Subgroups — C₆₇×#21402 #30700653

Order 17152.

1226 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#529×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2283	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1991	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1958	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1893	C₂	1	2	yes	maximal
7	C₆₇×#1892	C₂	1	2	yes	maximal, complemented
8	C₄×#103×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₄×#102×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
11	#529×C₆₇	—	2	4	no	—
12	#529×C₆₇	C₄	1	8	yes	complemented
13	C₂×#126×C₆₇	—	2	1	no	—
14	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#123×C₆₇	—	2	1	no	—
17	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₆₇	—	2	1	no	—
21	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#117×C₆₇	—	2	1	no	—
23	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#103×C₆₇	C₂²	1	4	yes	—
25	C₂×#102×C₆₇	C₂²	1	4	yes	—
26	C₂×C₄×C₈×C₆₇	—	2	4	no	—
27	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
29	#383×C₆₇	—	2	1	no	—
30	#382×C₆₇	C₂²	1	2	yes	—
31	#381×C₆₇	—	2	1	no	—
32	#380×C₆₇	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
34	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
35	#376×C₆₇	—	2	1	no	—
36	#340×C₆₇	C₂²	1	4	yes	—
37	#340×C₆₇	C₂²	1	4	yes	—
38	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
39	C₂²×D₄×C₆₇	—	2	1	no	—
40	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	—	2	1	no	—
42	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	—	2	4	no	—
46	C₂²×C₈×C₆₇	—	2	8	no	—
47	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
48	#126×C₆₇	—	2	8	no	—
49	#125×C₆₇	—	2	2	no	—
50	#125×C₆₇	C₂×C₄	1	4	yes	—
51	#124×C₆₇	—	2	2	no	—
52	#124×C₆₇	C₂×C₄	1	4	yes	—
53	#123×C₆₇	—	2	8	no	—
54	C₂²⋊Q₈×C₆₇	—	2	2	no	—
55	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
56	#121×C₆₇	—	2	2	no	—
57	#121×C₆₇	C₂×C₄	1	4	yes	—
58	C₄×Q₈×C₆₇	—	2	2	no	—
59	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
60	C₄×D₄×C₆₇	—	2	8	no	—
61	C₄×D₄×C₆₇	—	2	2	no	—
62	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
63	#117×C₆₇	—	2	8	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
65	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
67	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
72	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
73	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
74	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
75	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
76	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
77	C₂×C₄²×C₆₇	—	2	1	no	—
78	C₂×C₄²×C₆₇	—	2	2	no	—
79	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
80	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
81	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
82	C₂×C₄²×C₆₇	D₄	1	4	yes	—
83	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
84	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
85	#103×C₆₇	—	2	8	no	—
86	#102×C₆₇	—	2	8	no	—
87	C₄×C₈×C₆₇	—	2	4	no	—
88	C₄×C₈×C₆₇	—	2	4	no	—
89	#95×C₆₇	—	2	2	no	—
90	#95×C₆₇	—	2	2	no	—
91	#95×C₆₇	—	2	2	no	—
92	#95×C₆₇	—	2	2	no	—
93	C₂⁴×C₆₇	—	2	1	no	—
94	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂×Q₈×C₆₇	—	2	1	no	—
96	C₂×Q₈×C₆₇	—	2	4	no	—
97	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
98	C₂×D₄×C₆₇	—	2	4	no	—
99	C₂×D₄×C₆₇	—	2	1	no	—
100	C₂×D₄×C₆₇	—	2	4	no	—
101	C₂×D₄×C₆₇	—	4	4	no	—
102	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₆₇	—	2	4	no	—
104	C₂²×C₄×C₆₇	—	2	4	no	—
105	C₂²×C₄×C₆₇	—	2	4	no	—
106	C₂²×C₄×C₆₇	—	2	1	no	—
107	C₂²×C₄×C₆₇	—	2	1	no	—
108	C₂²×C₄×C₆₇	—	2	1	no	—
109	C₂²×C₄×C₆₇	—	2	4	no	—
110	C₂²×C₄×C₆₇	—	2	4	no	—
111	C₂²×C₄×C₆₇	—	4	4	no	—
112	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
113	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
114	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
115	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
122	C₂×C₈×C₆₇	—	2	16	no	—
123	C₂×C₈×C₆₇	—	2	4	no	—
124	C₂×C₈×C₆₇	—	2	4	no	—
125	C₂×C₈×C₆₇	#41	1	8	yes	—
126	C₄⋊C₄×C₆₇	—	2	4	no	—
127	C₄⋊C₄×C₆₇	—	2	4	no	—
128	C₄⋊C₄×C₆₇	—	2	4	no	—
129	C₄⋊C₄×C₆₇	—	2	4	no	—
130	C₄⋊C₄×C₆₇	—	2	1	no	—
131	C₄⋊C₄×C₆₇	—	2	1	no	—
132	C₄⋊C₄×C₆₇	—	2	4	no	—
133	C₄⋊C₄×C₆₇	—	2	4	no	—
134	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
135	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
136	C₂²⋊C₄×C₆₇	—	2	4	no	—
137	C₂²⋊C₄×C₆₇	—	2	4	no	—
138	C₂²⋊C₄×C₆₇	—	2	4	no	—
139	C₂²⋊C₄×C₆₇	—	2	4	no	—
140	C₂²⋊C₄×C₆₇	—	2	4	no	—
141	C₂²⋊C₄×C₆₇	—	4	4	no	—
142	C₄²×C₆₇	—	2	4	no	—
143	C₄²×C₆₇	—	4	4	no	—
144	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
145	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
146	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
147	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
148	C₄²×C₆₇	C₂×D₄	1	2	yes	—
149	C₄²×C₆₇	C₂×D₄	1	2	yes	—
150	C₂³×C₆₇	—	2	2	no	—
151	C₂³×C₆₇	—	2	2	no	—
152	C₂³×C₆₇	—	2	2	no	—
153	C₂³×C₆₇	—	4	2	no	—
154	C₂³×C₆₇	—	4	2	no	—
155	C₂³×C₆₇	—	4	2	no	—
156	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
157	C₂³×C₆₇	#135	1	4	yes	—
158	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
159	Q₈×C₆₇	—	2	4	no	—
160	Q₈×C₆₇	—	4	2	no	—
161	D₄×C₆₇	—	2	4	no	—
162	D₄×C₆₇	—	4	8	no	—
163	D₄×C₆₇	—	4	2	no	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	2	8	no	—
169	C₂×C₄×C₆₇	—	2	4	no	—
170	C₂×C₄×C₆₇	—	2	4	no	—
171	C₂×C₄×C₆₇	—	2	8	no	—
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	2	2	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	4	2	no	—
176	C₂×C₄×C₆₇	—	4	2	no	—
177	C₂×C₄×C₆₇	—	4	2	no	—
178	C₂×C₄×C₆₇	—	4	8	no	—
179	C₂×C₄×C₆₇	—	4	4	no	—
180	C₂×C₄×C₆₇	—	4	4	no	—
181	C₂×C₄×C₆₇	—	4	4	no	—
182	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
183	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
184	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
185	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
186	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
187	C₂×C₄×C₆₇	#135	1	4	yes	—
188	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
189	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
191	C₈×C₆₇	—	2	8	no	—
192	C₂²×C₆₇	—	2	2	no	—
193	C₂²×C₆₇	—	2	8	no	—
194	C₂²×C₆₇	—	2	2	no	—
195	C₂²×C₆₇	—	4	2	no	—
196	C₂²×C₆₇	—	4	2	no	—
197	C₂²×C₆₇	—	4	2	no	—
198	C₂²×C₆₇	—	8	4	no	—
199	C₂²×C₆₇	#416	1	4	yes	—
200	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
201	C₂²×C₆₇	#532	1	2	yes	—
202	C₂²×C₆₇	C₂×#135	1	2	yes	—
203	C₂²×C₆₇	C₂×#135	1	2	yes	—
204	C₄×C₆₇	—	2	2	no	—
205	C₄×C₆₇	—	2	2	no	—
206	C₄×C₆₇	—	2	2	no	—
207	C₄×C₆₇	—	2	2	no	—
208	C₄×C₆₇	—	4	2	no	—
209	C₄×C₆₇	—	4	2	no	—
210	C₄×C₆₇	—	4	2	no	—
211	C₄×C₆₇	—	8	4	no	—
212	C₄×C₆₇	#416	1	4	yes	—
213	C₄×C₆₇	C₄×#41	1	2	yes	—
214	C₄×C₆₇	#525	1	2	yes	—
215	#21402	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
216	C₂×C₆₇	—	2	4	no	—
217	C₂×C₆₇	—	8	2	no	—
218	C₂×C₆₇	#2297	1	1	yes	characteristic
219	C₂×C₆₇	C₂×#416	1	1	yes	characteristic
220	C₂×C₆₇	C₂×#416	1	1	yes	characteristic
221	C₂×C₆₇	#3168	1	4	yes	—
222	C₂×#529	C₂×C₆₇	1	1	yes	characteristic
223	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
224	#2283	C₂×C₆₇	1	1	yes	characteristic, complemented
225	#1991	C₂×C₆₇	1	2	yes	complemented
226	#1958	C₂×C₆₇	1	2	yes	complemented
227	#1893	C₂×C₆₇	1	2	yes	—
228	#1892	C₂×C₆₇	1	2	yes	complemented
229	C₄×#103	C₂×C₆₇	1	2	yes	complemented
230	C₄×#102	C₂×C₆₇	1	2	yes	complemented
231	C₆₇	#21402	1	1	yes	characteristic, direct product factor, Sylow
232	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
233	#529	—	2	4	no	—
234	#529	C₄×C₆₇	1	8	yes	complemented
235	C₂×#126	—	2	1	no	—
236	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
237	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
238	C₂×#123	—	2	1	no	—
239	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
240	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
241	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
242	C₂×C₄×D₄	—	2	1	no	—
243	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
244	C₂×#117	—	2	1	no	—
245	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
246	C₂×#103	C₂²×C₆₇	1	4	yes	—
247	C₂×#102	C₂²×C₆₇	1	4	yes	—
248	C₂×C₄×C₈	—	2	4	no	—
249	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
250	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
251	#383	—	2	1	no	—
252	#382	C₂²×C₆₇	1	2	yes	—
253	#381	—	2	1	no	—
254	#380	C₂²×C₆₇	1	2	yes	—
255	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
256	C₄×C₂²⋊C₄	—	2	1	no	—
257	#376	—	2	1	no	—
258	#340	C₂²×C₆₇	1	4	yes	—
259	#340	C₂²×C₆₇	1	4	yes	—
260	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
261	C₂²×D₄	—	2	1	no	—
262	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
263	C₂³×C₄	—	2	1	no	—
264	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
265	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
266	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
267	C₂²×C₈	—	2	4	no	—
268	C₂²×C₈	—	2	8	no	—
269	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
270	#126	—	2	8	no	—
271	#125	—	2	2	no	—
272	#125	C₂×C₄×C₆₇	1	4	yes	—
273	#124	—	2	2	no	—
274	#124	C₂×C₄×C₆₇	1	4	yes	—
275	#123	—	2	8	no	—
276	C₂²⋊Q₈	—	2	2	no	—
277	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
278	#121	—	2	2	no	—
279	#121	C₂×C₄×C₆₇	1	4	yes	—
280	C₄×Q₈	—	2	2	no	—
281	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
282	C₄×D₄	—	2	8	no	—
283	C₄×D₄	—	2	2	no	—
284	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
285	#117	—	2	8	no	—
286	C₂×C₄⋊C₄	—	2	1	no	—
287	C₂×C₄⋊C₄	—	2	1	no	—
288	C₂×C₄⋊C₄	—	2	1	no	—
289	C₂×C₄⋊C₄	—	2	1	no	—
290	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
291	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
292	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
293	C₂×C₂²⋊C₄	—	2	1	no	—
294	C₂×C₂²⋊C₄	—	2	1	no	—
295	C₂×C₂²⋊C₄	—	2	1	no	—
296	C₂×C₂²⋊C₄	—	2	1	no	—
297	C₂×C₂²⋊C₄	—	2	1	no	—
298	C₂×C₂²⋊C₄	—	2	2	no	—
299	C₂×C₄²	—	2	1	no	—
300	C₂×C₄²	—	2	2	no	—
301	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
302	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
303	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
304	C₂×C₄²	D₄×C₆₇	1	4	yes	—
305	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
306	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
307	#103	—	2	8	no	—
308	#102	—	2	8	no	—
309	C₄×C₈	—	2	4	no	—
310	C₄×C₈	—	2	4	no	—
311	#95	—	2	2	no	—
312	#95	—	2	2	no	—
313	#95	—	2	2	no	—
314	#95	—	2	2	no	—
315	C₂⁴	—	2	1	no	—
316	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
317	C₂×Q₈	—	2	1	no	—
318	C₂×Q₈	—	2	4	no	—
319	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
320	C₂×D₄	—	2	4	no	—
321	C₂×D₄	—	2	1	no	—
322	C₂×D₄	—	2	4	no	—
323	C₂×D₄	—	4	4	no	—
324	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
325	C₂²×C₄	—	2	4	no	—
326	C₂²×C₄	—	2	4	no	—
327	C₂²×C₄	—	2	4	no	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	1	no	—
330	C₂²×C₄	—	2	1	no	—
331	C₂²×C₄	—	2	4	no	—
332	C₂²×C₄	—	2	4	no	—
333	C₂²×C₄	—	4	4	no	—
334	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
335	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
336	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
337	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
338	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
339	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
340	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
341	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
342	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
343	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
344	C₂×C₈	—	2	16	no	—
345	C₂×C₈	—	2	4	no	—
346	C₂×C₈	—	2	4	no	—
347	C₂×C₈	#41×C₆₇	1	8	yes	—
348	C₄⋊C₄	—	2	4	no	—
349	C₄⋊C₄	—	2	4	no	—
350	C₄⋊C₄	—	2	4	no	—
351	C₄⋊C₄	—	2	4	no	—
352	C₄⋊C₄	—	2	1	no	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	2	4	no	—
355	C₄⋊C₄	—	2	4	no	—
356	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
357	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
358	C₂²⋊C₄	—	2	4	no	—
359	C₂²⋊C₄	—	2	4	no	—
360	C₂²⋊C₄	—	2	4	no	—
361	C₂²⋊C₄	—	2	4	no	—
362	C₂²⋊C₄	—	2	4	no	—
363	C₂²⋊C₄	—	4	4	no	—
364	C₄²	—	2	4	no	—
365	C₄²	—	4	4	no	—
366	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
367	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
368	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
369	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
370	C₄²	C₂×D₄×C₆₇	1	2	yes	—
371	C₄²	C₂×D₄×C₆₇	1	2	yes	—
372	C₂³	—	2	2	no	—
373	C₂³	—	2	2	no	—
374	C₂³	—	2	2	no	—
375	C₂³	—	4	2	no	—
376	C₂³	—	4	2	no	—
377	C₂³	—	4	2	no	—
378	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
379	C₂³	#135×C₆₇	1	4	yes	—
380	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
381	Q₈	—	2	4	no	—
382	Q₈	—	4	2	no	—
383	D₄	—	2	4	no	—
384	D₄	—	4	8	no	—
385	D₄	—	4	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	8	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	2	4	no	—
393	C₂×C₄	—	2	8	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	8	no	—
401	C₂×C₄	—	4	4	no	—
402	C₂×C₄	—	4	4	no	—
403	C₂×C₄	—	4	4	no	—
404	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
405	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
406	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
407	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
408	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
409	C₂×C₄	#135×C₆₇	1	4	yes	—
410	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
411	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
412	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
413	C₈	—	2	8	no	—
414	C₂²	—	2	2	no	—
415	C₂²	—	2	2	no	—
416	C₂²	—	2	8	no	—
417	C₂²	—	4	2	no	—
418	C₂²	—	4	2	no	—
419	C₂²	—	4	2	no	—
420	C₂²	—	8	4	no	—
421	C₂²	#416×C₆₇	1	4	yes	—
422	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
423	C₂²	#532×C₆₇	1	2	yes	—
424	C₂²	C₂×#135×C₆₇	1	2	yes	—
425	C₂²	C₂×#135×C₆₇	1	2	yes	—
426	C₄	—	2	2	no	—
427	C₄	—	2	2	no	—
428	C₄	—	2	2	no	—
429	C₄	—	2	2	no	—
430	C₄	—	4	2	no	—
431	C₄	—	4	2	no	—
432	C₄	—	4	2	no	—
433	C₄	—	8	4	no	—
434	C₄	#416×C₆₇	1	4	yes	—
435	C₄	C₄×#41×C₆₇	1	2	yes	—
436	C₄	#525×C₆₇	1	2	yes	—
437	C₂	—	2	4	no	—
438	C₂	—	8	2	no	—
439	C₂	C₆₇×#2297	1	1	yes	characteristic
440	C₂	C₂×#416×C₆₇	1	1	yes	characteristic
441	C₂	C₂×#416×C₆₇	1	1	yes	characteristic
442	C₂	C₆₇×#3168	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂×#121×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×C₈×C₆₇	C₂	semidirect
3. C₆₇×#2283	C₂	semidirect
4. C₆₇×#1991	C₂	semidirect
5. C₆₇×#1958	C₂	semidirect
7. C₆₇×#1892	C₂	semidirect
8. C₄×#103×C₆₇	C₂	semidirect
9. C₄×#102×C₆₇	C₂	semidirect
12. #529×C₆₇	C₄	semidirect
215. #21402	C₆₇	direct (N × H)
223. C₂²×C₄×C₈	C₂×C₆₇	semidirect
224. #2283	C₂×C₆₇	semidirect
225. #1991	C₂×C₆₇	semidirect
226. #1958	C₂×C₆₇	semidirect
228. #1892	C₂×C₆₇	semidirect
229. C₄×#103	C₂×C₆₇	semidirect
230. C₄×#102	C₂×C₆₇	semidirect
231. C₆₇	#21402	direct (N × H)
234. #529	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#529×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₆₇×#2283	2
C₆₇×#1991	2
C₆₇×#1958	2
C₆₇×#1893	2
C₆₇×#1892	2
C₄×#103×C₆₇	2
C₄×#102×C₆₇	2
#21402	67

Subgroups — C67×#21402 #30700653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21402 #30700653