Subgroups — C₂×#93995 #72319941

Order 1024.

665 conjugacy classes of subgroups.

The subgroup lattice has 329 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#1398	C₂	1	2	yes	maximal
2	C₂×#11405	C₂	1	1	yes	maximal, characteristic
3	#93995	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₆₄	C₄	1	4	yes	—
5	C₂×C₄×M₃₂	C₄	1	1	yes	characteristic
6	C₂²×#346	C₂²	1	1	yes	characteristic
7	C₂×#1450	C₄	1	1	yes	characteristic
8	C₂×#1398	—	2	2	no	—
9	C₂×#1398	—	2	2	no	—
10	C₂×#1398	—	2	2	no	—
11	C₂×#1398	—	2	2	no	—
12	C₂×#1398	C₄	1	4	yes	—
13	C₂×#1398	C₂²	1	4	yes	—
14	#11405	—	2	2	no	—
15	#11405	C₄	1	2	yes	—
16	#11405	C₂²	1	2	yes	—
17	C₂×#1313	—	2	2	no	—
18	C₂²×M₃₂	—	2	1	no	—
19	C₂³×C₁₆	C₂×C₄	1	2	yes	—
20	C₂×#431	—	2	1	no	—
21	C₂²×C₄×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×M₆₄	—	2	4	no	—
23	C₂×M₆₄	—	2	4	no	—
24	C₂×M₆₄	—	2	4	no	—
25	C₂×M₆₄	—	2	4	no	—
26	C₂×M₆₄	C₈	1	8	yes	complemented
27	C₂×M₆₄	C₂×C₄	1	8	yes	—
28	C₂²×C₃₂	C₈	1	8	yes	complemented
29	C₄×M₃₂	—	2	1	no	—
30	C₄×M₃₂	—	2	1	no	—
31	C₄×M₃₂	C₈	1	2	yes	—
32	C₄×M₃₂	C₂×C₄	1	2	yes	—
33	C₂×#346	—	2	1	no	—
34	C₂×#346	C₂×C₄	1	2	yes	—
35	C₂×#346	C₂×C₄	1	1	yes	characteristic
36	C₂×#346	D₄	1	2	yes	—
37	C₂×#346	D₄	1	2	yes	—
38	C₂×#346	D₄	1	1	yes	characteristic
39	C₂×#346	D₄	1	1	yes	characteristic
40	C₂×#346	Q₈	1	1	yes	characteristic
41	C₂×#346	Q₈	1	1	yes	characteristic
42	C₂×#346	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₁₆	C₈	1	2	yes	—
44	#1450	—	2	1	no	—
45	#1450	—	2	1	no	—
46	#1450	C₈	1	2	yes	—
47	#1450	C₂×C₄	1	2	yes	—
48	C₂×#322	—	2	1	no	—
49	C₂×C₈²	C₈	1	2	yes	—
50	#1398	—	2	4	no	—
51	#1398	—	2	4	no	—
52	#1398	—	2	4	no	—
53	#1398	—	2	4	no	—
54	#1313	—	2	4	no	—
55	#1313	—	2	4	no	—
56	C₂³×C₈	C₄²	1	1	yes	characteristic
57	C₂³×C₈	M₁₆	1	1	yes	characteristic
58	C₂²×C₄²	M₁₆	1	1	yes	characteristic
59	C₂×M₃₂	—	2	2	no	—
60	C₂×M₃₂	—	2	2	no	—
61	C₂×M₃₂	—	4	2	no	—
62	C₂×M₃₂	—	4	2	no	—
63	C₂²×C₁₆	—	2	2	no	—
64	C₂²×C₁₆	C₄²	1	2	yes	—
65	C₂²×C₁₆	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₁₆	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₁₆	C₄⋊C₄	1	2	yes	—
68	C₂²×C₁₆	C₄⋊C₄	1	2	yes	—
69	C₂²×C₁₆	C₂×C₈	1	4	yes	—
70	C₂²×C₁₆	C₂×C₈	1	4	yes	—
71	C₂²×C₁₆	M₁₆	1	4	yes	—
72	C₂²×C₁₆	M₁₆	1	4	yes	—
73	C₂²×C₁₆	C₂²×C₄	1	2	yes	—
74	#431	—	2	2	no	—
75	#431	—	2	2	no	—
76	#431	—	4	1	no	—
77	#431	—	4	1	no	—
78	C₂×C₄⋊C₈	—	4	1	no	—
79	C₂×C₂²⋊C₈	—	4	1	no	—
80	C₂×C₄×C₈	—	2	2	no	—
81	C₂×C₄×C₈	C₄²	1	1	yes	characteristic
82	C₂×C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₈	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄×C₈	C₂×C₈	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂×C₈	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂×C₈	1	1	yes	characteristic
90	C₂×C₄×C₈	M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₈	M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₈	C₂²×C₄	1	1	yes	characteristic
93	M₆₄	—	2	8	no	—
94	M₆₄	—	2	8	no	—
95	M₆₄	—	2	8	no	—
96	M₆₄	—	2	8	no	—
97	C₂×C₃₂	—	2	8	no	—
98	C₂×C₃₂	—	2	8	no	—
99	C₂×C₃₂	C₂×C₈	1	16	yes	complemented
100	#346	—	2	2	no	—
101	#346	—	2	2	no	—
102	#346	C₂²⋊C₄	1	2	yes	—
103	#346	C₂²⋊C₄	1	2	yes	—
104	#346	C₂²⋊C₄	1	2	yes	—
105	#346	C₄⋊C₄	1	2	yes	—
106	#346	C₄⋊C₄	1	2	yes	—
107	#346	C₄⋊C₄	1	1	yes	characteristic
108	#346	C₄⋊C₄	1	1	yes	characteristic
109	#346	C₂×D₄	1	2	yes	—
110	#346	C₂×D₄	1	1	yes	characteristic
111	#346	C₂×Q₈	1	1	yes	characteristic
112	C₄×C₁₆	—	2	2	no	—
113	C₄×C₁₆	C₂×C₈	1	4	yes	—
114	#322	—	2	2	no	—
115	#322	—	2	2	no	—
116	C₈²	—	2	2	no	—
117	C₈²	C₂×C₈	1	4	yes	—
118	C₂³×C₄	—	2	1	no	—
119	C₂³×C₄	#97	1	1	yes	characteristic
120	C₂²×C₈	—	2	2	no	—
121	C₂²×C₈	—	2	2	no	—
122	C₂²×C₈	—	4	2	no	—
123	C₂²×C₈	#95	1	1	yes	characteristic
124	C₂²×C₈	#95	1	1	yes	characteristic
125	C₂²×C₈	C₄×C₈	1	1	yes	characteristic
126	C₂²×C₈	C₄×C₈	1	2	yes	—
127	C₂²×C₈	C₄×C₈	1	1	yes	characteristic
128	C₂²×C₈	C₄×C₈	1	2	yes	—
129	C₂²×C₈	#97	1	1	yes	characteristic
130	C₂²×C₈	#97	1	1	yes	characteristic
131	C₂²×C₈	#97	1	2	yes	—
132	C₂²×C₈	#97	1	2	yes	—
133	C₂²×C₈	C₂²⋊C₈	1	1	yes	characteristic
134	C₂²×C₈	C₂²⋊C₈	1	1	yes	characteristic
135	C₂²×C₈	C₄⋊C₈	1	1	yes	characteristic
136	C₂²×C₈	C₄⋊C₈	1	1	yes	characteristic
137	C₂²×C₈	C₂×C₄²	1	1	yes	characteristic
138	C₂²×C₈	C₂×M₁₆	1	1	yes	characteristic
139	C₂×C₄²	—	2	1	no	—
140	C₂×C₄²	—	2	1	no	—
141	C₂×C₄²	—	2	1	no	—
142	C₂×C₄²	#97	1	1	yes	characteristic
143	C₂×C₄²	C₂²⋊C₈	1	1	yes	characteristic
144	C₂×C₄²	C₂²⋊C₈	1	1	yes	characteristic
145	C₂×C₄²	C₄⋊C₈	1	1	yes	characteristic
146	C₂×C₄²	C₄⋊C₈	1	1	yes	characteristic
147	C₂×C₄²	C₂×M₁₆	1	1	yes	characteristic
148	M₃₂	—	4	4	no	—
149	M₃₂	—	4	4	no	—
150	C₂×C₁₆	—	2	4	no	—
151	C₂×C₁₆	—	4	2	no	—
152	C₂×C₁₆	—	4	2	no	—
153	C₂×C₁₆	#95	1	2	yes	—
154	C₂×C₁₆	#95	1	2	yes	—
155	C₂×C₁₆	C₄×C₈	1	4	yes	—
156	C₂×C₁₆	#97	1	4	yes	—
157	C₂×C₁₆	C₂²⋊C₈	1	4	yes	—
158	C₂×C₁₆	C₂²⋊C₈	1	4	yes	—
159	C₂×C₁₆	C₂²⋊C₈	1	4	yes	—
160	C₂×C₁₆	C₂²⋊C₈	1	4	yes	—
161	C₂×C₁₆	C₄⋊C₈	1	4	yes	—
162	C₂×C₁₆	C₄⋊C₈	1	4	yes	—
163	C₂×C₁₆	C₄⋊C₈	1	4	yes	—
164	C₂×C₁₆	C₄⋊C₈	1	4	yes	—
165	C₂×C₁₆	C₂×C₂²⋊C₄	1	2	yes	—
166	C₂×C₁₆	C₂×C₄⋊C₄	1	2	yes	—
167	C₂×C₁₆	C₂²×C₈	1	4	yes	—
168	C₂×C₁₆	C₂×M₁₆	1	4	yes	—
169	C₄⋊C₈	—	4	2	no	—
170	C₄⋊C₈	—	4	2	no	—
171	C₂²⋊C₈	—	4	2	no	—
172	C₂²⋊C₈	—	4	2	no	—
173	C₄×C₈	—	2	4	no	—
174	C₄×C₈	—	4	2	no	—
175	C₄×C₈	#95	1	1	yes	characteristic
176	C₄×C₈	#95	1	1	yes	characteristic
177	C₄×C₈	C₄×C₈	1	1	yes	characteristic
178	C₄×C₈	C₄×C₈	1	1	yes	characteristic
179	C₄×C₈	C₂²⋊C₈	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊C₈	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊C₈	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊C₈	1	1	yes	characteristic
183	C₄×C₈	C₄⋊C₈	1	1	yes	characteristic
184	C₄×C₈	C₄⋊C₈	1	1	yes	characteristic
185	C₄×C₈	C₄⋊C₈	1	1	yes	characteristic
186	C₄×C₈	C₄⋊C₈	1	1	yes	characteristic
187	C₄×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
188	C₄×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
189	C₄×C₈	C₂²×C₈	1	1	yes	characteristic
190	C₄×C₈	C₂²×C₈	1	1	yes	characteristic
191	C₃₂	—	2	16	no	—
192	C₂⁴	#347	1	1	yes	characteristic
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	8	2	no	—
200	C₂²×C₄	#322	1	2	yes	—
201	C₂²×C₄	#322	1	2	yes	—
202	C₂²×C₄	#336	1	1	yes	characteristic
203	C₂²×C₄	#336	1	1	yes	characteristic
204	C₂²×C₄	#347	1	1	yes	characteristic
205	C₂²×C₄	#347	1	1	yes	characteristic
206	C₂²×C₄	#347	1	1	yes	characteristic
207	C₂²×C₄	#347	1	2	yes	—
208	C₂²×C₄	#347	1	2	yes	—
209	C₂²×C₄	C₂×#97	1	1	yes	characteristic
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	4	4	no	—
219	C₂×C₈	—	8	4	no	—
220	C₂×C₈	C₈²	1	2	yes	—
221	C₂×C₈	C₈²	1	1	yes	characteristic
222	C₂×C₈	C₈²	1	1	yes	characteristic
223	C₂×C₈	#322	1	2	yes	—
224	C₂×C₈	#322	1	1	yes	characteristic
225	C₂×C₈	#322	1	1	yes	characteristic
226	C₂×C₈	#322	1	2	yes	—
227	C₂×C₈	#322	1	2	yes	—
228	C₂×C₈	#336	1	1	yes	characteristic
229	C₂×C₈	#336	1	1	yes	characteristic
230	C₂×C₈	#336	1	2	yes	—
231	C₂×C₈	#336	1	2	yes	—
232	C₂×C₈	#336	1	2	yes	—
233	C₂×C₈	#336	1	2	yes	—
234	C₂×C₈	#336	1	1	yes	characteristic
235	C₂×C₈	#336	1	1	yes	characteristic
236	C₂×C₈	C₂×#95	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄×C₈	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄×C₈	1	2	yes	—
239	C₂×C₈	C₂×#97	1	2	yes	—
240	C₂×C₈	C₂×C₂²⋊C₈	1	1	yes	characteristic
241	C₂×C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
242	C₄²	—	2	1	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	1	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	#336	1	1	yes	characteristic
249	C₄²	#336	1	1	yes	characteristic
250	C₄²	C₂×C₂²⋊C₈	1	1	yes	characteristic
251	C₄²	C₂×C₄⋊C₈	1	1	yes	characteristic
252	C₁₆	—	4	4	no	—
253	C₁₆	#336	1	4	yes	—
254	C₁₆	#336	1	4	yes	—
255	C₁₆	C₂×C₂²⋊C₈	1	4	yes	—
256	C₁₆	C₂×C₄⋊C₈	1	4	yes	—
257	C₂³	—	4	2	no	—
258	C₂³	#1313	1	2	yes	—
259	C₂³	#1313	1	2	yes	—
260	C₂³	#1375	1	1	yes	characteristic
261	C₂³	#1375	1	1	yes	characteristic
262	C₂³	C₂×#347	1	1	yes	characteristic
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	8	1	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	#1273	1	2	yes	—
274	C₂×C₄	#1273	1	1	yes	characteristic
275	C₂×C₄	#1273	1	1	yes	characteristic
276	C₂×C₄	#1313	1	2	yes	—
277	C₂×C₄	#1313	1	2	yes	—
278	C₂×C₄	#1313	1	2	yes	—
279	C₂×C₄	#1313	1	2	yes	—
280	C₂×C₄	#1313	1	2	yes	—
281	C₂×C₄	#1313	1	2	yes	—
282	C₂×C₄	#1375	1	1	yes	characteristic
283	C₂×C₄	#1375	1	2	yes	—
284	C₂×C₄	#1375	1	1	yes	characteristic
285	C₂×C₄	#1375	1	1	yes	characteristic
286	C₂×C₄	#1375	1	1	yes	characteristic
287	C₂×C₄	C₂×#322	1	2	yes	—
288	C₂×C₄	C₂×#336	1	1	yes	characteristic
289	C₂×C₄	C₂×#347	1	2	yes	—
290	C₂×C₄	C₂×#347	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	—	8	4	no	—
296	C₈	#1273	1	2	yes	—
297	C₈	#1273	1	1	yes	characteristic
298	C₈	#1273	1	1	yes	characteristic
299	C₈	C₂×C₈²	1	1	yes	characteristic
300	C₈	C₂×#322	1	1	yes	characteristic
301	C₈	C₂×#336	1	2	yes	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	4	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	2	no	—
307	C₂²	#7137	1	2	yes	—
308	C₂²	#7137	1	1	yes	characteristic
309	C₂²	#7137	1	1	yes	characteristic
310	C₂²	C₂×#1313	1	2	yes	—
311	C₂²	C₂×#1375	1	1	yes	characteristic
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	#7137	1	1	yes	characteristic
317	C₄	#7137	1	2	yes	—
318	C₄	#7137	1	1	yes	characteristic
319	C₄	C₂×#1273	1	1	yes	characteristic
320	C₄	C₂×#1313	1	2	yes	—
321	C₄	C₂×#1375	1	1	yes	characteristic
322	C₂	—	2	1	no	—
323	C₂	—	2	2	no	—
324	C₂	—	2	1	no	—
325	C₂	—	2	2	no	—
326	C₂	#93995	1	2	yes	direct product factor
327	C₂	C₂×#7137	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₈²
commutator	C₈	C₂×C₈²
Frattini, cosocle	C₂×#346	C₂³
socle	C₂²	#7137
prime radical	C₂⁴	#347
LCS	G ▷ C₈ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #93995	C₂	direct (N × H)
26. C₂×M₆₄	C₈	semidirect
28. C₂²×C₃₂	C₈	semidirect
99. C₂×C₃₂	C₂×C₈	semidirect
326. C₂	#93995	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×#1398	2
C₂×#11405	2
#93995	2

Subgroups — C2×#93995 #72319941

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#93995 #72319941