Subgroups — C₂×#22559734 #30838462

Order 17152.

618 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863688	C₂	1	2	yes	maximal
2	C₂×#16863526	C₂	1	1	yes	maximal, characteristic
3	#22559734	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863147	C₂	1	1	yes	maximal, characteristic
5	C₂×#16863145	C₂	1	2	yes	maximal
6	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
8	#16863688	—	2	4	no	—
9	#16863688	C₂²	1	8	yes	—
10	#16863526	—	2	2	no	—
11	#16863526	C₂²	1	4	yes	—
12	C₂×#14523976	—	2	2	no	—
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×Dic₆₇⋊C₈	—	2	1	no	—
16	C₂×#14523966	—	2	2	no	—
17	C₂×#14523953	—	2	1	no	—
18	#16863147	C₂²	1	4	yes	—
19	#16863145	C₂²	1	8	yes	—
20	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
21	#500×C₆₇	—	2	2	no	—
22	#500×C₆₇	C₂²	1	4	yes	—
23	C₂×#107×C₆₇	—	2	1	no	—
24	C₂×#107×C₆₇	—	2	1	no	—
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₃₄	D₄	1	1	yes	characteristic
27	C₂²×C₆₇⋊C₈	—	2	1	no	—
28	C₂²×C₆₇⋊C₈	—	2	2	no	—
29	C₄×Dic₁₃₄	—	2	1	no	—
30	C₄×Dic₁₃₄	—	2	1	no	—
31	C₄×Dic₁₃₄	D₄	1	2	yes	—
32	C₄×Dic₁₃₄	C₂³	1	2	yes	—
33	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
34	C₂×Dic₆₇⋊C₄	—	2	1	no	—
35	C₂×C₄×Dic₆₇	—	2	1	no	—
36	#14523976	—	2	8	no	—
37	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
38	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
39	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
40	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
41	Dic₆₇⋊C₈	—	2	4	no	—
42	#14523966	—	2	8	no	—
43	#14523953	—	2	4	no	—
44	C₂²×Q₈×C₆₇	—	2	2	no	—
45	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
47	C₄⋊Q₈×C₆₇	—	2	2	no	—
48	C₄⋊Q₈×C₆₇	—	2	2	no	—
49	C₄⋊Q₈×C₆₇	—	2	2	no	—
50	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
51	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
52	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	#107×C₆₇	—	2	4	no	—
55	#107×C₆₇	—	2	4	no	—
56	C₄×C₈×C₆₇	D₄	1	2	yes	—
57	C₄×C₈×C₆₇	C₂³	1	2	yes	—
58	C₂×Dic₁₃₄	—	2	4	no	—
59	C₂×Dic₁₃₄	—	2	1	no	—
60	C₂×Dic₁₃₄	C₂×D₄	1	2	yes	—
61	C₂²×Dic₆₇	—	2	1	no	—
62	C₂²×Dic₆₇	—	2	1	no	—
63	C₂×C₆₇⋊C₈	—	2	2	no	—
64	C₂×C₆₇⋊C₈	—	2	4	no	—
65	C₂×C₆₇⋊C₈	—	4	2	no	—
66	C₂×C₆₇⋊C₈	—	4	4	no	—
67	C₄⋊Dic₆₇	—	2	1	no	—
68	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
69	Dic₆₇⋊C₄	—	2	2	no	—
70	Dic₆₇⋊C₄	—	2	2	no	—
71	C₄×Dic₆₇	—	2	2	no	—
72	C₄×Dic₆₇	—	2	2	no	—
73	C₂×Q₈×C₆₇	—	2	4	no	—
74	C₂×Q₈×C₆₇	—	4	2	no	—
75	C₂×Q₈×C₆₇	—	8	4	no	—
76	C₂²×C₄×C₆₇	—	4	2	no	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
80	C₂×C₈×C₆₇	—	2	2	no	—
81	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
82	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
84	C₄⋊C₄×C₆₇	—	2	4	no	—
85	C₄⋊C₄×C₆₇	—	2	4	no	—
86	C₄⋊C₄×C₆₇	—	4	2	no	—
87	C₄⋊C₄×C₆₇	—	4	2	no	—
88	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₆₇	#41	1	1	yes	characteristic
91	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
92	C₂³×C₆₇	#121	1	1	yes	characteristic
93	Q₈×C₆₇	—	8	4	no	—
94	Q₈×C₆₇	—	8	4	no	—
95	C₂×C₄×C₆₇	—	2	2	no	—
96	C₂×C₄×C₆₇	—	4	4	no	—
97	C₂×C₄×C₆₇	—	8	4	no	—
98	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
99	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
100	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
102	C₂×C₄×C₆₇	#137	1	2	yes	—
103	C₂×C₄×C₆₇	#137	1	4	yes	—
104	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
107	C₂×Dic₆₇	—	2	2	no	—
108	C₂×Dic₆₇	—	2	2	no	—
109	C₂×Dic₆₇	—	4	2	no	—
110	C₂×Dic₆₇	—	4	2	no	—
111	Dic₁₃₄	—	2	4	no	—
112	Dic₁₃₄	—	4	2	no	—
113	C₈×C₆₇	—	2	2	no	—
114	C₈×C₆₇	C₂×Q₁₆	1	4	yes	—
115	C₆₇⋊C₈	—	4	2	no	—
116	C₆₇⋊C₈	—	4	4	no	—
117	C₂²×C₆₇	#462	1	2	yes	—
118	C₂²×C₆₇	#470	1	2	yes	—
119	C₂²×C₆₇	#472	1	2	yes	—
120	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
121	C₄×C₆₇	—	2	1	no	—
122	C₄×C₆₇	—	2	1	no	—
123	C₄×C₆₇	—	8	4	no	—
124	C₄×C₆₇	#462	1	2	yes	—
125	C₄×C₆₇	#470	1	2	yes	—
126	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
127	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
128	C₄×C₆₇	C₂×#137	1	2	yes	—
129	Dic₆₇	—	4	2	no	—
130	Dic₆₇	—	4	2	no	—
131	C₂×#1696	—	67	1	no	maximal, Sylow
132	C₂×C₆₇	#1696	1	4	yes	complemented
133	C₂×C₆₇	C₂×#462	1	1	yes	characteristic
134	C₂×C₆₇	C₂×#470	1	1	yes	characteristic
135	C₂×C₆₇	C₂×#472	1	1	yes	characteristic
136	C₂×#500	D₆₇	1	1	yes	characteristic
137	C₂×#477	—	67	2	no	—
138	C₂×#446	—	67	1	no	—
139	#1696	—	67	8	no	—
140	C₂×#334	—	67	1	no	—
141	C₂×#333	—	67	2	no	—
142	C₆₇	C₂×#1696	1	1	yes	characteristic, complemented, Sylow
143	C₂×C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
144	C₂×C₄×Q₈	—	67	1	no	—
145	#500	—	2	2	no	—
146	#500	C₂×D₆₇	1	4	yes	—
147	#477	—	67	8	no	—
148	#477	—	134	4	no	—
149	#446	—	67	4	no	—
150	#446	—	134	2	no	—
151	C₂×#107	—	2	1	no	—
152	C₂×#107	—	2	1	no	—
153	C₂×#107	—	134	2	no	—
154	C₂×C₄⋊C₈	—	67	2	no	—
155	C₂×C₄⋊C₈	—	67	1	no	—
156	C₂×C₄⋊C₈	—	134	1	no	—
157	C₂×#103	—	134	2	no	—
158	C₂×#97	—	134	1	no	—
159	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
160	#334	—	67	4	no	—
161	#333	—	67	8	no	—
162	C₂²×Q₈	—	2	2	no	—
163	C₂²×Q₈	—	67	1	no	—
164	C₂²×C₈	—	134	2	no	—
165	C₂²×C₈	—	134	1	no	—
166	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
167	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
168	C₄⋊Q₈	—	2	2	no	—
169	C₄⋊Q₈	—	2	2	no	—
170	C₄⋊Q₈	—	2	2	no	—
171	C₄⋊Q₈	C₂²×D₆₇	1	4	yes	—
172	C₄×Q₈	—	67	2	no	—
173	C₄×Q₈	—	67	2	no	—
174	C₄×Q₈	—	134	1	no	—
175	C₄×Q₈	—	134	1	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	67	1	no	—
179	C₂×C₄⋊C₄	—	134	1	no	—
180	C₂×C₄²	—	134	1	no	—
181	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
182	#107	—	2	4	no	—
183	#107	—	2	4	no	—
184	#107	—	134	8	no	—
185	C₄⋊C₈	—	67	2	no	—
186	C₄⋊C₈	—	67	4	no	—
187	C₄⋊C₈	—	134	4	no	—
188	C₄⋊C₈	—	134	2	no	—
189	C₄⋊C₈	—	134	1	no	—
190	#103	—	134	8	no	—
191	#97	—	134	4	no	—
192	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
193	C₄×C₈	C₂²×D₆₇	1	2	yes	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	—	4	2	no	—
196	C₂×Q₈	—	8	4	no	—
197	C₂×Q₈	—	67	2	no	—
198	C₂×Q₈	—	134	4	no	—
199	C₂×Q₈	—	134	1	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	134	1	no	—
202	C₂²×C₄	—	134	1	no	—
203	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
205	C₂²×C₄	#11761627	1	1	yes	characteristic
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	268	2	no	—
208	C₂×C₈	—	268	4	no	—
209	C₂×C₈	—	134	4	no	—
210	C₂×C₈	—	134	2	no	—
211	C₂×C₈	#11761618	1	4	yes	—
212	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
213	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	—	67	2	no	—
219	C₄⋊C₄	—	134	1	no	—
220	C₄⋊C₄	—	134	2	no	—
221	C₄⋊C₄	—	134	2	no	—
222	C₄²	—	134	2	no	—
223	C₄²	—	134	2	no	—
224	C₄²	D₄×D₆₇	1	1	yes	characteristic
225	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
226	C₄²	#11761627	1	1	yes	characteristic
227	C₄²	C₂³×D₆₇	1	1	yes	characteristic
228	C₂³	#14524014	1	1	yes	characteristic
229	Q₈	—	8	4	no	—
230	Q₈	—	8	4	no	—
231	Q₈	—	268	2	no	—
232	Q₈	—	134	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	8	4	no	—
236	C₂×C₄	—	268	2	no	—
237	C₂×C₄	—	268	2	no	—
238	C₂×C₄	—	134	2	no	—
239	C₂×C₄	—	134	2	no	—
240	C₂×C₄	#14524014	1	1	yes	characteristic
241	C₂×C₄	#14524014	1	1	yes	characteristic
242	C₂×C₄	#14524014	1	1	yes	characteristic
243	C₂×C₄	C₂×#11761618	1	2	yes	—
244	C₂×C₄	#14524074	1	2	yes	—
245	C₂×C₄	#14524075	1	4	yes	—
246	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
247	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
248	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	268	2	no	—
251	C₈	—	268	4	no	—
252	C₈	C₂×#11761618	1	4	yes	—
253	C₂²	#16863603	1	2	yes	—
254	C₂²	#16863651	1	2	yes	—
255	C₂²	#16863663	1	2	yes	—
256	C₂²	C₂×#14524014	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	8	4	no	—
260	C₄	—	268	2	no	—
261	C₄	—	268	2	no	—
262	C₄	#16863603	1	2	yes	—
263	C₄	#16863651	1	2	yes	—
264	C₄	C₂²×#11761618	1	1	yes	characteristic
265	C₄	C₂×#14524074	1	1	yes	characteristic
266	C₄	C₂×#14524075	1	2	yes	—
267	C₂	#22559734	1	4	yes	direct product factor
268	C₂	C₂×#16863603	1	1	yes	characteristic
269	C₂	C₂×#16863651	1	1	yes	characteristic
270	C₂	C₂×#16863663	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524014
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#500×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#121
hypercenter	C₂×#500	D₆₇
hypocenter	C₆₇	C₂×#1696
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #22559734	C₂	direct (N × H)
132. C₂×C₆₇	#1696	semidirect
142. C₆₇	C₂×#1696	semidirect
267. C₂	#22559734	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×#16863688	2
C₂×#16863526	2
#22559734	2
C₂×#16863147	2
C₂×#16863145	2
C₂×#500×C₆₇	2
C₂×#1696	67

Subgroups — C2×#22559734 #30838462

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22559734 #30838462