Subgroups — C₂²×#17988935 #33955170

Order 18688.

3086 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556588	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988935	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#14556511	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄×#10596831	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₄⋊Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₂₉₂	—	2	1	no	—
7	C₂³×D₂₉₂	—	2	1	no	—
8	C₂³×D₂₉₂	C₂²	1	2	yes	—
9	C₂³×#10596831	C₂²	1	2	yes	—
10	C₂×#14556588	—	2	6	no	—
11	C₂×#14556588	C₂²	1	12	yes	—
12	#17988935	C₂²	1	64	yes	direct product factor
13	C₂×#14556511	C₂²	1	48	yes	complemented
14	C₂×C₄×#10596831	C₂²	1	12	yes	complemented
15	C₂³×Q₈×C₇₃	—	2	1	no	—
16	C₂×C₄⋊Q₈×C₇₃	—	2	6	no	—
17	C₂×C₄⋊Q₈×C₇₃	C₂²	1	12	yes	complemented
18	C₂²×C₄⋊C₄×C₇₃	—	2	1	no	—
19	C₂²×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
20	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
21	C₂⁴×D₇₃	—	2	2	no	—
22	C₂²×D₂₉₂	—	2	6	no	—
23	C₂²×D₂₉₂	—	2	12	no	—
24	C₂²×D₂₉₂	—	2	12	no	—
25	C₂²×D₂₉₂	—	2	16	no	—
26	C₂²×D₂₉₂	—	4	8	no	—
27	C₂²×D₂₉₂	—	4	8	no	—
28	C₂²×D₂₉₂	C₂³	1	12	yes	—
29	C₂²×#10596831	—	2	8	no	—
30	C₂²×#10596831	C₂³	1	12	yes	—
31	#14556588	—	2	24	no	—
32	#14556588	C₂³	1	16	yes	—
33	#14556511	C₂³	1	64	yes	complemented
34	C₄×#10596831	C₂³	1	16	yes	complemented
35	C₂²×Q₈×C₇₃	—	2	12	no	—
36	C₂²×Q₈×C₇₃	—	4	8	no	—
37	C₂³×C₄×C₇₃	—	2	2	no	—
38	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₇₃	—	2	24	no	—
42	C₄⋊Q₈×C₇₃	C₂³	1	16	yes	complemented
43	C₂×C₄⋊C₄×C₇₃	—	2	12	no	—
44	C₂×C₄⋊C₄×C₇₃	—	2	6	no	—
45	C₂×C₄⋊C₄×C₇₃	C₂³	1	12	yes	—
46	C₂×C₄²×C₇₃	D₄	1	6	yes	—
47	C₂×C₄²×C₇₃	C₂³	1	6	yes	—
48	C₂³×D₇₃	—	2	12	no	—
49	C₂³×D₇₃	—	4	8	no	—
50	C₂³×D₇₃	—	4	8	no	—
51	C₂³×D₇₃	—	4	8	no	—
52	C₂×D₂₉₂	—	2	48	no	—
53	C₂×D₂₉₂	—	2	12	no	—
54	C₂×D₂₉₂	—	2	16	no	—
55	C₂×D₂₉₂	—	2	16	no	—
56	C₂×D₂₉₂	—	4	24	no	—
57	C₂×D₂₉₂	—	4	48	no	—
58	C₂×D₂₉₂	—	4	48	no	—
59	C₂×D₂₉₂	C₂⁴	1	8	yes	—
60	C₂×#10596831	—	2	24	no	—
61	C₂×#10596831	C₂⁴	1	8	yes	—
62	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
63	C₂×Q₈×C₇₃	—	2	16	no	—
64	C₂×Q₈×C₇₃	—	4	48	no	—
65	C₂²×C₄×C₇₃	—	2	12	no	—
66	C₂²×C₄×C₇₃	—	4	8	no	—
67	C₂²×C₄×C₇₃	D₈	1	8	yes	—
68	C₂²×C₄×C₇₃	SD₁₆	1	8	yes	—
69	C₂²×C₄×C₇₃	C₂×D₄	1	3	yes	—
70	C₂²×C₄×C₇₃	C₂×D₄	1	6	yes	—
71	C₂²×C₄×C₇₃	C₂×D₄	1	6	yes	—
72	C₂²×C₄×C₇₃	#41	1	8	yes	—
73	C₂²×C₄×C₇₃	C₂⁴	1	3	yes	—
74	C₄⋊C₄×C₇₃	—	2	16	no	—
75	C₄⋊C₄×C₇₃	—	2	12	no	—
76	C₄⋊C₄×C₇₃	C₂⁴	1	8	yes	—
77	C₄²×C₇₃	C₂×D₄	1	12	yes	—
78	C₄²×C₇₃	C₂⁴	1	4	yes	—
79	C₂³×C₇₃	#124	1	4	yes	—
80	C₂³×C₇₃	C₂×D₈	1	4	yes	—
81	C₂³×C₇₃	C₂×SD₁₆	1	4	yes	—
82	C₂³×C₇₃	C₂²×D₄	1	3	yes	—
83	C₂²×D₇₃	—	2	8	no	—
84	C₂²×D₇₃	—	4	24	no	—
85	C₂²×D₇₃	—	4	24	no	—
86	C₂²×D₇₃	—	4	24	no	—
87	C₂²×D₇₃	—	8	32	no	—
88	Q₈×C₇₃	—	4	32	no	—
89	C₂×C₄×C₇₃	—	2	8	no	—
90	C₂×C₄×C₇₃	—	4	24	no	—
91	C₂×C₄×C₇₃	#124	1	12	yes	—
92	C₂×C₄×C₇₃	C₂×D₈	1	24	yes	—
93	C₂×C₄×C₇₃	C₂×SD₁₆	1	24	yes	—
94	C₂×C₄×C₇₃	C₂²×D₄	1	3	yes	—
95	C₂×C₄×C₇₃	C₂²×D₄	1	4	yes	—
96	C₂×C₄×C₇₃	C₂²×D₄	1	4	yes	—
97	C₂×C₄×C₇₃	C₂×#41	1	12	yes	—
98	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
99	D₂₉₂	—	2	16	no	—
100	D₂₉₂	—	4	24	no	—
101	D₂₉₂	—	4	32	no	—
102	D₂₉₂	—	4	32	no	—
103	#10596831	—	2	8	no	—
104	C₂²×C₇₃	#486	1	16	yes	complemented
105	C₂²×C₇₃	C₂×#124	1	6	yes	—
106	C₂²×C₇₃	C₂²×D₈	1	6	yes	—
107	C₂²×C₇₃	C₂²×SD₁₆	1	6	yes	—
108	C₂²×C₇₃	C₂³×D₄	1	1	yes	characteristic
109	C₂×D₇₃	—	4	8	no	—
110	C₂×D₇₃	—	4	8	no	—
111	C₂×D₇₃	—	4	8	no	—
112	C₂×D₇₃	—	8	48	no	—
113	C₄×C₇₃	—	4	8	no	—
114	C₄×C₇₃	C₂×#124	1	6	yes	—
115	C₄×C₇₃	C₂²×D₈	1	8	yes	—
116	C₄×C₇₃	C₂²×SD₁₆	1	8	yes	—
117	C₄×C₇₃	C₂²×#41	1	2	yes	—
118	C₂²×#486	—	73	1	no	maximal, Sylow
119	C₂×C₇₃	C₂×#486	1	12	yes	complemented
120	C₂×C₇₃	C₂²×#124	1	1	yes	characteristic
121	C₂×C₇₃	C₂³×D₈	1	1	yes	characteristic
122	C₂×C₇₃	C₂³×SD₁₆	1	1	yes	characteristic
123	D₇₃	—	8	8	no	—
124	C₂²×C₄⋊Q₈	D₇₃	1	1	yes	characteristic, complemented
125	C₂²×#127	—	73	1	no	—
126	C₂×#486	—	73	24	no	—
127	C₂²×#102	—	73	4	no	—
128	C₂²×C₄×C₈	—	73	1	no	—
129	C₇₃	C₂²×#486	1	1	yes	characteristic, complemented, Sylow
130	C₂³×Q₈	—	2	1	no	—
131	C₂³×D₄	—	73	2	no	—
132	C₂³×D₄	—	146	1	no	—
133	C₂³×D₄	—	146	1	no	—
134	C₂³×C₈	—	73	2	no	—
135	C₂×C₄⋊Q₈	—	2	6	no	—
136	C₂×C₄⋊Q₈	C₂×D₇₃	1	12	yes	complemented
137	C₂×#127	—	73	12	no	—
138	C₂×#127	—	146	6	no	—
139	C₂²×C₄⋊C₄	—	2	1	no	—
140	C₂²×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
141	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
142	#486	—	73	64	no	—
143	C₂×#102	—	73	48	no	—
144	C₂×C₄×C₈	—	73	12	no	—
145	C₂⁵	—	146	2	no	—
146	C₂²×Q₈	—	2	12	no	—
147	C₂²×Q₈	—	4	8	no	—
148	C₂²×D₄	—	292	8	no	—
149	C₂²×D₄	—	292	8	no	—
150	C₂²×D₄	—	73	12	no	—
151	C₂²×D₄	—	146	12	no	—
152	C₂²×D₄	—	146	12	no	—
153	C₂²×D₄	—	146	6	no	—
154	C₂²×D₄	—	146	16	no	—
155	C₂³×C₄	—	2	2	no	—
156	C₂³×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
157	C₂³×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
158	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
159	C₂²×C₈	—	73	12	no	—
160	C₂²×C₈	—	146	8	no	—
161	C₄⋊Q₈	—	2	24	no	—
162	C₄⋊Q₈	C₂²×D₇₃	1	16	yes	complemented
163	#127	—	73	16	no	—
164	#127	—	146	24	no	—
165	C₂×C₄⋊C₄	—	2	12	no	—
166	C₂×C₄⋊C₄	—	2	6	no	—
167	C₂×C₄⋊C₄	C₂²×D₇₃	1	12	yes	—
168	C₂×C₄²	C₂²⋊D₇₃	1	6	yes	—
169	C₂×C₄²	C₂²×D₇₃	1	6	yes	—
170	#102	—	73	64	no	—
171	C₄×C₈	—	73	16	no	—
172	C₂⁴	—	292	8	no	—
173	C₂⁴	—	292	8	no	—
174	C₂⁴	—	292	8	no	—
175	C₂⁴	—	146	12	no	—
176	C₂⁴	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
177	C₂×Q₈	—	2	16	no	—
178	C₂×Q₈	—	4	48	no	—
179	C₂×D₄	—	292	24	no	—
180	C₂×D₄	—	292	48	no	—
181	C₂×D₄	—	292	48	no	—
182	C₂×D₄	—	73	8	no	—
183	C₂×D₄	—	146	16	no	—
184	C₂×D₄	—	146	16	no	—
185	C₂×D₄	—	146	12	no	—
186	C₂×D₄	—	146	48	no	—
187	C₂²×C₄	—	2	12	no	—
188	C₂²×C₄	—	4	8	no	—
189	C₂²×C₄	#11924013	1	8	yes	—
190	C₂²×C₄	#11924015	1	8	yes	—
191	C₂²×C₄	C₂×C₂²⋊D₇₃	1	3	yes	—
192	C₂²×C₄	C₂×C₂²⋊D₇₃	1	6	yes	—
193	C₂²×C₄	C₂×C₂²⋊D₇₃	1	6	yes	—
194	C₂²×C₄	#11924029	1	8	yes	—
195	C₂²×C₄	C₂³×D₇₃	1	3	yes	—
196	C₂×C₈	—	73	8	no	—
197	C₂×C₈	—	146	24	no	—
198	C₄⋊C₄	—	2	16	no	—
199	C₄⋊C₄	—	2	12	no	—
200	C₄⋊C₄	C₂³×D₇₃	1	8	yes	—
201	C₄²	C₂×C₂²⋊D₇₃	1	12	yes	—
202	C₄²	C₂³×D₇₃	1	4	yes	—
203	C₂³	—	292	24	no	—
204	C₂³	—	292	24	no	—
205	C₂³	—	292	24	no	—
206	C₂³	—	584	32	no	—
207	C₂³	—	146	8	no	—
208	C₂³	#14556580	1	4	yes	—
209	C₂³	C₂×#11924013	1	4	yes	—
210	C₂³	C₂×#11924015	1	4	yes	—
211	C₂³	C₂²×C₂²⋊D₇₃	1	3	yes	—
212	Q₈	—	4	32	no	—
213	D₄	—	292	32	no	—
214	D₄	—	292	32	no	—
215	D₄	—	292	24	no	—
216	D₄	—	146	16	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	4	24	no	—
219	C₂×C₄	#14556580	1	12	yes	—
220	C₂×C₄	C₂×#11924013	1	24	yes	—
221	C₂×C₄	C₂×#11924015	1	24	yes	—
222	C₂×C₄	C₂²×C₂²⋊D₇₃	1	4	yes	—
223	C₂×C₄	C₂²×C₂²⋊D₇₃	1	4	yes	—
224	C₂×C₄	C₂²×C₂²⋊D₇₃	1	3	yes	—
225	C₂×C₄	C₂×#11924029	1	12	yes	—
226	C₂×C₄	C₂⁴×D₇₃	1	1	yes	characteristic
227	C₈	—	146	8	no	—
228	C₂²	—	292	8	no	—
229	C₂²	—	292	8	no	—
230	C₂²	—	292	8	no	—
231	C₂²	—	584	48	no	—
232	C₂²	#17988935	1	16	yes	direct product factor
233	C₂²	C₂×#14556580	1	6	yes	—
234	C₂²	C₂²×#11924013	1	6	yes	—
235	C₂²	C₂²×#11924015	1	6	yes	—
236	C₂²	C₂³×C₂²⋊D₇₃	1	1	yes	characteristic
237	C₄	—	4	8	no	—
238	C₄	C₂×#14556580	1	6	yes	—
239	C₄	C₂²×#11924013	1	8	yes	—
240	C₄	C₂²×#11924015	1	8	yes	—
241	C₄	C₂²×#11924029	1	2	yes	—
242	C₂	—	584	8	no	—
243	C₂	C₂×#17988935	1	12	yes	direct product factor
244	C₂	C₂²×#14556580	1	1	yes	characteristic
245	C₂	C₂³×#11924013	1	1	yes	characteristic
246	C₂	C₂³×#11924015	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₇₃
commutator, cosocle	C₂×C₄×C₇₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇₃
Fitting	C₂²×C₄⋊Q₈×C₇₃	C₂
socle	C₂⁴×C₇₃	C₂×D₄
prime radical	C₂²×#14556588	C₂
hypercenter	C₂²×C₄⋊Q₈	D₇₃
hypocenter	C₇₃	C₂²×#486
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×#17988935	C₂	direct (N × H)
3. C₂²×#14556511	C₂	semidirect
4. C₂²×C₄×#10596831	C₂	semidirect
5. C₂²×C₄⋊Q₈×C₇₃	C₂	semidirect
12. #17988935	C₂²	direct (N × H)
13. C₂×#14556511	C₂²	semidirect
14. C₂×C₄×#10596831	C₂²	semidirect
17. C₂×C₄⋊Q₈×C₇₃	C₂²	semidirect
33. #14556511	C₂³	semidirect
34. C₄×#10596831	C₂³	semidirect
42. C₄⋊Q₈×C₇₃	C₂³	semidirect
104. C₂²×C₇₃	#486	semidirect
119. C₂×C₇₃	C₂×#486	semidirect
124. C₂²×C₄⋊Q₈	D₇₃	semidirect
129. C₇₃	C₂²×#486	semidirect
136. C₂×C₄⋊Q₈	C₂×D₇₃	semidirect
162. C₄⋊Q₈	C₂²×D₇₃	semidirect
232. C₂²	#17988935	direct (N × H)
243. C₂	C₂×#17988935	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14556588	2
C₂×#17988935	2
C₂²×#14556511	2
C₂²×C₄×#10596831	2
C₂²×C₄⋊Q₈×C₇₃	2
C₂²×#486	73

Subgroups — C22×#17988935 #33955170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17988935 #33955170