Subgroups — C₂×C₆₇×#1676 #30698368

Order 17152.

638 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#477×C₆₇	C₂	1	2	yes	maximal
3	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1676	C₂	1	8	yes	maximal, direct product factor
5	C₂×#335×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#333×C₆₇	C₂	1	2	yes	maximal
7	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
8	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
9	#500×C₆₇	—	2	2	no	—
10	#500×C₆₇	C₂²	1	4	yes	—
11	#477×C₆₇	—	2	4	no	—
12	#477×C₆₇	C₂²	1	8	yes	—
13	C₈×Q₈×C₆₇	—	2	2	no	—
14	C₈×Q₈×C₆₇	C₂²	1	4	yes	—
15	C₂×#107×C₆₇	—	2	1	no	—
16	C₂×#107×C₆₇	—	2	1	no	—
17	C₂×#107×C₆₇	—	2	2	no	—
18	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#103×C₆₇	—	2	2	no	—
22	C₂×C₄×C₈×C₆₇	—	2	1	no	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	#335×C₆₇	C₂²	1	4	yes	—
25	#333×C₆₇	C₂²	1	8	yes	—
26	C₂²×Q₈×C₆₇	—	2	2	no	—
27	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	—	2	1	no	—
29	C₂²×C₈×C₆₇	—	2	2	no	—
30	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₄⋊Q₈×C₆₇	—	2	2	no	—
33	C₄⋊Q₈×C₆₇	—	2	2	no	—
34	C₄⋊Q₈×C₆₇	—	2	2	no	—
35	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
36	C₄×Q₈×C₆₇	—	2	1	no	—
37	C₄×Q₈×C₆₇	—	2	1	no	—
38	C₄×Q₈×C₆₇	D₄	1	2	yes	—
39	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	—	2	1	no	—
45	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
46	#107×C₆₇	—	2	4	no	—
47	#107×C₆₇	—	2	4	no	—
48	#107×C₆₇	—	2	8	no	—
49	C₄⋊C₈×C₆₇	—	2	4	no	—
50	C₄⋊C₈×C₆₇	—	2	2	no	—
51	C₄⋊C₈×C₆₇	—	2	1	no	—
52	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
53	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
54	#103×C₆₇	—	2	8	no	—
55	C₄×C₈×C₆₇	—	2	4	no	—
56	C₄×C₈×C₆₇	D₄	1	2	yes	—
57	C₄×C₈×C₆₇	C₂³	1	2	yes	—
58	C₂×Q₈×C₆₇	—	2	1	no	—
59	C₂×Q₈×C₆₇	—	2	2	no	—
60	C₂×Q₈×C₆₇	—	2	4	no	—
61	C₂×Q₈×C₆₇	—	4	2	no	—
62	C₂×Q₈×C₆₇	—	8	4	no	—
63	C₂×Q₈×C₆₇	Q₁₆	1	4	yes	—
64	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₆₇	—	2	1	no	—
66	C₂²×C₄×C₆₇	—	2	1	no	—
67	C₂²×C₄×C₆₇	—	4	2	no	—
68	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
71	C₂×C₈×C₆₇	—	2	2	no	—
72	C₂×C₈×C₆₇	—	2	4	no	—
73	C₂×C₈×C₆₇	—	2	2	no	—
74	C₂×C₈×C₆₇	—	4	4	no	—
75	C₂×C₈×C₆₇	—	4	2	no	—
76	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
77	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
79	C₄⋊C₄×C₆₇	—	2	2	no	—
80	C₄⋊C₄×C₆₇	—	2	1	no	—
81	C₄⋊C₄×C₆₇	—	2	2	no	—
82	C₄⋊C₄×C₆₇	—	2	4	no	—
83	C₄⋊C₄×C₆₇	—	2	4	no	—
84	C₄⋊C₄×C₆₇	—	4	2	no	—
85	C₄⋊C₄×C₆₇	—	4	2	no	—
86	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
87	C₄²×C₆₇	—	2	2	no	—
88	C₄²×C₆₇	—	2	2	no	—
89	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₆₇	#41	1	1	yes	characteristic
92	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
93	C₂³×C₆₇	#121	1	1	yes	characteristic
94	Q₈×C₆₇	—	2	2	no	—
95	Q₈×C₆₇	—	2	4	no	—
96	Q₈×C₆₇	—	8	4	no	—
97	Q₈×C₆₇	—	8	4	no	—
98	Q₈×C₆₇	C₂×Q₁₆	1	4	yes	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	2	4	no	—
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	—	2	2	no	—
103	C₂×C₄×C₆₇	—	4	2	no	—
104	C₂×C₄×C₆₇	—	4	4	no	—
105	C₂×C₄×C₆₇	—	8	4	no	—
106	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
107	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
108	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
110	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
111	C₂×C₄×C₆₇	#135	1	2	yes	—
112	C₂×C₄×C₆₇	#137	1	2	yes	—
113	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
116	C₈×C₆₇	—	2	2	no	—
117	C₈×C₆₇	—	4	4	no	—
118	C₈×C₆₇	—	4	2	no	—
119	C₈×C₆₇	C₂×Q₁₆	1	4	yes	—
120	C₂²×C₆₇	#462	1	2	yes	—
121	C₂²×C₆₇	#467	1	2	yes	—
122	C₂²×C₆₇	#472	1	2	yes	—
123	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
124	C₄×C₆₇	—	2	4	no	—
125	C₄×C₆₇	—	2	1	no	—
126	C₄×C₆₇	—	2	1	no	—
127	C₄×C₆₇	—	4	2	no	—
128	C₄×C₆₇	—	8	4	no	—
129	C₄×C₆₇	#462	1	2	yes	—
130	C₄×C₆₇	#467	1	2	yes	—
131	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
132	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
133	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
134	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
135	C₂×#1676	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂×C₆₇	#1676	1	4	yes	direct product factor
137	C₂×C₆₇	C₂×#462	1	1	yes	characteristic
138	C₂×C₆₇	C₂×#467	1	1	yes	characteristic
139	C₂×C₆₇	C₂×#472	1	1	yes	characteristic
140	C₂×#500	C₂×C₆₇	1	1	yes	characteristic
141	C₂×#477	C₂×C₆₇	1	2	yes	—
142	C₂×C₈×Q₈	C₂×C₆₇	1	1	yes	characteristic
143	#1676	C₂×C₆₇	1	8	yes	direct product factor
144	C₂×#335	C₂×C₆₇	1	1	yes	characteristic
145	C₂×#333	C₂×C₆₇	1	2	yes	—
146	C₆₇	C₂×#1676	1	1	yes	characteristic, direct product factor, Sylow
147	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
148	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
149	#500	—	2	2	no	—
150	#500	C₂²×C₆₇	1	4	yes	—
151	#477	—	2	4	no	—
152	#477	C₂²×C₆₇	1	8	yes	—
153	C₈×Q₈	—	2	2	no	—
154	C₈×Q₈	C₂²×C₆₇	1	4	yes	—
155	C₂×#107	—	2	1	no	—
156	C₂×#107	—	2	1	no	—
157	C₂×#107	—	2	2	no	—
158	C₂×C₄⋊C₈	—	2	1	no	—
159	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
160	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
161	C₂×#103	—	2	2	no	—
162	C₂×C₄×C₈	—	2	1	no	—
163	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
164	#335	C₂²×C₆₇	1	4	yes	—
165	#333	C₂²×C₆₇	1	8	yes	—
166	C₂²×Q₈	—	2	2	no	—
167	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
168	C₂²×C₈	—	2	1	no	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
171	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
172	C₄⋊Q₈	—	2	2	no	—
173	C₄⋊Q₈	—	2	2	no	—
174	C₄⋊Q₈	—	2	2	no	—
175	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
176	C₄×Q₈	—	2	1	no	—
177	C₄×Q₈	—	2	1	no	—
178	C₄×Q₈	D₄×C₆₇	1	2	yes	—
179	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
184	C₂×C₄²	—	2	1	no	—
185	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
186	#107	—	2	4	no	—
187	#107	—	2	4	no	—
188	#107	—	2	8	no	—
189	C₄⋊C₈	—	2	4	no	—
190	C₄⋊C₈	—	2	1	no	—
191	C₄⋊C₈	—	2	2	no	—
192	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
193	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
194	#103	—	2	8	no	—
195	C₄×C₈	—	2	4	no	—
196	C₄×C₈	D₄×C₆₇	1	2	yes	—
197	C₄×C₈	C₂³×C₆₇	1	2	yes	—
198	C₂×Q₈	—	2	1	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	—	2	4	no	—
201	C₂×Q₈	—	4	2	no	—
202	C₂×Q₈	—	8	4	no	—
203	C₂×Q₈	Q₁₆×C₆₇	1	4	yes	—
204	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
210	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	4	4	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	Q₁₆×C₆₇	1	4	yes	—
217	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
218	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
230	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
231	C₄²	#41×C₆₇	1	1	yes	characteristic
232	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
233	C₂³	#121×C₆₇	1	1	yes	characteristic
234	Q₈	—	2	4	no	—
235	Q₈	—	2	2	no	—
236	Q₈	—	8	4	no	—
237	Q₈	—	8	4	no	—
238	Q₈	C₂×Q₁₆×C₆₇	1	4	yes	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
247	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
248	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
249	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
250	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
251	C₂×C₄	#135×C₆₇	1	2	yes	—
252	C₂×C₄	#137×C₆₇	1	2	yes	—
253	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
254	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
255	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	4	4	no	—
258	C₈	—	4	2	no	—
259	C₈	C₂×Q₁₆×C₆₇	1	4	yes	—
260	C₂²	#462×C₆₇	1	2	yes	—
261	C₂²	#467×C₆₇	1	2	yes	—
262	C₂²	#472×C₆₇	1	2	yes	—
263	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
264	C₄	—	2	4	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	4	2	no	—
268	C₄	—	8	4	no	—
269	C₄	#462×C₆₇	1	2	yes	—
270	C₄	#467×C₆₇	1	2	yes	—
271	C₄	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
272	C₄	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
273	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
274	C₄	C₂×#137×C₆₇	1	1	yes	characteristic
275	C₂	C₆₇×#1676	1	4	yes	direct product factor
276	C₂	C₂×#462×C₆₇	1	1	yes	characteristic
277	C₂	C₂×#467×C₆₇	1	1	yes	characteristic
278	C₂	C₂×#472×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1676	C₂	direct (N × H)
135. C₂×#1676	C₆₇	direct (N × H)
136. C₂×C₆₇	#1676	direct (N × H)
143. #1676	C₂×C₆₇	direct (N × H)
146. C₆₇	C₂×#1676	direct (N × H)
275. C₂	C₆₇×#1676	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×#500×C₆₇	2
C₂×#477×C₆₇	2
C₂×C₈×Q₈×C₆₇	2
C₆₇×#1676	2
C₂×#335×C₆₇	2
C₂×#333×C₆₇	2
C₂×#1676	67

Subgroups — C2×C67×#1676 #30698368

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1676 #30698368