Subgroups — C₆₇×#33137 #30712388

Order 17152.

852 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#465×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2625	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2624	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1968	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1941	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1936	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1935	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1895	C₂	1	1	yes	maximal, characteristic
10	C₆₇×#1894	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1807	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
18	#465×C₆₇	—	2	4	no	—
19	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#106×C₆₇	C₂²	1	2	yes	—
21	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#103×C₆₇	C₂²	1	2	yes	—
23	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#102×C₆₇	C₂²	1	2	yes	—
25	#405×C₆₇	—	2	4	no	—
26	C₂×#97×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
28	#400×C₆₇	—	2	1	no	—
29	#399×C₆₇	—	2	1	no	—
30	#398×C₆₇	—	2	1	no	—
31	#397×C₆₇	—	2	1	no	—
32	#397×C₆₇	—	2	2	no	—
33	#394×C₆₇	—	2	2	no	—
34	#391×C₆₇	C₂²	1	2	yes	—
35	#390×C₆₇	C₂²	1	2	yes	—
36	#389×C₆₇	C₂²	1	2	yes	—
37	#389×C₆₇	C₂²	1	2	yes	—
38	#380×C₆₇	C₂²	1	1	yes	characteristic
39	#380×C₆₇	C₂²	1	1	yes	characteristic
40	#340×C₆₇	C₂²	1	2	yes	—
41	#340×C₆₇	C₂²	1	2	yes	—
42	#340×C₆₇	C₂²	1	2	yes	—
43	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₆₇	—	2	1	no	—
45	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
47	C₂×SD₁₆×C₆₇	—	2	2	no	—
48	C₂×SD₁₆×C₆₇	—	2	4	no	—
49	C₂×M₁₆×C₆₇	—	2	2	no	—
50	C₂×M₁₆×C₆₇	—	2	4	no	—
51	C₂²×C₈×C₆₇	—	2	4	no	—
52	C₂²×C₈×C₆₇	D₄	1	4	yes	—
53	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²⋊Q₈×C₆₇	—	2	2	no	—
57	C₂²⋊Q₈×C₆₇	—	2	2	no	—
58	#121×C₆₇	—	2	2	no	—
59	#121×C₆₇	—	2	2	no	—
60	#117×C₆₇	—	2	2	no	—
61	#117×C₆₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
65	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
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	2	no	—
76	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
78	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
79	#106×C₆₇	—	2	2	no	—
80	#106×C₆₇	—	2	4	no	—
81	#103×C₆₇	—	2	2	no	—
82	#103×C₆₇	—	2	4	no	—
83	#102×C₆₇	—	2	2	no	—
84	#102×C₆₇	—	2	4	no	—
85	#97×C₆₇	—	2	4	no	—
86	C₄×C₈×C₆₇	—	2	4	no	—
87	#95×C₆₇	—	2	1	no	—
88	#95×C₆₇	—	2	1	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	2	no	—
97	C₂×Q₈×C₆₇	—	2	1	no	—
98	C₂×Q₈×C₆₇	—	2	2	no	—
99	C₂×D₄×C₆₇	—	2	1	no	—
100	C₂×D₄×C₆₇	—	2	2	no	—
101	C₂×D₄×C₆₇	—	2	1	no	—
102	C₂×D₄×C₆₇	—	2	2	no	—
103	C₂×D₄×C₆₇	—	4	2	no	—
104	C₂×D₄×C₆₇	—	8	2	no	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	2	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	2	1	no	—
109	C₂²×C₄×C₆₇	—	2	1	no	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	—	2	1	no	—
112	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
116	C₂²×C₄×C₆₇	#41	1	2	yes	—
117	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
118	SD₁₆×C₆₇	—	4	4	no	—
119	M₁₆×C₆₇	—	2	8	no	—
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	—	2	2	no	—
123	C₂×C₈×C₆₇	—	2	8	no	—
124	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
126	C₂×C₈×C₆₇	#41	1	4	yes	—
127	C₂×C₈×C₆₇	#41	1	4	yes	—
128	C₄⋊C₄×C₆₇	—	2	2	no	—
129	C₄⋊C₄×C₆₇	—	2	2	no	—
130	C₄⋊C₄×C₆₇	—	2	1	no	—
131	C₄⋊C₄×C₆₇	—	2	1	no	—
132	C₄⋊C₄×C₆₇	—	2	2	no	—
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	—	2	2	no	—
135	C₄⋊C₄×C₆₇	—	2	2	no	—
136	C₄⋊C₄×C₆₇	—	2	1	no	—
137	C₄⋊C₄×C₆₇	—	2	1	no	—
138	C₄⋊C₄×C₆₇	—	4	2	no	—
139	C₄⋊C₄×C₆₇	—	8	2	no	—
140	C₄⋊C₄×C₆₇	—	8	2	no	—
141	C₂²⋊C₄×C₆₇	—	2	4	no	—
142	C₂²⋊C₄×C₆₇	—	4	2	no	—
143	C₂²⋊C₄×C₆₇	—	4	2	no	—
144	C₂²⋊C₄×C₆₇	—	8	2	no	—
145	C₂²⋊C₄×C₆₇	—	8	2	no	—
146	C₂²⋊C₄×C₆₇	—	8	1	no	—
147	C₂²⋊C₄×C₆₇	—	8	1	no	—
148	C₄²×C₆₇	—	2	2	no	—
149	C₄²×C₆₇	—	2	2	no	—
150	C₂³×C₆₇	—	2	2	no	—
151	C₂³×C₆₇	—	2	1	no	—
152	C₂³×C₆₇	—	2	1	no	—
153	C₂³×C₆₇	—	2	2	no	—
154	C₂³×C₆₇	—	4	1	no	—
155	C₂³×C₆₇	—	4	2	no	—
156	C₂³×C₆₇	—	4	1	no	—
157	C₂³×C₆₇	—	4	2	no	—
158	C₂³×C₆₇	—	4	1	no	—
159	C₂³×C₆₇	#121	1	2	yes	—
160	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
161	Q₈×C₆₇	—	4	1	no	—
162	Q₈×C₆₇	—	4	1	no	—
163	Q₈×C₆₇	—	4	2	no	—
164	D₄×C₆₇	—	4	1	no	—
165	D₄×C₆₇	—	4	1	no	—
166	D₄×C₆₇	—	4	2	no	—
167	D₄×C₆₇	—	8	4	no	—
168	C₂×C₄×C₆₇	—	2	2	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	2	2	no	—
171	C₂×C₄×C₆₇	—	2	4	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	—	2	2	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	4	1	no	—
179	C₂×C₄×C₆₇	—	4	1	no	—
180	C₂×C₄×C₆₇	—	4	1	no	—
181	C₂×C₄×C₆₇	—	4	2	no	—
182	C₂×C₄×C₆₇	—	4	2	no	—
183	C₂×C₄×C₆₇	—	4	2	no	—
184	C₂×C₄×C₆₇	—	4	2	no	—
185	C₂×C₄×C₆₇	—	4	2	no	—
186	C₂×C₄×C₆₇	—	4	2	no	—
187	C₂×C₄×C₆₇	—	4	1	no	—
188	C₂×C₄×C₆₇	—	4	2	no	—
189	C₂×C₄×C₆₇	—	4	2	no	—
190	C₂×C₄×C₆₇	—	4	1	no	—
191	C₂×C₄×C₆₇	#121	1	2	yes	—
192	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
194	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
195	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
196	C₈×C₆₇	—	2	2	no	—
197	C₈×C₆₇	—	2	4	no	—
198	C₈×C₆₇	#123	1	4	yes	—
199	C₂²×C₆₇	—	2	2	no	—
200	C₂²×C₆₇	—	2	2	no	—
201	C₂²×C₆₇	—	2	2	no	—
202	C₂²×C₆₇	—	4	2	no	—
203	C₂²×C₆₇	—	4	2	no	—
204	C₂²×C₆₇	—	8	1	no	—
205	C₂²×C₆₇	—	8	2	no	—
206	C₂²×C₆₇	—	8	2	no	—
207	C₂²×C₆₇	—	8	1	no	—
208	C₂²×C₆₇	—	8	1	no	—
209	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
210	C₂²×C₆₇	#532	1	2	yes	—
211	C₂²×C₆₇	#577	1	1	yes	characteristic
212	C₂²×C₆₇	#577	1	2	yes	—
213	C₂²×C₆₇	#577	1	1	yes	characteristic
214	C₄×C₆₇	—	2	2	no	—
215	C₄×C₆₇	—	4	2	no	—
216	C₄×C₆₇	—	4	2	no	—
217	C₄×C₆₇	—	8	1	no	—
218	C₄×C₆₇	—	8	2	no	—
219	C₄×C₆₇	—	8	2	no	—
220	C₄×C₆₇	—	8	1	no	—
221	C₄×C₆₇	—	8	1	no	—
222	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
223	C₄×C₆₇	#525	1	1	yes	characteristic
224	C₄×C₆₇	#529	1	2	yes	—
225	#33137	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
226	C₂×C₆₇	—	4	2	no	—
227	C₂×C₆₇	—	16	1	no	—
228	C₂×C₆₇	#2854	1	1	yes	characteristic
229	C₂×C₆₇	#3057	1	1	yes	characteristic
230	C₂×C₆₇	#3060	1	1	yes	characteristic
231	C₂×C₆₇	#3173	1	2	yes	—
232	C₂×C₆₇	#3174	1	2	yes	—
233	C₂×#465	C₂×C₆₇	1	1	yes	characteristic
234	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
235	#2625	C₂×C₆₇	1	1	yes	characteristic, complemented
236	#2624	C₂×C₆₇	1	1	yes	characteristic
237	#1968	C₂×C₆₇	1	2	yes	complemented
238	#1941	C₂×C₆₇	1	2	yes	complemented
239	#1936	C₂×C₆₇	1	2	yes	complemented
240	#1935	C₂×C₆₇	1	2	yes	complemented
241	#1895	C₂×C₆₇	1	1	yes	characteristic
242	#1894	C₂×C₆₇	1	1	yes	characteristic, complemented
243	#1807	C₂×C₆₇	1	1	yes	characteristic, complemented
244	C₆₇	#33137	1	1	yes	characteristic, direct product factor, Sylow
245	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
246	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
247	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
248	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
249	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
250	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
251	#465	—	2	4	no	—
252	C₂×#106	C₂²×C₆₇	1	2	yes	—
253	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
254	C₂×#103	C₂²×C₆₇	1	2	yes	—
255	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
256	C₂×#102	C₂²×C₆₇	1	2	yes	—
257	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
258	#405	—	2	4	no	—
259	C₂×#97	C₂²×C₆₇	1	2	yes	—
260	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
261	#400	—	2	1	no	—
262	#399	—	2	1	no	—
263	#398	—	2	1	no	—
264	#397	—	2	1	no	—
265	#397	—	2	2	no	—
266	#394	—	2	2	no	—
267	#391	C₂²×C₆₇	1	2	yes	—
268	#390	C₂²×C₆₇	1	2	yes	—
269	#389	C₂²×C₆₇	1	2	yes	—
270	#389	C₂²×C₆₇	1	2	yes	—
271	#380	C₂²×C₆₇	1	1	yes	characteristic
272	#380	C₂²×C₆₇	1	1	yes	characteristic
273	#340	C₂²×C₆₇	1	2	yes	—
274	#340	C₂²×C₆₇	1	2	yes	—
275	#340	C₂²×C₆₇	1	2	yes	—
276	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
277	C₂²×D₄	—	2	1	no	—
278	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
279	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
280	C₂×SD₁₆	—	2	2	no	—
281	C₂×SD₁₆	—	2	4	no	—
282	C₂×M₁₆	—	2	2	no	—
283	C₂×M₁₆	—	2	4	no	—
284	C₂²×C₈	—	2	4	no	—
285	C₂²×C₈	D₄×C₆₇	1	4	yes	—
286	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
287	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
288	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
289	C₂²⋊Q₈	—	2	2	no	—
290	C₂²⋊Q₈	—	2	2	no	—
291	#121	—	2	2	no	—
292	#121	—	2	2	no	—
293	#117	—	2	2	no	—
294	#117	—	2	2	no	—
295	C₂×C₄⋊C₄	—	2	1	no	—
296	C₂×C₄⋊C₄	—	2	2	no	—
297	C₂×C₄⋊C₄	—	2	2	no	—
298	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
299	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
300	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
301	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
302	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
303	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
304	C₂×C₂²⋊C₄	—	2	1	no	—
305	C₂×C₂²⋊C₄	—	2	1	no	—
306	C₂×C₂²⋊C₄	—	2	1	no	—
307	C₂×C₂²⋊C₄	—	2	1	no	—
308	C₂×C₂²⋊C₄	—	2	2	no	—
309	C₂×C₂²⋊C₄	—	2	2	no	—
310	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
311	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
312	#106	—	2	2	no	—
313	#106	—	2	4	no	—
314	#103	—	2	2	no	—
315	#103	—	2	4	no	—
316	#102	—	2	2	no	—
317	#102	—	2	4	no	—
318	#97	—	2	4	no	—
319	C₄×C₈	—	2	4	no	—
320	#95	—	2	1	no	—
321	#95	—	2	1	no	—
322	#95	—	2	2	no	—
323	#95	—	2	2	no	—
324	#95	—	2	2	no	—
325	#95	—	2	2	no	—
326	C₂⁴	—	2	1	no	—
327	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
328	C₂×Q₈	—	2	1	no	—
329	C₂×Q₈	—	2	2	no	—
330	C₂×Q₈	—	2	1	no	—
331	C₂×Q₈	—	2	2	no	—
332	C₂×D₄	—	2	1	no	—
333	C₂×D₄	—	2	2	no	—
334	C₂×D₄	—	2	1	no	—
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	—	4	2	no	—
337	C₂×D₄	—	8	2	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
346	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
347	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
348	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
349	C₂²×C₄	#41×C₆₇	1	2	yes	—
350	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
351	SD₁₆	—	4	4	no	—
352	M₁₆	—	2	8	no	—
353	C₂×C₈	—	2	2	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	8	no	—
357	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
358	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
359	C₂×C₈	#41×C₆₇	1	4	yes	—
360	C₂×C₈	#41×C₆₇	1	4	yes	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	1	no	—
370	C₄⋊C₄	—	2	1	no	—
371	C₄⋊C₄	—	4	2	no	—
372	C₄⋊C₄	—	8	2	no	—
373	C₄⋊C₄	—	8	2	no	—
374	C₂²⋊C₄	—	2	4	no	—
375	C₂²⋊C₄	—	4	2	no	—
376	C₂²⋊C₄	—	4	2	no	—
377	C₂²⋊C₄	—	8	2	no	—
378	C₂²⋊C₄	—	8	2	no	—
379	C₂²⋊C₄	—	8	1	no	—
380	C₂²⋊C₄	—	8	1	no	—
381	C₄²	—	2	2	no	—
382	C₄²	—	2	2	no	—
383	C₂³	—	2	2	no	—
384	C₂³	—	2	1	no	—
385	C₂³	—	2	1	no	—
386	C₂³	—	2	2	no	—
387	C₂³	—	4	1	no	—
388	C₂³	—	4	2	no	—
389	C₂³	—	4	1	no	—
390	C₂³	—	4	2	no	—
391	C₂³	—	4	1	no	—
392	C₂³	#121×C₆₇	1	2	yes	—
393	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
394	Q₈	—	4	1	no	—
395	Q₈	—	4	1	no	—
396	Q₈	—	4	2	no	—
397	D₄	—	4	1	no	—
398	D₄	—	4	1	no	—
399	D₄	—	4	2	no	—
400	D₄	—	8	4	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	4	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	4	1	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	#121×C₆₇	1	2	yes	—
425	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
426	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
427	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
428	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
429	C₈	—	2	2	no	—
430	C₈	—	2	4	no	—
431	C₈	#123×C₆₇	1	4	yes	—
432	C₂²	—	2	2	no	—
433	C₂²	—	2	2	no	—
434	C₂²	—	2	2	no	—
435	C₂²	—	4	2	no	—
436	C₂²	—	4	2	no	—
437	C₂²	—	8	1	no	—
438	C₂²	—	8	1	no	—
439	C₂²	—	8	1	no	—
440	C₂²	—	8	2	no	—
441	C₂²	—	8	2	no	—
442	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
443	C₂²	#532×C₆₇	1	2	yes	—
444	C₂²	#577×C₆₇	1	2	yes	—
445	C₂²	#577×C₆₇	1	1	yes	characteristic
446	C₂²	#577×C₆₇	1	1	yes	characteristic
447	C₄	—	2	2	no	—
448	C₄	—	4	2	no	—
449	C₄	—	4	2	no	—
450	C₄	—	8	1	no	—
451	C₄	—	8	1	no	—
452	C₄	—	8	1	no	—
453	C₄	—	8	2	no	—
454	C₄	—	8	2	no	—
455	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
456	C₄	#525×C₆₇	1	1	yes	characteristic
457	C₄	#529×C₆₇	1	2	yes	—
458	C₂	—	4	2	no	—
459	C₂	—	16	1	no	—
460	C₂	C₆₇×#2854	1	1	yes	characteristic
461	C₂	C₆₇×#3057	1	1	yes	characteristic
462	C₂	C₆₇×#3060	1	1	yes	characteristic
463	C₂	C₆₇×#3173	1	2	yes	—
464	C₂	C₆₇×#3174	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#121×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#405×C₆₇	C₂	semidirect
3. C₆₇×#2625	C₂	semidirect
5. C₆₇×#1968	C₂	semidirect
6. C₆₇×#1941	C₂	semidirect
7. C₆₇×#1936	C₂	semidirect
8. C₆₇×#1935	C₂	semidirect
10. C₆₇×#1894	C₂	semidirect
11. C₆₇×#1807	C₂	semidirect
225. #33137	C₆₇	direct (N × H)
234. C₂×#405	C₂×C₆₇	semidirect
235. #2625	C₂×C₆₇	semidirect
237. #1968	C₂×C₆₇	semidirect
238. #1941	C₂×C₆₇	semidirect
239. #1936	C₂×C₆₇	semidirect
240. #1935	C₂×C₆₇	semidirect
242. #1894	C₂×C₆₇	semidirect
243. #1807	C₂×C₆₇	semidirect
244. C₆₇	#33137	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×#465×C₆₇	2
C₂×#405×C₆₇	2
C₆₇×#2625	2
C₆₇×#2624	2
C₆₇×#1968	2
C₆₇×#1941	2
C₆₇×#1936	2
C₆₇×#1935	2
C₆₇×#1895	2
C₆₇×#1894	2
C₆₇×#1807	2
#33137	67

Subgroups — C67×#33137 #30712388

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33137 #30712388