Subgroups — C₂×#22558227 #30766080

Order 17152.

446 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863161	C₂	1	1	yes	maximal, characteristic
3	#22558227	C₂	1	4	yes	maximal, direct product factor
4	C₂×#467×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761612	—	2	1	no	—
6	C₂²×C₆₇⋊C₁₆	—	2	1	no	—
7	C₂²⋊(C₆₇⋊C₁₆)	—	2	1	no	—
8	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
9	#16863161	—	2	1	no	—
10	#16863161	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
12	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
14	#467×C₆₇	—	2	1	no	—
15	#467×C₆₇	—	2	1	no	—
16	#467×C₆₇	C₄	1	2	yes	—
17	#467×C₆₇	C₂²	1	2	yes	—
18	C₂×#107×C₆₇	C₄	1	1	yes	characteristic
19	C₂×#103×C₆₇	—	2	1	no	—
20	C₂×#11761612	—	2	4	no	—
21	C₂×#11761612	—	2	2	no	—
22	C₂×#11761612	—	4	2	no	—
23	C₂×#11761612	—	4	1	no	—
24	C₂²×C₆₇⋊C₈	—	4	1	no	—
25	C₂×C₆₇⋊C₁₆	—	2	2	no	—
26	C₂×C₆₇⋊C₁₆	—	4	2	no	—
27	C₂²×Q₈×C₆₇	—	2	1	no	—
28	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
29	C₂×Q₁₆×C₆₇	—	2	1	no	—
30	C₂×Q₁₆×C₆₇	—	4	2	no	—
31	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₆₇	—	4	2	no	—
33	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	C₂²⋊Q₈×C₆₇	—	2	2	no	—
40	C₂²⋊Q₈×C₆₇	—	2	2	no	—
41	C₂²⋊Q₈×C₆₇	—	4	1	no	—
42	C₂²⋊Q₈×C₆₇	—	4	1	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
44	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
45	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
46	#107×C₆₇	—	2	1	no	—
47	#107×C₆₇	C₂×C₄	1	2	yes	—
48	#103×C₆₇	—	2	2	no	—
49	#103×C₆₇	—	2	2	no	—
50	#11761612	—	2	4	no	—
51	#11761612	—	4	4	no	—
52	#11761612	—	4	2	no	—
53	C₂×C₆₇⋊C₈	—	4	4	no	—
54	C₂×C₆₇⋊C₈	—	4	2	no	—
55	C₆₇⋊C₁₆	—	4	2	no	—
56	C₂⁴×C₆₇	SD₁₆	1	1	yes	characteristic
57	C₂×Q₈×C₆₇	—	2	2	no	—
58	C₂×Q₈×C₆₇	—	4	1	no	—
59	C₂×Q₈×C₆₇	—	4	2	no	—
60	C₂×Q₈×C₆₇	—	4	2	no	—
61	C₂²×C₄×C₆₇	—	2	2	no	—
62	C₂²×C₄×C₆₇	—	2	2	no	—
63	C₂²×C₄×C₆₇	—	4	1	no	—
64	C₂²×C₄×C₆₇	—	4	1	no	—
65	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
71	Q₁₆×C₆₇	—	4	2	no	—
72	Q₁₆×C₆₇	—	4	2	no	—
73	C₂×C₈×C₆₇	—	2	2	no	—
74	C₂×C₈×C₆₇	—	2	2	no	—
75	C₂×C₈×C₆₇	—	4	1	no	—
76	C₂×C₈×C₆₇	—	4	1	no	—
77	C₂×C₈×C₆₇	—	4	1	no	—
78	C₂×C₈×C₆₇	—	4	1	no	—
79	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₄⋊C₄×C₆₇	—	2	2	no	—
84	C₄⋊C₄×C₆₇	—	4	1	no	—
85	C₄⋊C₄×C₆₇	—	4	2	no	—
86	C₄⋊C₄×C₆₇	—	4	2	no	—
87	C₂²⋊C₄×C₆₇	—	4	2	no	—
88	C₂²⋊C₄×C₆₇	—	4	2	no	—
89	C₂³×C₆₇	—	2	2	no	—
90	C₂³×C₆₇	—	2	2	no	—
91	C₂³×C₆₇	—	2	2	no	—
92	C₂³×C₆₇	#102	1	1	yes	characteristic
93	C₂³×C₆₇	#102	1	1	yes	characteristic
94	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
95	Q₈×C₆₇	—	4	2	no	—
96	Q₈×C₆₇	—	4	2	no	—
97	Q₈×C₆₇	—	8	2	no	—
98	C₂×C₄×C₆₇	—	2	2	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	2	2	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	—	4	2	no	—
106	C₂×C₄×C₆₇	—	4	2	no	—
107	C₂×C₄×C₆₇	—	8	2	no	—
108	C₂×C₄×C₆₇	—	8	2	no	—
109	C₂×C₄×C₆₇	#99	1	2	yes	—
110	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
111	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
112	C₂×C₄×C₆₇	#104	1	2	yes	—
113	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
115	C₈×C₆₇	—	2	1	no	—
116	C₈×C₆₇	—	2	1	no	—
117	C₈×C₆₇	—	2	1	no	—
118	C₈×C₆₇	—	2	1	no	—
119	C₆₇⋊C₈	—	4	4	no	—
120	C₂²×C₆₇	—	2	1	no	—
121	C₂²×C₆₇	—	2	1	no	—
122	C₂²×C₆₇	—	2	1	no	—
123	C₂²×C₆₇	—	2	1	no	—
124	C₂²×C₆₇	—	2	1	no	—
125	C₂²×C₆₇	—	2	1	no	—
126	C₂²×C₆₇	—	4	4	no	—
127	C₂²×C₆₇	#327	1	2	yes	—
128	C₂²×C₆₇	#359	1	2	yes	—
129	C₂²×C₆₇	#362	1	2	yes	—
130	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
131	C₄×C₆₇	—	2	1	no	—
132	C₄×C₆₇	—	2	1	no	—
133	C₄×C₆₇	—	8	2	no	—
134	C₄×C₆₇	—	8	2	no	—
135	C₄×C₆₇	#327	1	2	yes	—
136	C₄×C₆₇	C₂×#99	1	1	yes	characteristic
137	C₄×C₆₇	C₂×#104	1	1	yes	characteristic
138	C₂×#1350	—	67	1	no	maximal, Sylow
139	C₂×C₆₇	—	4	1	no	—
140	C₂×C₆₇	—	4	1	no	—
141	C₂×C₆₇	#1350	1	4	yes	complemented
142	C₂×C₆₇	C₂×#327	1	1	yes	characteristic
143	C₂×C₆₇	C₂×#359	1	1	yes	characteristic
144	C₂×C₆₇	C₂×#362	1	1	yes	characteristic
145	C₂×#467	D₆₇	1	1	yes	characteristic
146	C₂×C₂²⋊C₁₆	—	67	1	no	—
147	C₂×#341	—	67	1	no	—
148	#1350	—	67	4	no	—
149	C₆₇	C₂×#1350	1	1	yes	characteristic, complemented, Sylow
150	C₂²×Q₁₆	Dic₆₇	1	1	yes	characteristic
151	C₂²×M₁₆	—	134	1	no	—
152	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
153	C₂×C₂²⋊Q₈	—	2	1	no	—
154	C₂²×C₁₆	—	134	1	no	—
155	#467	—	2	1	no	—
156	#467	—	2	1	no	—
157	#467	Dic₆₇	1	2	yes	—
158	#467	C₂×D₆₇	1	2	yes	—
159	C₂×#107	Dic₆₇	1	1	yes	characteristic
160	C₂×#103	—	2	1	no	—
161	C₂²⋊C₁₆	—	67	2	no	—
162	C₂²⋊C₁₆	—	134	1	no	—
163	#341	—	67	2	no	—
164	#341	—	134	1	no	—
165	C₂²×Q₈	—	2	1	no	—
166	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
167	C₂×Q₁₆	—	2	1	no	—
168	C₂×Q₁₆	—	4	2	no	—
169	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
170	C₂×M₁₆	—	268	2	no	—
171	C₂×M₁₆	—	268	1	no	—
172	C₂×M₁₆	—	134	4	no	—
173	C₂×M₁₆	—	134	2	no	—
174	C₂²×C₈	—	4	2	no	—
175	C₂²×C₈	—	268	1	no	—
176	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
177	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
178	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
179	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
180	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
182	C₂²⋊Q₈	—	2	2	no	—
183	C₂²⋊Q₈	—	2	2	no	—
184	C₂²⋊Q₈	—	4	1	no	—
185	C₂²⋊Q₈	—	4	1	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	4	1	no	—
188	C₂×C₂²⋊C₄	—	4	1	no	—
189	C₂×C₁₆	—	268	2	no	—
190	C₂×C₁₆	—	134	2	no	—
191	#107	—	2	1	no	—
192	#107	C₂×Dic₆₇	1	2	yes	—
193	#103	—	2	2	no	—
194	#103	—	2	2	no	—
195	C₂⁴	#11761616	1	1	yes	characteristic
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	—	4	1	no	—
198	C₂×Q₈	—	4	2	no	—
199	C₂×Q₈	—	4	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	4	1	no	—
203	C₂²×C₄	—	4	1	no	—
204	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
205	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
206	C₂²×C₄	#11761613	1	1	yes	characteristic
207	C₂²×C₄	#11761613	1	1	yes	characteristic
208	C₂²×C₄	#11761616	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
210	Q₁₆	—	4	2	no	—
211	Q₁₆	—	4	2	no	—
212	M₁₆	—	268	2	no	—
213	M₁₆	—	268	4	no	—
214	M₁₆	—	134	4	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	268	4	no	—
222	C₂×C₈	—	268	2	no	—
223	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
224	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
225	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
226	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₂²⋊C₄	—	4	2	no	—
232	C₂²⋊C₄	—	4	2	no	—
233	C₁₆	—	268	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	2	no	—
237	C₂³	#14523964	1	1	yes	characteristic
238	C₂³	#14523964	1	1	yes	characteristic
239	C₂³	C₂×#11761616	1	1	yes	characteristic
240	Q₈	—	4	2	no	—
241	Q₈	—	4	2	no	—
242	Q₈	—	8	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	#14523958	1	2	yes	—
255	C₂×C₄	#14523964	1	1	yes	characteristic
256	C₂×C₄	#14523964	1	1	yes	characteristic
257	C₂×C₄	#14523970	1	2	yes	—
258	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
259	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	268	4	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	4	4	no	—
272	C₂²	#16863130	1	2	yes	—
273	C₂²	#16863203	1	2	yes	—
274	C₂²	#16863212	1	2	yes	—
275	C₂²	C₂×#14523964	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	8	2	no	—
279	C₄	—	8	2	no	—
280	C₄	#16863130	1	2	yes	—
281	C₄	C₂×#14523958	1	1	yes	characteristic
282	C₄	C₂×#14523970	1	1	yes	characteristic
283	C₂	—	4	1	no	—
284	C₂	—	4	1	no	—
285	C₂	#22558227	1	4	yes	direct product factor
286	C₂	C₂×#16863130	1	1	yes	characteristic
287	C₂	C₂×#16863203	1	1	yes	characteristic
288	C₂	C₂×#16863212	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14523964
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#467×C₆₇	C₂
socle	C₂³×C₆₇	#102
prime radical	C₂⁴×C₆₇	SD₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#467	D₆₇
hypocenter	C₆₇	C₂×#1350
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#467

Complements

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

N	H ≅ G/N	type
3. #22558227	C₂	direct (N × H)
141. C₂×C₆₇	#1350	semidirect
149. C₆₇	C₂×#1350	semidirect
285. C₂	#22558227	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×C₂²⋊(C₆₇⋊C₁₆)	2
C₂×#16863161	2
#22558227	2
C₂×#467×C₆₇	2
C₂×#1350	67

Subgroups — C2×#22558227 #30766080

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558227 #30766080