Subgroups — C₂×#22558308 #30770732

Order 17152.

614 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523979	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14523975	C₂	1	1	yes	maximal, characteristic
3	#22558308	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761612	—	2	1	no	—
6	C₂²×C₄⋊Dic₆₇	—	2	1	no	—
7	C₂×#14523979	—	2	1	no	—
8	C₂×#14523979	—	2	2	no	—
9	C₂×#14523979	C₄	1	4	yes	complemented
10	C₂×#14523979	C₂²	1	2	yes	—
11	C₂×#14523975	—	2	1	no	—
12	C₂×#14523975	—	2	2	no	—
13	C₂×#14523975	C₄	1	4	yes	—
14	C₂×#14523975	C₂²	1	2	yes	—
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×M₃₂×C₆₇	—	2	2	no	—
17	C₂×M₃₂×C₆₇	—	2	2	no	—
18	C₂×M₃₂×C₆₇	C₄	1	2	yes	complemented
19	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₁₆×C₆₇	C₄	1	2	yes	complemented
21	C₂³×Dic₆₇	—	4	1	no	—
22	C₂×#11761612	—	2	4	no	—
23	C₂×#11761612	—	2	2	no	—
24	C₂×#11761612	—	4	2	no	—
25	C₂×#11761612	—	4	1	no	—
26	C₂²×C₆₇⋊C₈	—	4	1	no	—
27	C₂×C₄⋊Dic₆₇	—	2	4	no	—
28	C₂×C₄⋊Dic₆₇	—	2	2	no	—
29	C₂×C₄⋊Dic₆₇	—	4	2	no	—
30	C₂×C₄⋊Dic₆₇	—	4	1	no	—
31	#14523979	—	2	4	no	—
32	#14523979	—	2	2	no	—
33	#14523979	C₂×C₄	1	4	yes	complemented
34	#14523975	—	2	4	no	—
35	#14523975	—	2	2	no	—
36	#14523975	C₂×C₄	1	4	yes	—
37	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	D₄	1	2	yes	—
44	C₂²×C₈×C₆₇	D₄	1	2	yes	—
45	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	M₃₂×C₆₇	—	2	4	no	—
49	M₃₂×C₆₇	—	2	2	no	—
50	M₃₂×C₆₇	—	2	2	no	—
51	C₂×C₁₆×C₆₇	—	2	4	no	—
52	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
53	C₂²×Dic₆₇	—	4	4	no	—
54	C₂²×Dic₆₇	—	4	2	no	—
55	C₂²×Dic₆₇	—	8	4	no	—
56	#11761612	—	2	4	no	—
57	#11761612	—	4	4	no	—
58	#11761612	—	4	2	no	—
59	C₂×C₆₇⋊C₈	—	4	4	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₄⋊Dic₆₇	—	2	4	no	—
62	C₄⋊Dic₆₇	—	4	4	no	—
63	C₄⋊Dic₆₇	—	4	2	no	—
64	C₂⁴×C₆₇	SD₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	D₈	1	2	yes	—
70	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₆₇	C₄²	1	2	yes	—
76	C₂×C₈×C₆₇	C₄²	1	2	yes	—
77	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
82	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
83	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
84	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
85	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
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	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
93	C₁₆×C₆₇	—	2	4	no	—
94	C₂³×C₆₇	—	2	4	no	—
95	C₂³×C₆₇	#102	1	2	yes	—
96	C₂³×C₆₇	#103	1	2	yes	—
97	C₂³×C₆₇	#106	1	1	yes	characteristic
98	C₂³×C₆₇	#106	1	1	yes	characteristic
99	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₆₇	#95	1	2	yes	—
101	C₂×C₄×C₆₇	#95	1	2	yes	—
102	C₂×C₄×C₆₇	#102	1	2	yes	—
103	C₂×C₄×C₆₇	#102	1	2	yes	—
104	C₂×C₄×C₆₇	#102	1	2	yes	—
105	C₂×C₄×C₆₇	#103	1	2	yes	—
106	C₂×C₄×C₆₇	#103	1	2	yes	—
107	C₂×C₄×C₆₇	#103	1	2	yes	—
108	C₂×C₄×C₆₇	#106	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#106	1	1	yes	characteristic
110	C₂×C₄×C₆₇	#107	1	2	yes	—
111	C₂×C₄×C₆₇	#107	1	2	yes	—
112	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
118	C₂×Dic₆₇	—	4	4	no	—
119	C₂×Dic₆₇	—	8	8	no	—
120	C₂×Dic₆₇	—	8	4	no	—
121	C₈×C₆₇	#95	1	2	yes	—
122	C₈×C₆₇	#95	1	2	yes	—
123	C₈×C₆₇	C₂×C₄²	1	1	yes	characteristic
124	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₆₇⋊C₈	—	4	4	no	—
128	C₂²×C₆₇	—	2	4	no	—
129	C₂²×C₆₇	—	2	4	no	—
130	C₂²×C₆₇	—	2	2	no	—
131	C₂²×C₆₇	—	2	2	no	—
132	C₂²×C₆₇	#340	1	2	yes	—
133	C₂²×C₆₇	#340	1	2	yes	—
134	C₂²×C₆₇	#365	1	4	yes	—
135	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
136	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
137	C₂²×C₆₇	C₂×#106	1	1	yes	characteristic
138	C₄×C₆₇	#340	1	2	yes	—
139	C₄×C₆₇	#340	1	2	yes	—
140	C₄×C₆₇	C₂×#95	1	1	yes	characteristic
141	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
142	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
143	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
144	Dic₆₇	—	8	4	no	—
145	C₂×#1383	—	67	1	no	maximal, Sylow
146	C₂×C₆₇	—	2	1	no	—
147	C₂×C₆₇	—	2	2	no	—
148	C₂×C₆₇	—	2	1	no	—
149	C₂×C₆₇	#1383	1	4	yes	complemented
150	C₂×C₆₇	C₂×#340	1	1	yes	characteristic
151	C₂×C₆₇	C₂×#365	1	2	yes	—
152	C₂²×M₃₂	D₆₇	1	1	yes	characteristic
153	C₂²×#108	—	67	1	no	—
154	C₂²×#106	—	67	1	no	—
155	#1383	—	67	4	no	—
156	C₆₇	C₂×#1383	1	1	yes	characteristic, complemented, Sylow
157	C₂²×M₁₆	—	134	1	no	—
158	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
159	C₂²×C₄⋊C₄	—	134	1	no	—
160	C₂×M₃₂	—	2	2	no	—
161	C₂×M₃₂	—	2	2	no	—
162	C₂×M₃₂	Dic₆₇	1	2	yes	complemented
163	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
164	C₂²×C₁₆	Dic₆₇	1	2	yes	complemented
165	C₂×#108	—	67	4	no	—
166	C₂×#108	—	67	2	no	—
167	C₂×#108	—	134	1	no	—
168	C₂×#108	—	134	2	no	—
169	C₂×#106	—	67	4	no	—
170	C₂×#106	—	67	2	no	—
171	C₂×#106	—	134	1	no	—
172	C₂×#106	—	134	2	no	—
173	C₂³×C₄	—	268	1	no	—
174	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
175	C₂×M₁₆	—	268	2	no	—
176	C₂×M₁₆	—	268	1	no	—
177	C₂×M₁₆	—	134	4	no	—
178	C₂×M₁₆	—	134	2	no	—
179	C₂²×C₈	—	268	1	no	—
180	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
181	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
182	C₂²×C₈	C₄×D₆₇	1	2	yes	—
183	C₂²×C₈	C₄×D₆₇	1	2	yes	—
184	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
185	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
186	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
187	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
188	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
189	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
190	C₂×C₄⋊C₄	—	268	2	no	—
191	C₂×C₄⋊C₄	—	268	1	no	—
192	C₂×C₄⋊C₄	—	134	4	no	—
193	C₂×C₄⋊C₄	—	134	2	no	—
194	M₃₂	—	2	4	no	—
195	M₃₂	—	2	2	no	—
196	M₃₂	—	2	2	no	—
197	C₂×C₁₆	—	2	4	no	—
198	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	complemented
199	#108	—	67	4	no	—
200	#108	—	134	4	no	—
201	#108	—	134	2	no	—
202	#106	—	67	4	no	—
203	#106	—	134	4	no	—
204	#106	—	134	2	no	—
205	C₂⁴	#11761617	1	1	yes	characteristic
206	C₂²×C₄	—	268	4	no	—
207	C₂²×C₄	—	268	2	no	—
208	C₂²×C₄	—	536	4	no	—
209	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
210	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
211	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
212	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
213	C₂²×C₄	D₅₃₆	1	2	yes	—
214	C₂²×C₄	#11761617	1	1	yes	characteristic
215	C₂²×C₄	#11761617	1	1	yes	characteristic
216	C₂²×C₄	#11761617	1	1	yes	characteristic
217	C₂²×C₄	Dic₂₆₈	1	2	yes	—
218	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
219	M₁₆	—	268	4	no	—
220	M₁₆	—	268	2	no	—
221	M₁₆	—	134	4	no	—
222	C₂×C₈	—	268	4	no	—
223	C₂×C₈	—	268	2	no	—
224	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
225	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
226	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
227	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
228	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
229	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
230	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
231	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
232	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
233	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
234	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
235	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
236	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
238	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
239	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
240	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
241	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
242	C₄⋊C₄	—	268	4	no	—
243	C₄⋊C₄	—	268	2	no	—
244	C₄⋊C₄	—	134	4	no	—
245	C₁₆	—	2	4	no	—
246	C₂³	—	2	4	no	—
247	C₂³	#14523965	1	2	yes	—
248	C₂³	#14523968	1	2	yes	—
249	C₂³	#14523975	1	1	yes	characteristic
250	C₂³	#14523975	1	1	yes	characteristic
251	C₂³	C₂×#11761617	1	1	yes	characteristic
252	C₂×C₄	—	268	4	no	—
253	C₂×C₄	—	536	8	no	—
254	C₂×C₄	—	536	4	no	—
255	C₂×C₄	C₆₇⋊#95	1	2	yes	—
256	C₂×C₄	C₆₇⋊#95	1	2	yes	—
257	C₂×C₄	#14523965	1	2	yes	—
258	C₂×C₄	#14523965	1	2	yes	—
259	C₂×C₄	#14523965	1	2	yes	—
260	C₂×C₄	#14523968	1	2	yes	—
261	C₂×C₄	#14523968	1	2	yes	—
262	C₂×C₄	#14523968	1	2	yes	—
263	C₂×C₄	#14523975	1	1	yes	characteristic
264	C₂×C₄	#14523975	1	1	yes	characteristic
265	C₂×C₄	#14523977	1	2	yes	—
266	C₂×C₄	#14523977	1	2	yes	—
267	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
268	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
269	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
270	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
271	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
272	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
273	C₈	—	268	4	no	—
274	C₈	C₆₇⋊#95	1	2	yes	—
275	C₈	C₆₇⋊#95	1	2	yes	—
276	C₈	C₂×C₄×Dic₆₇	1	1	yes	characteristic
277	C₈	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
278	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
279	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
280	C₂²	—	2	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	#16863159	1	2	yes	—
285	C₂²	#16863159	1	2	yes	—
286	C₂²	#16863219	1	4	yes	—
287	C₂²	C₂×#14523965	1	1	yes	characteristic
288	C₂²	C₂×#14523968	1	1	yes	characteristic
289	C₂²	C₂×#14523975	1	1	yes	characteristic
290	C₄	—	536	4	no	—
291	C₄	#16863159	1	2	yes	—
292	C₄	#16863159	1	2	yes	—
293	C₄	C₂×C₆₇⋊#95	1	1	yes	characteristic
294	C₄	C₂×#14523965	1	1	yes	characteristic
295	C₄	C₂×#14523968	1	1	yes	characteristic
296	C₄	C₂×#14523977	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	#22558308	1	4	yes	direct product factor
301	C₂	C₂×#16863159	1	1	yes	characteristic
302	C₂	C₂×#16863219	1	2	yes	—

	subgroup	G/H
center	C₂³	#14523975
commutator	C₈×C₆₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂²×M₃₂×C₆₇	C₂
socle	C₂³×C₆₇	#106
prime radical	C₂⁴×C₆₇	SD₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂²×M₃₂	D₆₇
hypocenter	C₆₇	C₂×#1383
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #22558308	C₂	direct (N × H)
9. C₂×#14523979	C₄	semidirect
18. C₂×M₃₂×C₆₇	C₄	semidirect
20. C₂²×C₁₆×C₆₇	C₄	semidirect
33. #14523979	C₂×C₄	semidirect
52. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
149. C₂×C₆₇	#1383	semidirect
156. C₆₇	C₂×#1383	semidirect
162. C₂×M₃₂	Dic₆₇	semidirect
164. C₂²×C₁₆	Dic₆₇	semidirect
198. C₂×C₁₆	C₂×Dic₆₇	semidirect
300. C₂	#22558308	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×#14523979	2
C₂²×#14523975	2
#22558308	2
C₂²×M₃₂×C₆₇	2
C₂×#1383	67

Subgroups — C2×#22558308 #30770732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558308 #30770732