Subgroups — C₂×#34036851 #49172333

Order 37504.

1112 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#34036851	C₂	1	16	yes	maximal, direct product factor
2	C₂×#22627622	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#22627619	C₂	1	2	yes	maximal, complemented
4	C₂×#22627618	C₂	1	2	yes	maximal, complemented
5	C₂×#22627613	C₂	1	1	yes	maximal, characteristic
6	C₂×#22627612	C₂	1	1	yes	maximal, characteristic
7	C₂×#22627602	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×Dic₅₈₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#22627589	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#22627586	C₂	1	1	yes	maximal, characteristic
11	C₂×#22627581	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×Q₈×C₂₉₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₅₈₆	C₂²	1	1	yes	characteristic
14	C₂²×Dic₅₈₆	C₂²	1	1	yes	characteristic
15	C₂²×D₁₁₇₂	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₂₉₃	C₂²	1	2	yes	—
17	#22627622	C₂²	1	8	yes	complemented
18	#22627619	C₂²	1	16	yes	complemented
19	#22627618	C₂²	1	16	yes	complemented
20	#22627613	C₂²	1	8	yes	—
21	#22627612	C₂²	1	8	yes	—
22	#22627602	C₂²	1	16	yes	complemented
23	C₄×Dic₅₈₆	C₂²	1	8	yes	complemented
24	#22627589	C₂²	1	8	yes	complemented
25	#22627586	C₂²	1	8	yes	—
26	#22627581	C₂²	1	16	yes	complemented
27	C₂×C₄⋊Dic₂₉₃	C₂²	1	1	yes	characteristic
28	C₂×#17989781	C₂²	1	1	yes	characteristic
29	C₂×#17989781	C₂²	1	2	yes	—
30	C₂×#17989781	C₂²	1	2	yes	—
31	C₂×#17989781	C₂²	1	1	yes	characteristic
32	C₂×#17989781	C₂²	1	2	yes	—
33	C₂×#17989779	C₂²	1	2	yes	—
34	C₂×#17989779	C₂²	1	2	yes	—
35	C₂×#17989779	C₂²	1	2	yes	—
36	C₂×#17989779	C₂²	1	2	yes	—
37	C₂×#17989779	C₂²	1	2	yes	—
38	C₂×C₄×Dic₂₉₃	C₂²	1	2	yes	—
39	C₂×C₄×Dic₂₉₃	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₂₉₃	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₂₉₃	C₂²	1	1	yes	characteristic
42	C₄×Q₈×C₂₉₃	C₂²	1	8	yes	complemented
43	C₂×C₄⋊C₄×C₂₉₃	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₂₉₃	C₂²	1	2	yes	—
45	C₂×C₄²×C₂₉₃	C₂²	1	2	yes	—
46	C₂×C₄²×C₂₉₃	C₂²	1	1	yes	characteristic
47	C₂³×D₂₉₃	C₂³	1	2	yes	—
48	C₂×Dic₅₈₆	—	2	4	no	—
49	C₂×Dic₅₈₆	—	2	4	no	—
50	C₂×Dic₅₈₆	C₂³	1	4	yes	—
51	C₂×Dic₅₈₆	C₂³	1	4	yes	—
52	C₂×D₁₁₇₂	—	2	4	no	—
53	C₂×D₁₁₇₂	C₂³	1	4	yes	—
54	C₂²×Dic₂₉₃	C₂³	1	2	yes	—
55	C₂²×Dic₂₉₃	C₂³	1	1	yes	characteristic
56	C₂²×Dic₂₉₃	C₂³	1	1	yes	characteristic
57	C₂²×Dic₂₉₃	C₂³	1	1	yes	characteristic
58	C₂²×Dic₂₉₃	C₂³	1	1	yes	characteristic
59	C₂×C₄×D₂₉₃	—	2	8	no	—
60	C₂×C₄×D₂₉₃	C₂³	1	8	yes	—
61	C₄⋊Dic₂₉₃	C₂³	1	4	yes	—
62	#17989781	C₂³	1	8	yes	—
63	#17989781	C₂³	1	4	yes	—
64	#17989781	C₂³	1	4	yes	—
65	#17989781	C₂³	1	8	yes	—
66	#17989781	C₂³	1	8	yes	—
67	#17989779	C₂³	1	8	yes	—
68	#17989779	C₂³	1	8	yes	—
69	#17989779	C₂³	1	8	yes	—
70	#17989779	C₂³	1	8	yes	—
71	#17989779	C₂³	1	8	yes	—
72	C₄×Dic₂₉₃	C₂³	1	4	yes	—
73	C₄×Dic₂₉₃	C₂³	1	4	yes	—
74	C₄×Dic₂₉₃	C₂³	1	8	yes	—
75	C₂×Q₈×C₂₉₃	—	2	4	no	—
76	C₂×Q₈×C₂₉₃	C₂³	1	4	yes	—
77	C₂²×C₄×C₂₉₃	C₂³	1	2	yes	—
78	C₂²×C₄×C₂₉₃	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₂₉₃	C₂³	1	1	yes	characteristic
80	C₂²×C₄×C₂₉₃	C₂³	1	2	yes	—
81	C₂²×C₄×C₂₉₃	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₂₉₃	C₂³	1	4	yes	—
83	C₄⋊C₄×C₂₉₃	C₂³	1	8	yes	—
84	C₄²×C₂₉₃	C₂³	1	4	yes	—
85	C₄²×C₂₉₃	C₂³	1	8	yes	—
86	C₂²×D₂₉₃	—	2	4	no	—
87	C₂²×D₂₉₃	—	2	4	no	—
88	C₂²×D₂₉₃	—	2	4	no	—
89	C₂²×D₂₉₃	C₂⁴	1	4	yes	—
90	Dic₅₈₆	—	2	8	no	—
91	Dic₅₈₆	—	2	8	no	—
92	D₁₁₇₂	—	2	8	no	—
93	C₂×Dic₂₉₃	—	2	2	no	—
94	C₂×Dic₂₉₃	—	2	4	no	—
95	C₂×Dic₂₉₃	—	2	2	no	—
96	C₂×Dic₂₉₃	—	2	2	no	—
97	C₂×Dic₂₉₃	#41	1	4	yes	—
98	C₂×Dic₂₉₃	C₂⁴	1	2	yes	—
99	C₂×Dic₂₉₃	C₂⁴	1	2	yes	—
100	C₂×Dic₂₉₃	C₂⁴	1	2	yes	—
101	C₂×Dic₂₉₃	C₂⁴	1	4	yes	—
102	C₂×Dic₂₉₃	C₂⁴	1	2	yes	—
103	C₄×D₂₉₃	—	2	16	no	—
104	C₂³×C₂₉₃	C₂⁴	1	1	yes	characteristic
105	Q₈×C₂₉₃	—	2	8	no	—
106	C₂×C₄×C₂₉₃	—	2	4	no	—
107	C₂×C₄×C₂₉₃	—	2	4	no	—
108	C₂×C₄×C₂₉₃	—	2	2	no	—
109	C₂×C₄×C₂₉₃	—	2	2	no	—
110	C₂×C₄×C₂₉₃	#41	1	4	yes	—
111	C₂×C₄×C₂₉₃	C₂⁴	1	2	yes	—
112	C₂×C₄×C₂₉₃	C₂⁴	1	4	yes	—
113	C₂×C₄×C₂₉₃	C₂⁴	1	2	yes	—
114	C₂×C₄×C₂₉₃	C₂⁴	1	4	yes	—
115	C₂×C₄×C₂₉₃	C₂⁴	1	2	yes	—
116	C₂²×C₂₉₃	C₂×#41	1	2	yes	—
117	C₂²×C₂₉₃	C₂×#41	1	2	yes	—
118	C₂²×C₂₉₃	2₋¹⁺⁴	1	2	yes	—
119	C₂²×C₂₉₃	C₂⁵	1	1	yes	characteristic
120	C₂×D₂₉₃	—	2	4	no	—
121	C₂×D₂₉₃	—	2	4	no	—
122	C₂×D₂₉₃	—	2	4	no	—
123	C₂×D₂₉₃	—	4	8	no	—
124	C₄×C₂₉₃	—	2	4	no	—
125	C₄×C₂₉₃	—	2	4	no	—
126	C₄×C₂₉₃	—	2	2	no	—
127	C₄×C₂₉₃	—	2	2	no	—
128	C₄×C₂₉₃	C₂×#41	1	4	yes	—
129	Dic₂₉₃	—	2	2	no	—
130	Dic₂₉₃	—	2	4	no	—
131	Dic₂₉₃	—	2	2	no	—
132	Dic₂₉₃	—	2	2	no	—
133	Dic₂₉₃	C₂×#41	1	4	yes	—
134	C₂×C₂₉₃	#556	1	4	yes	complemented
135	C₂×C₂₉₃	C₂²×#41	1	1	yes	characteristic
136	C₂×C₂₉₃	C₂²×#41	1	1	yes	characteristic
137	C₂×C₂₉₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
138	D₂₉₃	—	4	4	no	—
139	C₂₉₃	C₂×#556	1	1	yes	characteristic, complemented, Sylow
140	C₂×#556	—	293	1	no	maximal, Sylow
141	#556	—	293	16	no	—
142	C₂×C₄⋊Q₈	—	293	1	no	—
143	C₂×#126	—	293	2	no	—
144	C₂×#126	—	293	2	no	—
145	C₂×#124	—	293	1	no	—
146	C₂×#124	—	293	1	no	—
147	C₂×C₂²⋊Q₈	—	293	2	no	—
148	C₂×C₄×Q₈	—	293	1	no	—
149	C₂×C₄×Q₈	—	293	1	no	—
150	C₂×C₄×Q₈	D₂₉₃	1	1	yes	characteristic, complemented
151	C₂×C₄×D₄	—	293	1	no	—
152	C₂×#117	—	293	2	no	—
153	C₂²×Q₈	—	293	1	no	—
154	C₂²×Q₈	—	293	1	no	—
155	C₂²×Q₈	C₂×D₂₉₃	1	1	yes	characteristic
156	C₂²×D₄	—	293	1	no	—
157	C₂³×C₄	—	293	2	no	—
158	C₄⋊Q₈	—	293	8	no	—
159	#126	—	293	16	no	—
160	#126	—	293	16	no	—
161	#124	—	293	8	no	—
162	#124	—	293	8	no	—
163	C₂²⋊Q₈	—	293	16	no	—
164	C₄×Q₈	—	293	8	no	—
165	C₄×Q₈	—	293	8	no	—
166	C₄×Q₈	C₂×D₂₉₃	1	8	yes	complemented
167	C₄×D₄	—	293	8	no	—
168	#117	—	293	16	no	—
169	C₂×C₄⋊C₄	—	293	1	no	—
170	C₂×C₄⋊C₄	—	293	2	no	—
171	C₂×C₄⋊C₄	—	293	2	no	—
172	C₂×C₄⋊C₄	—	293	1	no	—
173	C₂×C₄⋊C₄	—	293	1	no	—
174	C₂×C₄⋊C₄	—	293	2	no	—
175	C₂×C₄⋊C₄	C₂×D₂₉₃	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂×D₂₉₃	1	2	yes	—
177	C₂×C₂²⋊C₄	—	293	2	no	—
178	C₂×C₂²⋊C₄	—	293	2	no	—
179	C₂×C₂²⋊C₄	—	293	2	no	—
180	C₂×C₂²⋊C₄	—	293	2	no	—
181	C₂×C₂²⋊C₄	—	293	2	no	—
182	C₂×C₄²	—	293	2	no	—
183	C₂×C₄²	—	293	1	no	—
184	C₂×C₄²	—	293	1	no	—
185	C₂×C₄²	C₂×D₂₉₃	1	2	yes	—
186	C₂×C₄²	C₂×D₂₉₃	1	1	yes	characteristic
187	C₂⁴	—	293	2	no	—
188	C₂×Q₈	—	2	4	no	—
189	C₂×Q₈	—	293	4	no	—
190	C₂×Q₈	—	293	4	no	—
191	C₂×Q₈	—	586	4	no	—
192	C₂×Q₈	—	586	4	no	—
193	C₂×Q₈	C₂²×D₂₉₃	1	4	yes	—
194	C₂×D₄	—	293	4	no	—
195	C₂×D₄	—	586	4	no	—
196	C₂²×C₄	—	293	1	no	—
197	C₂²×C₄	—	293	1	no	—
198	C₂²×C₄	—	293	1	no	—
199	C₂²×C₄	—	293	1	no	—
200	C₂²×C₄	—	293	2	no	—
201	C₂²×C₄	—	293	8	no	—
202	C₂²×C₄	—	586	8	no	—
203	C₂²×C₄	C₂²×D₂₉₃	1	2	yes	—
204	C₂²×C₄	C₂²×D₂₉₃	1	1	yes	characteristic
205	C₂²×C₄	C₂²×D₂₉₃	1	2	yes	—
206	C₂²×C₄	C₂²×D₂₉₃	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₂₉₃	1	1	yes	characteristic
208	C₄⋊C₄	—	293	4	no	—
209	C₄⋊C₄	—	293	8	no	—
210	C₄⋊C₄	—	293	8	no	—
211	C₄⋊C₄	—	293	4	no	—
212	C₄⋊C₄	—	293	4	no	—
213	C₄⋊C₄	—	293	8	no	—
214	C₄⋊C₄	C₂²×D₂₉₃	1	4	yes	—
215	C₄⋊C₄	C₂²×D₂₉₃	1	8	yes	—
216	C₂²⋊C₄	—	293	8	no	—
217	C₂²⋊C₄	—	293	8	no	—
218	C₂²⋊C₄	—	293	8	no	—
219	C₂²⋊C₄	—	293	8	no	—
220	C₂²⋊C₄	—	293	8	no	—
221	C₄²	—	293	8	no	—
222	C₄²	—	293	4	no	—
223	C₄²	—	293	4	no	—
224	C₄²	C₂²×D₂₉₃	1	8	yes	—
225	C₄²	C₂²×D₂₉₃	1	4	yes	—
226	C₂³	—	293	4	no	—
227	C₂³	—	586	4	no	—
228	C₂³	—	586	4	no	—
229	C₂³	—	586	4	no	—
230	C₂³	C₂³×D₂₉₃	1	1	yes	characteristic
231	Q₈	—	2	8	no	—
232	Q₈	—	586	8	no	—
233	Q₈	—	586	8	no	—
234	D₄	—	586	8	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	293	4	no	—
240	C₂×C₄	—	293	2	no	—
241	C₂×C₄	—	293	2	no	—
242	C₂×C₄	—	293	2	no	—
243	C₂×C₄	—	293	2	no	—
244	C₂×C₄	—	293	4	no	—
245	C₂×C₄	—	586	16	no	—
246	C₂×C₄	—	586	2	no	—
247	C₂×C₄	—	586	4	no	—
248	C₂×C₄	—	586	2	no	—
249	C₂×C₄	—	586	2	no	—
250	C₂×C₄	#17989802	1	4	yes	—
251	C₂×C₄	C₂³×D₂₉₃	1	4	yes	—
252	C₂×C₄	C₂³×D₂₉₃	1	2	yes	—
253	C₂×C₄	C₂³×D₂₉₃	1	4	yes	—
254	C₂×C₄	C₂³×D₂₉₃	1	2	yes	—
255	C₂×C₄	C₂³×D₂₉₃	1	2	yes	—
256	C₂²	—	586	4	no	—
257	C₂²	—	586	4	no	—
258	C₂²	—	586	4	no	—
259	C₂²	—	1172	8	no	—
260	C₂²	#41×D₂₉₃	1	2	yes	—
261	C₂²	C₂×#17989802	1	2	yes	—
262	C₂²	#22627672	1	2	yes	—
263	C₂²	C₂⁴×D₂₉₃	1	1	yes	characteristic
264	C₄	—	2	4	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	4	no	—
267	C₄	—	2	2	no	—
268	C₄	—	293	4	no	—
269	C₄	—	586	2	no	—
270	C₄	—	586	4	no	—
271	C₄	—	586	2	no	—
272	C₄	—	586	2	no	—
273	C₄	C₂×#17989802	1	4	yes	—
274	C₂	—	1172	4	no	—
275	C₂	#34036851	1	4	yes	direct product factor
276	C₂	C₂×#41×D₂₉₃	1	1	yes	characteristic
277	C₂	C₂²×#17989802	1	1	yes	characteristic
278	C₂	C₂×#22627672	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₂₉₃
commutator, cosocle	C₂²×C₂₉₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₉₃
Fitting	C₂×C₄×Q₈×C₂₉₃	C₂
socle	C₂³×C₂₉₃	C₂⁴
prime radical	C₂²×D₁₁₇₂	C₂²
hypercenter	C₂×C₄×Q₈	D₂₉₃
hypocenter	C₂₉₃	C₂×#556
LCS	G ▷ C₂²×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #34036851	C₂	direct (N × H)
2. C₂×#22627622	C₂	semidirect
3. C₂×#22627619	C₂	semidirect
4. C₂×#22627618	C₂	semidirect
7. C₂×#22627602	C₂	semidirect
8. C₂×C₄×Dic₅₈₆	C₂	semidirect
9. C₂×#22627589	C₂	semidirect
11. C₂×#22627581	C₂	semidirect
12. C₂×C₄×Q₈×C₂₉₃	C₂	semidirect
17. #22627622	C₂²	semidirect
18. #22627619	C₂²	semidirect
19. #22627618	C₂²	semidirect
22. #22627602	C₂²	semidirect
23. C₄×Dic₅₈₆	C₂²	semidirect
24. #22627589	C₂²	semidirect
26. #22627581	C₂²	semidirect
42. C₄×Q₈×C₂₉₃	C₂²	semidirect
134. C₂×C₂₉₃	#556	semidirect
139. C₂₉₃	C₂×#556	semidirect
150. C₂×C₄×Q₈	D₂₉₃	semidirect
166. C₄×Q₈	C₂×D₂₉₃	semidirect
275. C₂	#34036851	direct (N × H)

Maximal subgroups

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

subgroup	index
#34036851	2
C₂×#22627622	2
C₂×#22627619	2
C₂×#22627618	2
C₂×#22627613	2
C₂×#22627612	2
C₂×#22627602	2
C₂×C₄×Dic₅₈₆	2
C₂×#22627589	2
C₂×#22627586	2
C₂×#22627581	2
C₂×C₄×Q₈×C₂₉₃	2
C₂×#556	293

Subgroups — C2×#34036851 #49172333

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#34036851 #49172333