Subgroups — C₆₇×#21325 #30700576

Order 17152.

1166 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	C₂²×#125×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1992	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1991	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1736	C₂	1	4	yes	maximal
7	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
8	C₂×#125×C₆₇	—	2	2	no	—
9	C₂×#125×C₆₇	—	2	4	no	—
10	C₂×#125×C₆₇	C₄	1	8	yes	complemented
11	C₂×#125×C₆₇	C₂²	1	2	yes	—
12	C₂×#125×C₆₇	C₂²	1	2	yes	—
13	C₂×#117×C₆₇	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₆₇	—	2	2	no	—
15	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	C₄×M₁₆×C₆₇	—	2	4	no	—
18	C₂×#97×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
20	#384×C₆₇	C₂²	1	2	yes	—
21	#382×C₆₇	C₂²	1	2	yes	—
22	#380×C₆₇	—	2	2	no	—
23	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
24	#340×C₆₇	C₂²	1	16	yes	—
25	C₂³×C₄×C₆₇	—	2	2	no	—
26	C₂³×C₄×C₆₇	D₄	1	2	yes	—
27	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₆₇	—	2	4	no	—
29	C₂×M₁₆×C₆₇	—	2	4	no	—
30	C₂×M₁₆×C₆₇	Q₈	1	8	yes	—
31	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
32	#125×C₆₇	—	2	16	no	—
33	#125×C₆₇	—	2	4	no	—
34	#125×C₆₇	—	2	4	no	—
35	#125×C₆₇	C₂×C₄	1	8	yes	—
36	#125×C₆₇	C₂×C₄	1	8	yes	—
37	#117×C₆₇	—	2	8	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	16	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
44	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
45	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
47	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
48	C₂×C₄²×C₆₇	—	2	2	no	—
49	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₆₇	D₄	1	2	yes	—
51	C₂×C₄²×C₆₇	D₄	1	4	yes	—
52	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	#97×C₆₇	—	2	4	no	—
55	C₄×C₈×C₆₇	—	2	4	no	—
56	#95×C₆₇	—	2	4	no	—
57	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	—	2	8	no	—
59	C₂²×C₄×C₆₇	—	2	4	no	—
60	C₂²×C₄×C₆₇	—	2	4	no	—
61	C₂²×C₄×C₆₇	—	4	4	no	—
62	C₂²×C₄×C₆₇	—	4	8	no	—
63	C₂²×C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
70	C₂²×C₄×C₆₇	#41	1	4	yes	—
71	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
72	M₁₆×C₆₇	—	2	16	no	—
73	C₂×C₈×C₆₇	—	2	4	no	—
74	C₂×C₈×C₆₇	—	2	4	no	—
75	C₂×C₈×C₆₇	C₂×Q₈	1	8	yes	—
76	C₄⋊C₄×C₆₇	—	2	16	no	—
77	C₄⋊C₄×C₆₇	—	2	16	no	—
78	C₄⋊C₄×C₆₇	—	2	8	no	—
79	C₄⋊C₄×C₆₇	—	2	4	no	—
80	C₄⋊C₄×C₆₇	—	2	8	no	—
81	C₄⋊C₄×C₆₇	—	4	8	no	—
82	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
83	C₂²⋊C₄×C₆₇	—	4	4	no	—
84	C₄²×C₆₇	—	4	4	no	—
85	C₄²×C₆₇	C₂²⋊C₄	1	8	yes	—
86	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
87	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
88	C₄²×C₆₇	C₂×D₄	1	2	yes	—
89	C₄²×C₆₇	C₂×D₄	1	2	yes	—
90	C₂³×C₆₇	—	2	4	no	—
91	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
92	C₂³×C₆₇	#124	1	2	yes	—
93	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
94	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	—	2	4	no	—
96	C₂×C₄×C₆₇	—	2	4	no	—
97	C₂×C₄×C₆₇	—	2	8	no	—
98	C₂×C₄×C₆₇	—	4	8	no	—
99	C₂×C₄×C₆₇	—	4	16	no	—
100	C₂×C₄×C₆₇	—	4	8	no	—
101	C₂×C₄×C₆₇	—	4	4	no	—
102	C₂×C₄×C₆₇	—	4	8	no	—
103	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
104	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
105	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
106	C₂×C₄×C₆₇	C₄×Q₈	1	8	yes	—
107	C₂×C₄×C₆₇	C₂²⋊Q₈	1	8	yes	—
108	C₂×C₄×C₆₇	#124	1	2	yes	—
109	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
110	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
113	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
114	C₈×C₆₇	—	2	8	no	—
115	C₂²×C₆₇	—	2	4	no	—
116	C₂²×C₆₇	—	2	4	no	—
117	C₂²×C₆₇	—	2	4	no	—
118	C₂²×C₆₇	#391	1	4	yes	—
119	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
120	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
121	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₂²×C₆₇	#575	1	4	yes	—
123	C₄×C₆₇	—	2	4	no	—
124	C₄×C₆₇	—	2	4	no	—
125	C₄×C₆₇	—	4	8	no	—
126	C₄×C₆₇	—	8	4	no	—
127	C₄×C₆₇	#391	1	4	yes	—
128	C₄×C₆₇	C₂×C₄×Q₈	1	2	yes	—
129	C₄×C₆₇	C₂×C₂²⋊Q₈	1	2	yes	—
130	#21325	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂×C₆₇	—	2	4	no	—
132	C₂×C₆₇	C₂×#391	1	1	yes	characteristic
133	C₂×C₆₇	#2896	1	2	yes	—
134	C₂×C₆₇	#3142	1	2	yes	—
135	C₂×C₆₇	#3166	1	2	yes	—
136	C₂²×#125	C₂×C₆₇	1	1	yes	characteristic
137	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic
138	#2285	C₂×C₆₇	1	1	yes	characteristic
139	#1992	C₂×C₆₇	1	4	yes	complemented
140	#1991	C₂×C₆₇	1	4	yes	complemented
141	#1736	C₂×C₆₇	1	4	yes	—
142	C₆₇	#21325	1	1	yes	characteristic, direct product factor, Sylow
143	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
144	C₂×#125	—	2	2	no	—
145	C₂×#125	—	2	4	no	—
146	C₂×#125	C₄×C₆₇	1	8	yes	complemented
147	C₂×#125	C₂²×C₆₇	1	2	yes	—
148	C₂×#125	C₂²×C₆₇	1	2	yes	—
149	C₂×#117	C₂²×C₆₇	1	2	yes	—
150	C₂²×C₄⋊C₄	—	2	2	no	—
151	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
152	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
153	C₄×M₁₆	—	2	4	no	—
154	C₂×#97	C₂²×C₆₇	1	2	yes	—
155	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
156	#384	C₂²×C₆₇	1	2	yes	—
157	#382	C₂²×C₆₇	1	2	yes	—
158	#380	—	2	2	no	—
159	C₄×C₄⋊C₄	—	2	2	no	—
160	#340	C₂²×C₆₇	1	16	yes	—
161	C₂³×C₄	—	2	2	no	—
162	C₂³×C₄	D₄×C₆₇	1	2	yes	—
163	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	—	2	4	no	—
166	C₂×M₁₆	Q₈×C₆₇	1	8	yes	—
167	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
168	#125	—	2	16	no	—
169	#125	—	2	4	no	—
170	#125	—	2	4	no	—
171	#125	C₂×C₄×C₆₇	1	8	yes	—
172	#125	C₂×C₄×C₆₇	1	8	yes	—
173	#117	—	2	8	no	—
174	C₂×C₄⋊C₄	—	2	16	no	—
175	C₂×C₄⋊C₄	—	2	4	no	—
176	C₂×C₄⋊C₄	—	2	4	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	—	2	4	no	—
180	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	8	yes	—
181	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
182	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
183	C₂×C₂²⋊C₄	—	2	2	no	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
186	C₂×C₄²	D₄×C₆₇	1	2	yes	—
187	C₂×C₄²	D₄×C₆₇	1	4	yes	—
188	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
189	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
190	#97	—	2	4	no	—
191	C₄×C₈	—	2	4	no	—
192	#95	—	2	4	no	—
193	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
194	C₂²×C₄	—	2	8	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	—	4	8	no	—
199	C₂²×C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
200	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
201	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
202	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
203	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
205	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
206	C₂²×C₄	#41×C₆₇	1	4	yes	—
207	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
208	M₁₆	—	2	16	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	C₂×Q₈×C₆₇	1	8	yes	—
212	C₄⋊C₄	—	2	16	no	—
213	C₄⋊C₄	—	2	16	no	—
214	C₄⋊C₄	—	2	8	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	2	8	no	—
217	C₄⋊C₄	—	4	8	no	—
218	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
219	C₂²⋊C₄	—	4	4	no	—
220	C₄²	—	4	4	no	—
221	C₄²	C₂²⋊C₄×C₆₇	1	8	yes	—
222	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
223	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
224	C₄²	C₂×D₄×C₆₇	1	2	yes	—
225	C₄²	C₂×D₄×C₆₇	1	2	yes	—
226	C₂³	—	2	4	no	—
227	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
228	C₂³	#124×C₆₇	1	2	yes	—
229	C₂³	C₄⋊Q₈×C₆₇	1	2	yes	—
230	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	4	8	no	—
235	C₂×C₄	—	4	16	no	—
236	C₂×C₄	—	4	8	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	8	no	—
239	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
240	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
241	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
242	C₂×C₄	C₄×Q₈×C₆₇	1	8	yes	—
243	C₂×C₄	C₂²⋊Q₈×C₆₇	1	8	yes	—
244	C₂×C₄	#124×C₆₇	1	2	yes	—
245	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
246	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
247	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
248	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
249	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
250	C₈	—	2	8	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	#391×C₆₇	1	4	yes	—
255	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
256	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
257	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
258	C₂²	#575×C₆₇	1	4	yes	—
259	C₄	—	2	4	no	—
260	C₄	—	2	4	no	—
261	C₄	—	4	8	no	—
262	C₄	—	8	4	no	—
263	C₄	#391×C₆₇	1	4	yes	—
264	C₄	C₂×C₄×Q₈×C₆₇	1	2	yes	—
265	C₄	C₂×C₂²⋊Q₈×C₆₇	1	2	yes	—
266	C₂	—	2	4	no	—
267	C₂	C₂×#391×C₆₇	1	1	yes	characteristic
268	C₂	C₆₇×#2896	1	2	yes	—
269	C₂	C₆₇×#3142	1	2	yes	—
270	C₂	C₆₇×#3166	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1992	C₂	semidirect
5. C₆₇×#1991	C₂	semidirect
10. C₂×#125×C₆₇	C₄	semidirect
130. #21325	C₆₇	direct (N × H)
139. #1992	C₂×C₆₇	semidirect
140. #1991	C₂×C₆₇	semidirect
142. C₆₇	#21325	direct (N × H)
146. C₂×#125	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#125×C₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₆₇×#2285	2
C₆₇×#1992	2
C₆₇×#1991	2
C₆₇×#1736	2
#21325	67

Subgroups — C67×#21325 #30700576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21325 #30700576