Subgroups — C₆₇×#21146 #30700397

Order 17152.

1114 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#423×C₆₇	C₂	1	2	yes	maximal
2	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1940	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1939	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1728	C₂	1	2	yes	maximal
8	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
12	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
13	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
14	#423×C₆₇	—	2	8	no	—
15	#423×C₆₇	—	2	4	no	—
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	8	yes	—
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
19	C₄×M₁₆×C₆₇	—	2	4	no	—
20	C₂×#97×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
22	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
24	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
25	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
27	C₂×#95×C₆₇	C₄	1	2	yes	—
28	#336×C₆₇	C₂²	1	8	yes	—
29	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
30	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₆₇	—	2	2	no	—
34	C₂×M₁₆×C₆₇	—	2	2	no	—
35	C₂×M₁₆×C₆₇	—	2	2	no	—
36	C₂×M₁₆×C₆₇	—	2	2	no	—
37	C₂×M₁₆×C₆₇	—	2	4	no	—
38	C₂×M₁₆×C₆₇	—	2	4	no	—
39	C₂×M₁₆×C₆₇	—	2	4	no	—
40	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
41	C₂×M₁₆×C₆₇	Q₈	1	4	yes	—
42	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
43	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
44	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
46	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
48	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₆₇	—	2	4	no	—
50	C₂×C₄²×C₆₇	—	2	4	no	—
51	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	D₄	1	4	yes	—
56	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
57	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
58	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₆₇	—	2	16	no	—
61	C₄⋊C₈×C₆₇	—	2	4	no	—
62	C₄⋊C₈×C₆₇	—	2	4	no	—
63	#97×C₆₇	—	2	4	no	—
64	C₄×C₈×C₆₇	—	2	4	no	—
65	#95×C₆₇	C₂×C₄	1	2	yes	—
66	#95×C₆₇	C₂×C₄	1	4	yes	—
67	#95×C₆₇	C₂×C₄	1	2	yes	—
68	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	—	2	4	no	—
70	C₂²×C₄×C₆₇	—	2	8	no	—
71	C₂²×C₄×C₆₇	—	2	8	no	—
72	C₂²×C₄×C₆₇	—	4	2	no	—
73	C₂²×C₄×C₆₇	—	4	4	no	—
74	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
75	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
76	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
81	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
87	C₂²×C₄×C₆₇	#41	1	2	yes	—
88	C₂²×C₄×C₆₇	#41	1	2	yes	—
89	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
90	M₁₆×C₆₇	—	2	8	no	—
91	M₁₆×C₆₇	—	2	8	no	—
92	M₁₆×C₆₇	—	4	8	no	—
93	C₂×C₈×C₆₇	—	2	2	no	—
94	C₂×C₈×C₆₇	—	2	2	no	—
95	C₂×C₈×C₆₇	—	2	2	no	—
96	C₂×C₈×C₆₇	—	2	2	no	—
97	C₂×C₈×C₆₇	—	2	4	no	—
98	C₂×C₈×C₆₇	—	2	4	no	—
99	C₂×C₈×C₆₇	—	2	4	no	—
100	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
101	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
102	C₄⋊C₄×C₆₇	—	2	8	no	—
103	C₄⋊C₄×C₆₇	—	2	4	no	—
104	C₄⋊C₄×C₆₇	M₁₆	1	8	yes	complemented
105	C₄²×C₆₇	—	2	4	no	—
106	C₄²×C₆₇	—	2	4	no	—
107	C₄²×C₆₇	—	2	4	no	—
108	C₄²×C₆₇	—	2	4	no	—
109	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
110	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
111	C₄²×C₆₇	C₂×D₄	1	2	yes	—
112	C₄²×C₆₇	C₂×D₄	1	2	yes	—
113	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
114	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
115	C₂³×C₆₇	—	2	2	no	—
116	C₂³×C₆₇	—	2	2	no	—
117	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂³×C₆₇	C₄×D₄	1	2	yes	—
119	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
120	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	—	2	4	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	4	no	—
126	C₂×C₄×C₆₇	—	2	8	no	—
127	C₂×C₄×C₆₇	—	2	4	no	—
128	C₂×C₄×C₆₇	—	2	4	no	—
129	C₂×C₄×C₆₇	—	2	4	no	—
130	C₂×C₄×C₆₇	—	4	4	no	—
131	C₂×C₄×C₆₇	—	4	16	no	—
132	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
138	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
139	C₂×C₄×C₆₇	#121	1	4	yes	—
140	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
141	C₂×C₄×C₆₇	#125	1	4	yes	—
142	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
143	C₂×C₄×C₆₇	C₂×M₁₆	1	8	yes	—
144	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
145	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
148	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
151	C₈×C₆₇	—	2	4	no	—
152	C₈×C₆₇	—	2	4	no	—
153	C₈×C₆₇	—	4	4	no	—
154	C₂²×C₆₇	—	2	2	no	—
155	C₂²×C₆₇	—	2	2	no	—
156	C₂²×C₆₇	—	2	4	no	—
157	C₂²×C₆₇	—	2	4	no	—
158	C₂²×C₆₇	#389	1	4	yes	—
159	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
160	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
161	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
162	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
163	C₂²×C₆₇	#568	1	2	yes	—
164	C₄×C₆₇	—	2	2	no	—
165	C₄×C₆₇	—	2	2	no	—
166	C₄×C₆₇	—	2	2	no	—
167	C₄×C₆₇	—	2	8	no	—
168	C₄×C₆₇	—	4	4	no	—
169	C₄×C₆₇	#389	1	4	yes	—
170	C₄×C₆₇	#423	1	4	yes	complemented
171	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
172	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
173	C₄×C₆₇	C₂×#125	1	1	yes	characteristic
174	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
175	#21146	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂×C₆₇	—	2	4	no	—
177	C₂×C₆₇	C₂×#389	1	1	yes	characteristic
178	C₂×C₆₇	C₂×#423	1	1	yes	characteristic
179	C₂×C₆₇	#2905	1	1	yes	characteristic
180	C₂×C₆₇	D₄×M₁₆	1	2	yes	—
181	C₂×C₆₇	Q₈×M₁₆	1	2	yes	—
182	C₂×#423	C₂×C₆₇	1	1	yes	characteristic
183	C₂×#423	C₂×C₆₇	1	2	yes	—
184	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic
185	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
186	#1940	C₂×C₆₇	1	4	yes	complemented
187	#1939	C₂×C₆₇	1	4	yes	complemented
188	#1728	C₂×C₆₇	1	2	yes	—
189	C₆₇	#21146	1	1	yes	characteristic, direct product factor, Sylow
190	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
191	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
192	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
193	C₂²×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
194	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
195	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
196	#423	—	2	8	no	—
197	#423	—	2	4	no	—
198	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
199	C₂×C₄⋊C₈	C₂²×C₆₇	1	8	yes	—
200	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
201	C₄×M₁₆	—	2	4	no	—
202	C₂×#97	C₂²×C₆₇	1	2	yes	—
203	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
204	C₄×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
205	C₄×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
206	C₄×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
207	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
208	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
209	C₂×#95	C₄×C₆₇	1	2	yes	—
210	#336	C₂²×C₆₇	1	8	yes	—
211	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	—
212	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
213	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
214	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
215	C₂×M₁₆	—	2	2	no	—
216	C₂×M₁₆	—	2	2	no	—
217	C₂×M₁₆	—	2	2	no	—
218	C₂×M₁₆	—	2	2	no	—
219	C₂×M₁₆	—	2	4	no	—
220	C₂×M₁₆	—	2	4	no	—
221	C₂×M₁₆	—	2	4	no	—
222	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
223	C₂×M₁₆	Q₈×C₆₇	1	4	yes	—
224	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
225	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
226	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
227	C₂×C₄⋊C₄	—	2	8	no	—
228	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
229	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
230	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
231	C₂×C₄²	—	2	4	no	—
232	C₂×C₄²	—	2	4	no	—
233	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
234	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
235	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
236	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
237	C₂×C₄²	D₄×C₆₇	1	4	yes	—
238	C₂×C₄²	Q₈×C₆₇	1	4	yes	—
239	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
240	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
241	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
242	C₄⋊C₈	—	2	16	no	—
243	C₄⋊C₈	—	2	4	no	—
244	C₄⋊C₈	—	2	4	no	—
245	#97	—	2	4	no	—
246	C₄×C₈	—	2	4	no	—
247	#95	C₂×C₄×C₆₇	1	2	yes	—
248	#95	C₂×C₄×C₆₇	1	4	yes	—
249	#95	C₂×C₄×C₆₇	1	2	yes	—
250	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
251	C₂²×C₄	—	2	4	no	—
252	C₂²×C₄	—	2	8	no	—
253	C₂²×C₄	—	2	8	no	—
254	C₂²×C₄	—	4	2	no	—
255	C₂²×C₄	—	4	4	no	—
256	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
257	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
258	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
259	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
260	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
261	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
262	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
263	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
264	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
265	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
266	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
267	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
268	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
269	C₂²×C₄	#41×C₆₇	1	2	yes	—
270	C₂²×C₄	#41×C₆₇	1	2	yes	—
271	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
272	M₁₆	—	2	8	no	—
273	M₁₆	—	2	8	no	—
274	M₁₆	—	4	8	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	2	4	no	—
282	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
283	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
284	C₄⋊C₄	—	2	8	no	—
285	C₄⋊C₄	—	2	4	no	—
286	C₄⋊C₄	M₁₆×C₆₇	1	8	yes	complemented
287	C₄²	—	2	4	no	—
288	C₄²	—	2	4	no	—
289	C₄²	—	2	4	no	—
290	C₄²	—	2	4	no	—
291	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
292	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
293	C₄²	C₂×D₄×C₆₇	1	2	yes	—
294	C₄²	C₂×D₄×C₆₇	1	2	yes	—
295	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
296	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
297	C₂³	—	2	2	no	—
298	C₂³	—	2	2	no	—
299	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
300	C₂³	C₄×D₄×C₆₇	1	2	yes	—
301	C₂³	C₄×Q₈×C₆₇	1	2	yes	—
302	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	2	8	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	4	4	no	—
313	C₂×C₄	—	4	16	no	—
314	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
315	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
316	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
317	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
318	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
319	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
320	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
321	C₂×C₄	#121×C₆₇	1	4	yes	—
322	C₂×C₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
323	C₂×C₄	#125×C₆₇	1	4	yes	—
324	C₂×C₄	C₄⋊Q₈×C₆₇	1	4	yes	—
325	C₂×C₄	C₂×M₁₆×C₆₇	1	8	yes	—
326	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
327	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
328	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
329	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
330	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
331	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
332	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
333	C₈	—	2	4	no	—
334	C₈	—	2	4	no	—
335	C₈	—	4	4	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	4	no	—
339	C₂²	—	2	4	no	—
340	C₂²	#389×C₆₇	1	4	yes	—
341	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
342	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
343	C₂²	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
344	C₂²	C₂²×M₁₆×C₆₇	1	2	yes	—
345	C₂²	#568×C₆₇	1	2	yes	—
346	C₄	—	2	2	no	—
347	C₄	—	2	2	no	—
348	C₄	—	2	2	no	—
349	C₄	—	2	8	no	—
350	C₄	—	4	4	no	—
351	C₄	#389×C₆₇	1	4	yes	—
352	C₄	#423×C₆₇	1	4	yes	complemented
353	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
354	C₄	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
355	C₄	C₂×#125×C₆₇	1	1	yes	characteristic
356	C₄	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
357	C₂	—	2	4	no	—
358	C₂	C₂×#389×C₆₇	1	1	yes	characteristic
359	C₂	C₂×#423×C₆₇	1	1	yes	characteristic
360	C₂	C₆₇×#2905	1	1	yes	characteristic
361	C₂	D₄×M₁₆×C₆₇	1	2	yes	—
362	C₂	Q₈×M₁₆×C₆₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
5. C₆₇×#1940	C₂	semidirect
6. C₆₇×#1939	C₂	semidirect
104. C₄⋊C₄×C₆₇	M₁₆	semidirect
170. C₄×C₆₇	#423	semidirect
175. #21146	C₆₇	direct (N × H)
186. #1940	C₂×C₆₇	semidirect
187. #1939	C₂×C₆₇	semidirect
189. C₆₇	#21146	direct (N × H)
286. C₄⋊C₄	M₁₆×C₆₇	semidirect
352. C₄	#423×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#423×C₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₆₇×#1940	2
C₆₇×#1939	2
C₆₇×#1728	2
#21146	67

Subgroups — C67×#21146 #30700397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21146 #30700397