Subgroups — C₆₇×#29357 #30708608

Order 17152.

674 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2862	C₂	1	1	yes	maximal, characteristic
2	C₄×#125×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2074	C₂	1	2	yes	maximal
4	C₆₇×#2026	C₂	1	2	yes	maximal
5	C₆₇×#1987	C₂	1	2	yes	maximal
6	C₆₇×#1981	C₂	1	2	yes	maximal
7	C₆₇×#1935	C₂	1	4	yes	maximal
8	C₆₇×#1767	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
13	C₂×#103×C₆₇	C₂²	1	4	yes	—
14	C₂×#103×C₆₇	C₂²	1	4	yes	—
15	C₂×#97×C₆₇	C₂²	1	2	yes	—
16	#391×C₆₇	—	2	2	no	—
17	#391×C₆₇	C₂²	1	2	yes	—
18	#391×C₆₇	C₂²	1	2	yes	—
19	#389×C₆₇	C₂²	1	2	yes	—
20	#387×C₆₇	—	2	2	no	—
21	#382×C₆₇	—	2	2	no	—
22	#382×C₆₇	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
24	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
25	C₄³×C₆₇	C₂²	1	1	yes	characteristic
26	#340×C₆₇	C₂²	1	4	yes	—
27	#340×C₆₇	C₂²	1	4	yes	—
28	#336×C₆₇	C₂²	1	2	yes	—
29	C₂²×Q₈×C₆₇	—	2	1	no	—
30	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
31	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
32	C₄⋊Q₈×C₆₇	—	2	4	no	—
33	C₄⋊Q₈×C₆₇	—	2	4	no	—
34	#125×C₆₇	—	2	4	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
43	C₂×C₄²×C₆₇	—	2	2	no	—
44	C₂×C₄²×C₆₇	D₄	1	2	yes	—
45	C₂×C₄²×C₆₇	D₄	1	2	yes	—
46	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₆₇	—	2	4	no	—
50	#103×C₆₇	—	2	8	no	—
51	#103×C₆₇	—	4	4	no	—
52	#97×C₆₇	—	4	2	no	—
53	#95×C₆₇	—	2	2	no	—
54	#95×C₆₇	—	2	2	no	—
55	#95×C₆₇	—	2	2	no	—
56	#95×C₆₇	—	2	4	no	—
57	#95×C₆₇	—	2	4	no	—
58	C₂×Q₈×C₆₇	—	2	4	no	—
59	C₂×Q₈×C₆₇	—	2	4	no	—
60	C₂×Q₈×C₆₇	—	2	2	no	—
61	C₂×Q₈×C₆₇	—	2	2	no	—
62	C₂×Q₈×C₆₇	—	4	4	no	—
63	C₂×Q₈×C₆₇	—	4	2	no	—
64	C₂²×C₄×C₆₇	—	2	2	no	—
65	C₂²×C₄×C₆₇	—	2	2	no	—
66	C₂²×C₄×C₆₇	—	2	2	no	—
67	C₂²×C₄×C₆₇	—	2	2	no	—
68	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
73	C₂×C₈×C₆₇	—	2	4	no	—
74	C₂×C₈×C₆₇	—	2	4	no	—
75	C₂×C₈×C₆₇	—	2	4	no	—
76	C₄⋊C₄×C₆₇	—	2	8	no	—
77	C₄⋊C₄×C₆₇	—	2	2	no	—
78	C₄⋊C₄×C₆₇	—	2	4	no	—
79	C₄⋊C₄×C₆₇	—	2	2	no	—
80	C₄⋊C₄×C₆₇	—	2	2	no	—
81	C₄⋊C₄×C₆₇	—	4	4	no	—
82	C₄⋊C₄×C₆₇	—	4	2	no	—
83	C₄⋊C₄×C₆₇	—	4	2	no	—
84	C₄⋊C₄×C₆₇	—	4	2	no	—
85	C₄⋊C₄×C₆₇	#41	1	4	yes	—
86	C₄²×C₆₇	—	2	4	no	—
87	C₄²×C₆₇	—	2	2	no	—
88	C₄²×C₆₇	—	2	1	no	—
89	C₄²×C₆₇	—	2	1	no	—
90	C₄²×C₆₇	—	4	2	no	—
91	C₄²×C₆₇	—	4	4	no	—
92	C₄²×C₆₇	#41	1	4	yes	—
93	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
94	Q₈×C₆₇	—	4	4	no	—
95	Q₈×C₆₇	—	4	4	no	—
96	Q₈×C₆₇	—	8	4	no	—
97	C₂×C₄×C₆₇	—	2	2	no	—
98	C₂×C₄×C₆₇	—	2	2	no	—
99	C₂×C₄×C₆₇	—	2	4	no	—
100	C₂×C₄×C₆₇	—	2	2	no	—
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	—	2	2	no	—
103	C₂×C₄×C₆₇	—	2	2	no	—
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	—	4	2	no	—
106	C₂×C₄×C₆₇	—	4	2	no	—
107	C₂×C₄×C₆₇	—	4	2	no	—
108	C₂×C₄×C₆₇	—	4	2	no	—
109	C₂×C₄×C₆₇	—	4	2	no	—
110	C₂×C₄×C₆₇	—	4	2	no	—
111	C₂×C₄×C₆₇	—	4	2	no	—
112	C₂×C₄×C₆₇	—	4	2	no	—
113	C₂×C₄×C₆₇	—	4	2	no	—
114	C₂×C₄×C₆₇	—	4	4	no	—
115	C₂×C₄×C₆₇	#124	1	2	yes	—
116	C₂×C₄×C₆₇	#124	1	2	yes	—
117	C₂×C₄×C₆₇	#137	1	4	yes	—
118	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
119	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
120	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
121	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
122	C₈×C₆₇	—	4	4	no	—
123	C₂²×C₆₇	#525	1	1	yes	characteristic
124	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
125	C₂²×C₆₇	#537	1	1	yes	characteristic
126	C₂²×C₆₇	C₂×#137	1	2	yes	—
127	C₂²×C₆₇	#575	1	2	yes	—
128	C₄×C₆₇	—	2	2	no	—
129	C₄×C₆₇	—	4	2	no	—
130	C₄×C₆₇	—	4	4	no	—
131	C₄×C₆₇	—	4	2	no	—
132	C₄×C₆₇	—	8	2	no	—
133	C₄×C₆₇	—	8	2	no	—
134	C₄×C₆₇	—	8	2	no	—
135	C₄×C₆₇	#491	1	4	yes	—
136	C₄×C₆₇	#556	1	2	yes	—
137	C₄×C₆₇	#559	1	2	yes	—
138	#29357	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂×C₆₇	#2592	1	1	yes	characteristic
140	C₂×C₆₇	#3116	1	2	yes	—
141	C₂×C₆₇	C₂×#491	1	1	yes	characteristic
142	C₂×C₆₇	#3140	1	1	yes	characteristic
143	C₂×C₆₇	#3252	1	2	yes	—
144	#2862	C₂×C₆₇	1	1	yes	characteristic
145	C₄×#125	C₂×C₆₇	1	1	yes	characteristic
146	#2074	C₂×C₆₇	1	2	yes	—
147	#2026	C₂×C₆₇	1	2	yes	—
148	#1987	C₂×C₆₇	1	2	yes	—
149	#1981	C₂×C₆₇	1	2	yes	—
150	#1935	C₂×C₆₇	1	4	yes	—
151	#1767	C₂×C₆₇	1	1	yes	characteristic
152	C₆₇	#29357	1	1	yes	characteristic, direct product factor, Sylow
153	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
154	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
155	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
156	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
157	C₂×#103	C₂²×C₆₇	1	4	yes	—
158	C₂×#103	C₂²×C₆₇	1	4	yes	—
159	C₂×#97	C₂²×C₆₇	1	2	yes	—
160	#391	—	2	2	no	—
161	#391	C₂²×C₆₇	1	2	yes	—
162	#391	C₂²×C₆₇	1	2	yes	—
163	#389	C₂²×C₆₇	1	2	yes	—
164	#387	—	2	2	no	—
165	#382	—	2	2	no	—
166	#382	C₂²×C₆₇	1	1	yes	characteristic
167	C₄×C₄⋊C₄	—	2	2	no	—
168	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
169	C₄³	C₂²×C₆₇	1	1	yes	characteristic
170	#340	C₂²×C₆₇	1	4	yes	—
171	#340	C₂²×C₆₇	1	4	yes	—
172	#336	C₂²×C₆₇	1	2	yes	—
173	C₂²×Q₈	—	2	1	no	—
174	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
175	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
176	C₄⋊Q₈	—	2	4	no	—
177	C₄⋊Q₈	—	2	4	no	—
178	#125	—	2	4	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
185	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
186	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	D₄×C₆₇	1	2	yes	—
189	C₂×C₄²	D₄×C₆₇	1	2	yes	—
190	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
191	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
193	C₄⋊C₈	—	2	4	no	—
194	#103	—	2	8	no	—
195	#103	—	4	4	no	—
196	#97	—	4	2	no	—
197	#95	—	2	2	no	—
198	#95	—	2	2	no	—
199	#95	—	2	2	no	—
200	#95	—	2	4	no	—
201	#95	—	2	4	no	—
202	C₂×Q₈	—	2	4	no	—
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	—	4	4	no	—
207	C₂×Q₈	—	4	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
213	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
214	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
216	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₄⋊C₄	—	2	8	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	4	4	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₄⋊C₄	#41×C₆₇	1	4	yes	—
230	C₄²	—	2	4	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	4	2	no	—
235	C₄²	—	4	4	no	—
236	C₄²	#41×C₆₇	1	4	yes	—
237	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
238	Q₈	—	4	4	no	—
239	Q₈	—	4	4	no	—
240	Q₈	—	8	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	#124×C₆₇	1	2	yes	—
260	C₂×C₄	#124×C₆₇	1	2	yes	—
261	C₂×C₄	#137×C₆₇	1	4	yes	—
262	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
263	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
264	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
265	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
266	C₈	—	4	4	no	—
267	C₂²	#525×C₆₇	1	1	yes	characteristic
268	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
269	C₂²	#537×C₆₇	1	1	yes	characteristic
270	C₂²	C₂×#137×C₆₇	1	2	yes	—
271	C₂²	#575×C₆₇	1	2	yes	—
272	C₄	—	2	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	4	no	—
275	C₄	—	4	2	no	—
276	C₄	—	8	2	no	—
277	C₄	—	8	2	no	—
278	C₄	—	8	2	no	—
279	C₄	#491×C₆₇	1	4	yes	—
280	C₄	#556×C₆₇	1	2	yes	—
281	C₄	#559×C₆₇	1	2	yes	—
282	C₂	C₆₇×#2592	1	1	yes	characteristic
283	C₂	C₆₇×#3116	1	2	yes	—
284	C₂	C₂×#491×C₆₇	1	1	yes	characteristic
285	C₂	C₆₇×#3140	1	1	yes	characteristic
286	C₂	C₆₇×#3252	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
138. #29357	C₆₇	direct (N × H)
152. C₆₇	#29357	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2862	2
C₄×#125×C₆₇	2
C₆₇×#2074	2
C₆₇×#2026	2
C₆₇×#1987	2
C₆₇×#1981	2
C₆₇×#1935	2
C₆₇×#1767	2
#29357	67

Subgroups — C67×#29357 #30708608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#29357 #30708608