Subgroups — C₂×#22560358 #30856096

Order 17152.

730 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560358	C₂	1	8	yes	maximal, direct product factor
2	C₂×#16863167	C₂	1	2	yes	maximal
3	C₂×#16863156	C₂	1	2	yes	maximal
4	C₂×C₆₇⋊#338	C₂	1	1	yes	maximal, characteristic
5	C₂×#16863153	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761612	C₂²	1	2	yes	—
8	C₂×#14524002	C₂²	1	2	yes	—
9	#16863167	C₂²	1	8	yes	—
10	#16863156	C₂²	1	8	yes	—
11	C₆₇⋊#338	C₂²	1	4	yes	—
12	#16863153	C₂²	1	4	yes	—
13	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	C₄×M₁₆×C₆₇	C₄	1	4	yes	complemented
17	C₄×M₁₆×C₆₇	C₂²	1	4	yes	—
18	C₂×#97×C₆₇	C₄	1	2	yes	complemented
19	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	complemented
20	C₂×#11761612	—	2	2	no	—
21	C₂×#11761612	—	2	2	no	—
22	C₂×#11761612	—	2	2	no	—
23	C₂×#11761612	—	2	2	no	—
24	C₂×#11761612	—	2	2	no	—
25	C₂×#11761612	C₂³	1	4	yes	—
26	C₂²×C₆₇⋊C₈	—	2	2	no	—
27	#14524002	—	2	2	no	—
28	#14524002	—	2	2	no	—
29	#14524002	—	2	2	no	—
30	#14524002	C₂³	1	4	yes	—
31	C₂×C₄⋊Dic₆₇	—	2	2	no	—
32	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
33	C₂×C₄×Dic₆₇	—	2	2	no	—
34	C₂³×C₄×C₆₇	—	2	1	no	—
35	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₆₇	—	2	2	no	—
37	C₂×M₁₆×C₆₇	—	2	2	no	—
38	C₂×M₁₆×C₆₇	—	2	2	no	—
39	C₂×M₁₆×C₆₇	—	2	2	no	—
40	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
42	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
43	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
44	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₆₇	—	2	1	no	—
47	C₂×C₄²×C₆₇	—	2	1	no	—
48	C₂×C₄²×C₆₇	—	2	1	no	—
49	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
53	#97×C₆₇	C₂×C₄	1	8	yes	complemented
54	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
55	C₂²×Dic₆₇	—	4	2	no	—
56	#11761612	—	2	4	no	—
57	#11761612	—	2	4	no	—
58	#11761612	—	4	2	no	—
59	#11761612	—	4	2	no	—
60	C₂×C₆₇⋊C₈	—	2	4	no	—
61	C₂×C₆₇⋊C₈	—	4	2	no	—
62	C₂×C₆₇⋊C₈	—	4	2	no	—
63	C₄⋊Dic₆₇	—	2	4	no	—
64	C₄⋊Dic₆₇	—	4	2	no	—
65	C₂²⋊Dic₆₇	—	2	4	no	—
66	C₂²⋊Dic₆₇	—	4	2	no	—
67	C₄×Dic₆₇	—	2	4	no	—
68	C₄×Dic₆₇	—	4	2	no	—
69	C₂⁴×C₆₇	#41	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	—	2	1	no	—
71	C₂²×C₄×C₆₇	—	2	2	no	—
72	C₂²×C₄×C₆₇	—	2	1	no	—
73	C₂²×C₄×C₆₇	—	2	2	no	—
74	C₂²×C₄×C₆₇	—	4	4	no	—
75	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	#41	1	2	yes	—
79	C₂²×C₄×C₆₇	#41	1	2	yes	—
80	C₂²×C₄×C₆₇	#41	1	2	yes	—
81	C₂²×C₄×C₆₇	#41	1	2	yes	—
82	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
85	M₁₆×C₆₇	—	2	4	no	—
86	M₁₆×C₆₇	—	2	4	no	—
87	M₁₆×C₆₇	—	2	4	no	—
88	M₁₆×C₆₇	—	2	4	no	—
89	C₂×C₈×C₆₇	—	2	2	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
94	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
95	C₄²×C₆₇	—	2	1	no	—
96	C₄²×C₆₇	—	2	2	no	—
97	C₄²×C₆₇	—	2	1	no	—
98	C₄²×C₆₇	—	2	2	no	—
99	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
100	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
101	C₂³×C₆₇	—	4	2	no	—
102	C₂³×C₆₇	#117	1	1	yes	characteristic
103	C₂³×C₆₇	#117	1	1	yes	characteristic
104	C₂³×C₆₇	#126	1	2	yes	—
105	C₂³×C₆₇	#126	1	2	yes	—
106	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	4	4	no	—
111	C₂×C₄×C₆₇	—	4	4	no	—
112	C₂×C₄×C₆₇	—	4	2	no	—
113	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
114	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
115	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
116	C₂×C₄×C₆₇	#117	1	2	yes	—
117	C₂×C₄×C₆₇	#117	1	2	yes	—
118	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
119	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
120	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
121	C₂×C₄×C₆₇	#126	1	2	yes	—
122	C₂×C₄×C₆₇	#126	1	2	yes	—
123	C₂×C₄×C₆₇	#126	1	2	yes	—
124	C₂×C₄×C₆₇	#126	1	2	yes	—
125	C₂×C₄×C₆₇	#126	1	2	yes	—
126	C₂×C₄×C₆₇	#126	1	2	yes	—
127	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
129	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
130	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
131	C₂×Dic₆₇	—	4	4	no	—
132	C₂×Dic₆₇	—	8	4	no	—
133	C₈×C₆₇	—	2	4	no	—
134	C₈×C₆₇	—	2	4	no	—
135	C₆₇⋊C₈	—	4	4	no	—
136	C₂²×C₆₇	—	2	4	no	—
137	C₂²×C₆₇	—	2	2	no	—
138	C₂²×C₆₇	—	4	1	no	—
139	C₂²×C₆₇	—	4	1	no	—
140	C₂²×C₆₇	—	4	2	no	—
141	C₂²×C₆₇	#383	1	1	yes	characteristic
142	C₂²×C₆₇	#383	1	2	yes	—
143	C₂²×C₆₇	#383	1	1	yes	characteristic
144	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
145	C₂²×C₆₇	C₂×#126	1	2	yes	—
146	C₄×C₆₇	—	2	4	no	—
147	C₄×C₆₇	—	4	2	no	—
148	C₄×C₆₇	#383	1	1	yes	characteristic
149	C₄×C₆₇	#383	1	2	yes	—
150	C₄×C₆₇	#383	1	1	yes	characteristic
151	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
152	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
153	C₄×C₆₇	C₂×#126	1	2	yes	—
154	Dic₆₇	—	8	4	no	—
155	C₂×#1841	—	67	1	no	maximal, Sylow
156	C₂×C₆₇	—	2	2	no	—
157	C₂×C₆₇	—	2	1	no	—
158	C₂×C₆₇	—	2	2	no	—
159	C₂×C₆₇	—	2	1	no	—
160	C₂×C₆₇	#1841	1	2	yes	complemented
161	C₂×C₆₇	C₂×#383	1	1	yes	characteristic
162	C₂×C₄×M₁₆	D₆₇	1	1	yes	characteristic
163	#1841	—	67	8	no	—
164	C₂×#344	—	67	2	no	—
165	C₂×#339	—	67	2	no	—
166	C₂×#338	—	67	1	no	—
167	C₂×H(3,ℤ/4)	—	67	1	no	—
168	C₆₇	C₂×#1841	1	1	yes	characteristic, complemented, Sylow
169	C₂²×M₁₆	—	67	2	no	—
170	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
171	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
172	C₂×#117	—	67	2	no	—
173	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
174	C₄×M₁₆	Dic₆₇	1	4	yes	complemented
175	C₄×M₁₆	C₂×D₆₇	1	4	yes	—
176	C₂×#97	Dic₆₇	1	2	yes	complemented
177	C₂×C₄×C₈	Dic₆₇	1	2	yes	complemented
178	#344	—	67	8	no	—
179	#339	—	67	8	no	—
180	#338	—	67	4	no	—
181	H(3,ℤ/4)	—	67	4	no	—
182	C₂³×C₄	—	2	1	no	—
183	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	67	4	no	—
189	C₂×M₁₆	—	134	2	no	—
190	C₂×M₁₆	—	134	2	no	—
191	C₂×M₁₆	—	134	2	no	—
192	C₂×M₁₆	—	134	2	no	—
193	C₂×M₁₆	—	134	2	no	—
194	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
195	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
196	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
197	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
198	C₂²×C₈	—	134	2	no	—
199	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
200	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
201	#117	—	67	4	no	—
202	#117	—	134	2	no	—
203	#117	—	134	2	no	—
204	#117	—	134	2	no	—
205	C₂×C₄⋊C₄	—	134	2	no	—
206	C₂×C₂²⋊C₄	—	134	2	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	—	2	1	no	—
209	C₂×C₄²	—	2	1	no	—
210	C₂×C₄²	—	134	2	no	—
211	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
212	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
213	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
214	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
215	#97	C₂×Dic₆₇	1	8	yes	complemented
216	C₄×C₈	C₂×Dic₆₇	1	8	yes	complemented
217	C₂⁴	#11761628	1	1	yes	characteristic
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	268	2	no	—
224	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
225	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
226	C₂²×C₄	#11761628	1	1	yes	characteristic
227	C₂²×C₄	#11761628	1	2	yes	—
228	C₂²×C₄	#11761628	1	2	yes	—
229	C₂²×C₄	#11761628	1	2	yes	—
230	C₂²×C₄	#11761628	1	2	yes	—
231	C₂²×C₄	#11761628	1	1	yes	characteristic
232	C₂²×C₄	#11761628	1	1	yes	characteristic
233	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	268	2	no	—
239	M₁₆	—	268	2	no	—
240	M₁₆	—	134	4	no	—
241	M₁₆	—	134	4	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	268	2	no	—
247	C₂×C₈	—	268	2	no	—
248	C₂×C₈	—	134	4	no	—
249	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
250	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
251	C₄⋊C₄	—	268	2	no	—
252	C₄⋊C₄	—	134	4	no	—
253	C₂²⋊C₄	—	268	2	no	—
254	C₂²⋊C₄	—	134	4	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	268	2	no	—
260	C₄²	—	134	4	no	—
261	C₄²	C₂²×Dic₆₇	1	2	yes	—
262	C₄²	C₂²×Dic₆₇	1	2	yes	—
263	C₂³	—	4	2	no	—
264	C₂³	#14524002	1	1	yes	characteristic
265	C₂³	#14524002	1	1	yes	characteristic
266	C₂³	#14524037	1	2	yes	—
267	C₂³	#14524037	1	2	yes	—
268	C₂³	C₂×#11761628	1	1	yes	characteristic
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	268	4	no	—
276	C₂×C₄	—	536	4	no	—
277	C₂×C₄	#14524002	1	1	yes	characteristic
278	C₂×C₄	#14524002	1	1	yes	characteristic
279	C₂×C₄	#14524002	1	1	yes	characteristic
280	C₂×C₄	#14524002	1	1	yes	characteristic
281	C₂×C₄	#14524002	1	1	yes	characteristic
282	C₂×C₄	#14524002	1	2	yes	—
283	C₂×C₄	#14524002	1	2	yes	—
284	C₂×C₄	#14524002	1	1	yes	characteristic
285	C₂×C₄	#14524037	1	2	yes	—
286	C₂×C₄	#14524037	1	2	yes	—
287	C₂×C₄	#14524037	1	2	yes	—
288	C₂×C₄	#14524037	1	2	yes	—
289	C₂×C₄	#14524037	1	2	yes	—
290	C₂×C₄	#14524037	1	2	yes	—
291	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
292	C₂×C₄	C₂×#11761628	1	2	yes	—
293	C₂×C₄	C₂×#11761628	1	2	yes	—
294	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
295	C₈	—	2	4	no	—
296	C₈	—	2	4	no	—
297	C₈	—	268	4	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	2	no	—
303	C₂²	#16863264	1	1	yes	characteristic
304	C₂²	#16863264	1	2	yes	—
305	C₂²	#16863264	1	1	yes	characteristic
306	C₂²	C₂×#14524002	1	1	yes	characteristic
307	C₂²	C₂×#14524037	1	2	yes	—
308	C₄	—	2	4	no	—
309	C₄	—	4	2	no	—
310	C₄	—	536	4	no	—
311	C₄	#16863264	1	1	yes	characteristic
312	C₄	#16863264	1	2	yes	—
313	C₄	#16863264	1	1	yes	characteristic
314	C₄	C₂×#14524002	1	1	yes	characteristic
315	C₄	C₂×#14524002	1	1	yes	characteristic
316	C₄	C₂×#14524037	1	2	yes	—
317	C₂	—	2	1	no	—
318	C₂	—	2	2	no	—
319	C₂	—	2	1	no	—
320	C₂	—	2	2	no	—
321	C₂	#22560358	1	2	yes	direct product factor
322	C₂	C₂×#16863264	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524002
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×C₄×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	#383
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₆₇
hypocenter	C₆₇	C₂×#1841
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
1. #22560358	C₂	direct (N × H)
16. C₄×M₁₆×C₆₇	C₄	semidirect
18. C₂×#97×C₆₇	C₄	semidirect
19. C₂×C₄×C₈×C₆₇	C₄	semidirect
53. #97×C₆₇	C₂×C₄	semidirect
54. C₄×C₈×C₆₇	C₂×C₄	semidirect
160. C₂×C₆₇	#1841	semidirect
168. C₆₇	C₂×#1841	semidirect
174. C₄×M₁₆	Dic₆₇	semidirect
176. C₂×#97	Dic₆₇	semidirect
177. C₂×C₄×C₈	Dic₆₇	semidirect
215. #97	C₂×Dic₆₇	semidirect
216. C₄×C₈	C₂×Dic₆₇	semidirect
321. C₂	#22560358	direct (N × H)

Maximal subgroups

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

subgroup	index
#22560358	2
C₂×#16863167	2
C₂×#16863156	2
C₂×C₆₇⋊#338	2
C₂×#16863153	2
C₂×C₄×M₁₆×C₆₇	2
C₂×#1841	67

Subgroups — C2×#22560358 #30856096

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22560358 #30856096