10596833	1	2	yes	—
228	C₄	#17989645	1	4	yes	complemented
229	C₄	C₂×D₄×#64408	1	1	yes	characteristic
230	C₄	C₂×#14556678	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	—	4	4	no	—
233	C₂	C₂×D₄×#10596833	1	1	yes	characteristic
234	C₂	C₂×#17989645	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting, prime radical	#532×C₇₃	C₄
cosocle	C₂×#10596831	C₂⁴
hypercenter	#532	#64408
hypocenter	C₇₃	#33434
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
1. C₂×#17989645	C₂	semidirect
2. #22626315	C₂	semidirect
3. D₄×#11924031	C₂	semidirect
4. #22626263	C₂	semidirect
5. #22626161	C₂	semidirect
6. C₂×C₄×#11924031	C₂	semidirect
29. C₂×#11924031	D₄	semidirect
110. C₄×C₇₃	#504	semidirect
125. C₇₃	#33434	semidirect
228. C₄	#17989645	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989645	2
#22626315	2
D₄×#11924031	2
#22626263	2
#22626161	2
C₂×C₄×#11924031	2
#22618643	2
#33434	73