Subgroups — C₂×#22558159 #30765015

Order 17152.

410 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863419	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863178	C₂	1	1	yes	maximal, characteristic
3	#22558159	C₂	1	4	yes	maximal, direct product factor
4	C₂×#349×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761612	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₆₇	C₄	1	1	yes	characteristic
7	C₂×#14523983	—	2	1	no	—
8	#16863419	—	2	1	no	—
9	#16863419	—	2	1	no	—
10	#16863419	C₄	1	2	yes	—
11	#16863419	C₂²	1	2	yes	—
12	C₂×Dic₆₇⋊C₈	—	2	1	no	—
13	C₂×Dic₆₇⋊C₈	—	2	1	no	—
14	#16863178	—	2	1	no	—
15	#16863178	C₂²	1	2	yes	—
16	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×M₃₂×C₆₇	—	2	1	no	—
18	#349×C₆₇	—	2	1	no	—
19	#349×C₆₇	C₂²	1	2	yes	—
20	C₂³×Dic₆₇	—	2	1	no	—
21	C₂×#11761612	—	2	1	no	—
22	C₂×#11761612	—	4	1	no	—
23	C₂×#11761612	—	4	1	no	—
24	C₂×#11761612	C₂×C₄	1	2	yes	—
25	C₂²×C₆₇⋊C₈	—	2	1	no	—
26	C₂×C₄×Dic₆₇	—	2	1	no	—
27	C₂×C₄×Dic₆₇	—	2	1	no	—
28	C₂×C₄×Dic₆₇	—	2	1	no	—
29	C₂×C₄×Dic₆₇	C₈	1	1	yes	characteristic
30	C₂×C₄×Dic₆₇	C₈	1	1	yes	characteristic
31	C₂×C₄×Dic₆₇	C₈	1	2	yes	—
32	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
33	#14523983	—	2	2	no	—
34	#14523983	—	4	1	no	—
35	C₂×C₆₇⋊C₁₆	—	4	1	no	—
36	Dic₆₇⋊C₈	—	2	2	no	—
37	Dic₆₇⋊C₈	—	2	2	no	—
38	Dic₆₇⋊C₈	—	2	2	no	—
39	Dic₆₇⋊C₈	—	2	2	no	—
40	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂×M₁₆×C₆₇	—	4	1	no	—
42	C₂×M₁₆×C₆₇	—	4	1	no	—
43	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
45	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
46	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
49	M₃₂×C₆₇	—	2	2	no	—
50	M₃₂×C₆₇	—	4	1	no	—
51	C₂×C₁₆×C₆₇	—	4	1	no	—
52	C₂²×Dic₆₇	—	2	1	no	—
53	C₂²×Dic₆₇	—	2	2	no	—
54	C₂²×Dic₆₇	—	2	1	no	—
55	C₂²×Dic₆₇	—	2	2	no	—
56	C₂²×Dic₆₇	—	4	4	no	—
57	#11761612	—	4	2	no	—
58	#11761612	—	4	2	no	—
59	C₂×C₆₇⋊C₈	—	2	2	no	—
60	C₂×C₆₇⋊C₈	—	4	1	no	—
61	C₂×C₆₇⋊C₈	—	4	1	no	—
62	C₄×Dic₆₇	—	2	1	no	—
63	C₄×Dic₆₇	—	2	2	no	—
64	C₄×Dic₆₇	—	2	1	no	—
65	C₄×Dic₆₇	—	2	2	no	—
66	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
67	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
68	C₆₇⋊C₁₆	—	4	2	no	—
69	C₂⁴×C₆₇	M₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	—	2	1	no	—
71	C₂²×C₄×C₆₇	—	2	1	no	—
72	C₂²×C₄×C₆₇	—	2	1	no	—
73	C₂²×C₄×C₆₇	—	2	1	no	—
74	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	M₁₆×C₆₇	—	4	1	no	—
81	M₁₆×C₆₇	—	4	1	no	—
82	M₁₆×C₆₇	—	4	1	no	—
83	M₁₆×C₆₇	—	4	1	no	—
84	C₂×C₈×C₆₇	—	2	1	no	—
85	C₂×C₈×C₆₇	—	2	1	no	—
86	C₂×C₈×C₆₇	—	2	1	no	—
87	C₂×C₈×C₆₇	—	2	1	no	—
88	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₁₆×C₆₇	—	4	2	no	—
93	C₂³×C₆₇	—	2	1	no	—
94	C₂³×C₆₇	—	2	1	no	—
95	C₂³×C₆₇	—	4	2	no	—
96	C₂³×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
97	C₂³×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
98	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₆₇	—	2	1	no	—
100	C₂×C₄×C₆₇	—	2	1	no	—
101	C₂×C₄×C₆₇	—	2	1	no	—
102	C₂×C₄×C₆₇	—	2	1	no	—
103	C₂×C₄×C₆₇	—	2	1	no	—
104	C₂×C₄×C₆₇	—	2	1	no	—
105	C₂×C₄×C₆₇	—	2	1	no	—
106	C₂×C₄×C₆₇	—	2	1	no	—
107	C₂×C₄×C₆₇	—	2	1	no	—
108	C₂×C₄×C₆₇	—	2	1	no	—
109	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
111	C₂×C₄×C₆₇	#99	1	1	yes	characteristic
112	C₂×C₄×C₆₇	#99	1	1	yes	characteristic
113	C₂×C₄×C₆₇	#101	1	1	yes	characteristic
114	C₂×C₄×C₆₇	#101	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
117	C₂×Dic₆₇	—	2	4	no	—
118	C₂×Dic₆₇	—	2	2	no	—
119	C₂×Dic₆₇	—	2	2	no	—
120	C₂×Dic₆₇	—	4	4	no	—
121	C₂×Dic₆₇	—	4	4	no	—
122	C₂×Dic₆₇	—	4	2	no	—
123	C₈×C₆₇	—	2	1	no	—
124	C₈×C₆₇	—	2	1	no	—
125	C₈×C₆₇	—	2	1	no	—
126	C₈×C₆₇	—	2	1	no	—
127	C₆₇⋊C₈	—	4	2	no	—
128	C₂²×C₆₇	—	2	2	no	—
129	C₂²×C₆₇	—	2	1	no	—
130	C₂²×C₆₇	—	2	1	no	—
131	C₂²×C₆₇	—	4	1	no	—
132	C₂²×C₆₇	—	4	1	no	—
133	C₂²×C₆₇	—	4	2	no	—
134	C₂²×C₆₇	—	4	2	no	—
135	C₂²×C₆₇	#324	1	1	yes	characteristic
136	C₂²×C₆₇	#324	1	1	yes	characteristic
137	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
138	C₄×C₆₇	—	2	1	no	—
139	C₄×C₆₇	—	2	1	no	—
140	C₄×C₆₇	#324	1	1	yes	characteristic
141	C₄×C₆₇	#324	1	1	yes	characteristic
142	C₄×C₆₇	C₂×#99	1	1	yes	characteristic
143	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
144	Dic₆₇	—	2	4	no	—
145	Dic₆₇	—	4	2	no	—
146	C₂×#1327	—	67	1	no	maximal, Sylow
147	C₂×C₆₇	—	2	1	no	—
148	C₂×C₆₇	—	2	1	no	—
149	C₂×C₆₇	—	4	1	no	—
150	C₂×C₆₇	—	4	1	no	—
151	C₂×C₆₇	#1327	1	2	yes	complemented
152	C₂×C₆₇	C₂×#324	1	1	yes	characteristic
153	C₂×#423	—	67	1	no	—
154	C₂×#349	—	67	1	no	—
155	C₂×#349	D₆₇	1	1	yes	characteristic
156	#1327	—	67	4	no	—
157	C₆₇	C₂×#1327	1	1	yes	characteristic, complemented, Sylow
158	C₂²×M₁₆	—	67	1	no	—
159	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
160	C₂²×C₄²	—	67	1	no	—
161	C₂×M₃₂	—	2	1	no	—
162	C₂×M₃₂	—	134	1	no	—
163	#423	—	67	2	no	—
164	#423	—	67	2	no	—
165	#423	—	134	1	no	—
166	#423	—	134	1	no	—
167	C₂×C₄⋊C₈	—	134	1	no	—
168	C₂×C₄⋊C₈	—	134	1	no	—
169	#349	—	2	1	no	—
170	#349	—	67	2	no	—
171	#349	—	134	1	no	—
172	#349	C₂×D₆₇	1	2	yes	—
173	C₂³×C₄	—	134	1	no	—
174	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
175	C₂×M₁₆	—	4	1	no	—
176	C₂×M₁₆	—	4	1	no	—
177	C₂×M₁₆	—	268	1	no	—
178	C₂×M₁₆	—	268	1	no	—
179	C₂×M₁₆	—	67	2	no	—
180	C₂×M₁₆	—	134	1	no	—
181	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
182	C₂×M₁₆	D₂₆₈	1	1	yes	characteristic
183	C₂×M₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
184	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
185	C₂²×C₈	—	134	1	no	—
186	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
187	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
188	C₂×C₄²	—	67	1	no	—
189	C₂×C₄²	—	67	2	no	—
190	C₂×C₄²	—	67	1	no	—
191	C₂×C₄²	—	67	2	no	—
192	C₂×C₄²	—	134	1	no	—
193	C₂×C₄²	—	134	1	no	—
194	C₂×C₄²	—	134	1	no	—
195	M₃₂	—	2	2	no	—
196	M₃₂	—	4	1	no	—
197	M₃₂	—	268	1	no	—
198	M₃₂	—	134	2	no	—
199	C₂×C₁₆	—	4	1	no	—
200	C₂×C₁₆	—	268	1	no	—
201	C₄⋊C₈	—	134	2	no	—
202	C₄⋊C₈	—	134	2	no	—
203	C₄⋊C₈	—	134	2	no	—
204	C₄⋊C₈	—	134	2	no	—
205	C₂⁴	#11761611	1	1	yes	characteristic
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	268	4	no	—
211	C₂²×C₄	—	134	1	no	—
212	C₂²×C₄	—	134	2	no	—
213	C₂²×C₄	—	134	2	no	—
214	C₂²×C₄	—	134	1	no	—
215	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
216	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
217	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
218	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
219	C₂²×C₄	#11761611	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
221	M₁₆	—	4	1	no	—
222	M₁₆	—	4	1	no	—
223	M₁₆	—	4	1	no	—
224	M₁₆	—	4	1	no	—
225	M₁₆	—	268	2	no	—
226	M₁₆	—	268	2	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	268	1	no	—
232	C₂×C₈	—	268	1	no	—
233	C₂×C₈	—	134	2	no	—
234	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
235	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
236	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
237	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
238	C₄²	—	67	2	no	—
239	C₄²	—	67	2	no	—
240	C₄²	—	134	1	no	—
241	C₄²	—	134	2	no	—
242	C₄²	—	134	1	no	—
243	C₄²	—	134	2	no	—
244	C₁₆	—	4	2	no	—
245	C₁₆	—	268	2	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	4	2	no	—
249	C₂³	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
250	C₂³	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
251	C₂³	C₂×#11761611	1	1	yes	characteristic
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	268	2	no	—
263	C₂×C₄	—	268	4	no	—
264	C₂×C₄	—	268	4	no	—
265	C₂×C₄	—	134	2	no	—
266	C₂×C₄	—	134	4	no	—
267	C₂×C₄	—	134	2	no	—
268	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
269	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
270	C₂×C₄	#14523957	1	1	yes	characteristic
271	C₂×C₄	#14523957	1	1	yes	characteristic
272	C₂×C₄	#14523961	1	1	yes	characteristic
273	C₂×C₄	#14523961	1	1	yes	characteristic
274	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
275	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	2	1	no	—
280	C₈	—	268	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	#16863121	1	1	yes	characteristic
289	C₂²	#16863121	1	1	yes	characteristic
290	C₂²	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	268	2	no	—
294	C₄	—	134	4	no	—
295	C₄	#16863121	1	1	yes	characteristic
296	C₄	#16863121	1	1	yes	characteristic
297	C₄	C₂×#14523957	1	1	yes	characteristic
298	C₄	C₂×#14523961	1	1	yes	characteristic
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	4	1	no	—
302	C₂	—	4	1	no	—
303	C₂	#22558159	1	2	yes	direct product factor
304	C₂	C₂×#16863121	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14523957
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×M₁₆	C₂²×D₆₇
Fitting	C₂×#349×C₆₇	C₂
socle	C₂²×C₆₇	#324
prime radical	C₂⁴×C₆₇	M₁₆
cosocle	C₂×M₁₆×C₆₇	C₂³
hypercenter	C₂×#349	D₆₇
hypocenter	C₆₇	C₂×#1327
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ C₂×#349

Complements

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

N	H ≅ G/N	type
3. #22558159	C₂	direct (N × H)
151. C₂×C₆₇	#1327	semidirect
157. C₆₇	C₂×#1327	semidirect
303. C₂	#22558159	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16863419	2
C₂×#16863178	2
#22558159	2
C₂×#349×C₆₇	2
C₂×#1327	67

Subgroups — C2×#22558159 #30765015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558159 #30765015