Subgroups — C₆₇×#19388 #30698639

Order 17152.

662 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3210	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3110	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3100	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2972	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1681	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1679	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1669	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1665	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1572	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1553	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1544	C₂	1	1	yes	maximal, characteristic, complemented
12	#533×C₆₇	C₂²	1	1	yes	characteristic
13	#532×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	#494×C₆₇	C₂²	1	1	yes	characteristic, complemented
16	#492×C₆₇	C₂²	1	1	yes	characteristic, complemented
17	#487×C₆₇	C₂²	1	1	yes	characteristic, complemented
18	#486×C₆₇	C₂²	1	1	yes	characteristic, complemented
19	#484×C₆₇	—	2	1	no	—
20	#480×C₆₇	—	2	1	no	—
21	#474×C₆₇	C₂²	1	2	yes	—
22	#474×C₆₇	C₂²	1	2	yes	—
23	#464×C₆₇	—	2	2	no	—
24	#461×C₆₇	C₂²	1	2	yes	—
25	#459×C₆₇	C₂²	1	2	yes	—
26	#452×C₆₇	—	2	2	no	—
27	#450×C₆₇	—	2	1	no	—
28	#447×C₆₇	—	2	1	no	—
29	#440×C₆₇	—	2	2	no	—
30	C₄×D₈×C₆₇	—	2	2	no	—
31	C₈×D₄×C₆₇	—	2	2	no	—
32	C₈×D₄×C₆₇	C₂²	1	2	yes	—
33	C₈×D₄×C₆₇	C₂²	1	2	yes	—
34	#432×C₆₇	C₂²	1	1	yes	characteristic
35	#431×C₆₇	C₂²	1	1	yes	characteristic
36	#431×C₆₇	C₂²	1	1	yes	characteristic
37	#431×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
39	#335×C₆₇	C₂²	1	2	yes	complemented
40	#334×C₆₇	C₂²	1	2	yes	complemented
41	#332×C₆₇	C₂²	1	2	yes	complemented
42	#332×C₆₇	C₂²	1	2	yes	complemented
43	#332×C₆₇	C₂²	1	2	yes	complemented
44	#331×C₆₇	C₂²	1	2	yes	complemented
45	C₂×#41×C₆₇	—	2	1	no	—
46	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₆₇	—	2	1	no	—
48	C₂×Q₁₆×C₆₇	—	2	2	no	—
49	C₂×SD₁₆×C₆₇	—	2	2	no	—
50	C₂×SD₁₆×C₆₇	—	2	2	no	—
51	C₂×D₈×C₆₇	—	2	2	no	—
52	C₂²×C₈×C₆₇	—	2	2	no	—
53	C₂²×C₈×C₆₇	—	2	2	no	—
54	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
55	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
57	#127×C₆₇	C₂³	1	1	yes	characteristic
58	#125×C₆₇	—	2	1	no	—
59	#124×C₆₇	—	2	1	no	—
60	C₂²⋊Q₈×C₆₇	—	2	1	no	—
61	C₂²⋊Q₈×C₆₇	—	2	1	no	—
62	#121×C₆₇	—	2	1	no	—
63	#121×C₆₇	—	2	1	no	—
64	C₄×Q₈×C₆₇	—	2	1	no	—
65	C₄×Q₈×C₆₇	—	2	1	no	—
66	C₄×D₄×C₆₇	—	2	1	no	—
67	C₄×D₄×C₆₇	—	2	1	no	—
68	C₄×D₄×C₆₇	—	2	1	no	—
69	C₄×D₄×C₆₇	—	2	1	no	—
70	C₄×D₄×C₆₇	D₄	1	2	yes	—
71	C₄×D₄×C₆₇	C₂³	1	2	yes	—
72	#117×C₆₇	—	2	1	no	—
73	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
74	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	#107×C₆₇	—	2	2	no	—
77	#107×C₆₇	—	2	2	no	—
78	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
79	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
80	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
81	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
82	#103×C₆₇	—	2	2	no	—
83	#103×C₆₇	—	2	2	no	—
84	#102×C₆₇	—	2	2	no	—
85	#102×C₆₇	—	2	2	no	—
86	#102×C₆₇	—	2	2	no	—
87	#102×C₆₇	—	2	2	no	—
88	C₂²⋊C₈×C₆₇	—	2	1	no	—
89	C₂²⋊C₈×C₆₇	—	2	1	no	—
90	C₂²⋊C₈×C₆₇	—	2	1	no	—
91	C₂²⋊C₈×C₆₇	—	2	2	no	—
92	C₂²⋊C₈×C₆₇	—	2	2	no	—
93	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
94	#97×C₆₇	—	2	1	no	—
95	C₄×C₈×C₆₇	—	2	1	no	—
96	C₄×C₈×C₆₇	—	2	1	no	—
97	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
98	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
99	C₂⁴×C₆₇	—	2	1	no	—
100	#41×C₆₇	—	8	2	no	—
101	C₂×Q₈×C₆₇	—	2	1	no	—
102	C₂×Q₈×C₆₇	—	2	1	no	—
103	C₂×Q₈×C₆₇	—	2	1	no	—
104	C₂×D₄×C₆₇	—	2	4	no	—
105	C₂×D₄×C₆₇	—	2	1	no	—
106	C₂×D₄×C₆₇	—	2	1	no	—
107	C₂×D₄×C₆₇	—	2	1	no	—
108	C₂×D₄×C₆₇	—	2	1	no	—
109	C₂×D₄×C₆₇	—	4	1	no	—
110	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₆₇	—	2	1	no	—
112	C₂²×C₄×C₆₇	—	2	2	no	—
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	—	4	1	no	—
115	C₂²×C₄×C₆₇	—	8	2	no	—
116	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
119	Q₁₆×C₆₇	—	8	2	no	—
120	SD₁₆×C₆₇	—	4	4	no	—
121	SD₁₆×C₆₇	—	8	2	no	—
122	D₈×C₆₇	—	4	4	no	—
123	C₂×C₈×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	—	2	2	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	—	8	2	no	—
127	C₂×C₈×C₆₇	—	8	2	no	—
128	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
129	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
130	C₄⋊C₄×C₆₇	—	2	1	no	—
131	C₄⋊C₄×C₆₇	—	2	1	no	—
132	C₄⋊C₄×C₆₇	—	2	1	no	—
133	C₄⋊C₄×C₆₇	—	2	1	no	—
134	C₄⋊C₄×C₆₇	—	2	1	no	—
135	C₄⋊C₄×C₆₇	—	2	1	no	—
136	C₄⋊C₄×C₆₇	—	2	1	no	—
137	C₄⋊C₄×C₆₇	—	4	1	no	—
138	C₄⋊C₄×C₆₇	—	4	1	no	—
139	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
140	C₂²⋊C₄×C₆₇	—	2	2	no	—
141	C₂²⋊C₄×C₆₇	—	2	2	no	—
142	C₂²⋊C₄×C₆₇	—	4	1	no	—
143	C₂²⋊C₄×C₆₇	—	4	1	no	—
144	C₂²⋊C₄×C₆₇	—	4	1	no	—
145	C₄²×C₆₇	—	4	1	no	—
146	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
147	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
148	C₄²×C₆₇	#41	1	1	yes	characteristic
149	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
150	C₂³×C₆₇	—	2	4	no	—
151	C₂³×C₆₇	—	2	2	no	—
152	C₂³×C₆₇	—	4	1	no	—
153	C₂³×C₆₇	—	8	1	no	—
154	C₂³×C₆₇	—	8	1	no	—
155	C₂³×C₆₇	#121	1	1	yes	characteristic
156	Q₈×C₆₇	—	4	2	no	—
157	Q₈×C₆₇	—	4	2	no	—
158	Q₈×C₆₇	—	4	2	no	—
159	D₄×C₆₇	—	2	4	no	—
160	D₄×C₆₇	—	2	2	no	—
161	D₄×C₆₇	—	4	2	no	—
162	D₄×C₆₇	—	4	2	no	—
163	D₄×C₆₇	—	8	1	no	—
164	D₄×C₆₇	—	8	2	no	—
165	D₄×C₆₇	#135	1	4	yes	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	2	2	no	—
169	C₂×C₄×C₆₇	—	4	1	no	—
170	C₂×C₄×C₆₇	—	4	2	no	—
171	C₂×C₄×C₆₇	—	4	2	no	—
172	C₂×C₄×C₆₇	—	4	1	no	—
173	C₂×C₄×C₆₇	—	4	1	no	—
174	C₂×C₄×C₆₇	—	4	1	no	—
175	C₂×C₄×C₆₇	—	8	2	no	—
176	C₂×C₄×C₆₇	—	8	2	no	—
177	C₂×C₄×C₆₇	—	8	1	no	—
178	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
179	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
180	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
181	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
184	C₈×C₆₇	—	2	2	no	—
185	C₈×C₆₇	—	2	2	no	—
186	C₈×C₆₇	—	4	2	no	—
187	C₈×C₆₇	—	4	2	no	—
188	C₈×C₆₇	—	4	2	no	—
189	C₂²×C₆₇	—	2	1	no	—
190	C₂²×C₆₇	—	2	1	no	—
191	C₂²×C₆₇	—	2	4	no	—
192	C₂²×C₆₇	—	2	1	no	—
193	C₂²×C₆₇	—	4	2	no	—
194	C₂²×C₆₇	—	4	2	no	—
195	C₂²×C₆₇	—	8	4	no	—
196	C₂²×C₆₇	—	8	1	no	—
197	C₂²×C₆₇	—	8	1	no	—
198	C₂²×C₆₇	—	8	1	no	—
199	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
200	C₄×C₆₇	—	2	1	no	—
201	C₄×C₆₇	—	2	1	no	—
202	C₄×C₆₇	—	2	1	no	—
203	C₄×C₆₇	—	2	1	no	—
204	C₄×C₆₇	—	4	2	no	—
205	C₄×C₆₇	—	8	1	no	—
206	C₄×C₆₇	—	8	1	no	—
207	C₄×C₆₇	—	8	1	no	—
208	C₄×C₆₇	C₂×#135	1	2	yes	—
209	C₄×C₆₇	#576	1	2	yes	—
210	#19388	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
211	C₂×C₆₇	—	4	1	no	—
212	C₂×C₆₇	—	4	4	no	—
213	C₂×C₆₇	—	16	1	no	—
214	C₂×C₆₇	#3041	1	1	yes	characteristic
215	C₂×C₆₇	#3050	1	1	yes	characteristic
216	C₂×C₆₇	#3069	1	1	yes	characteristic
217	#3210	C₂×C₆₇	1	1	yes	characteristic, complemented
218	#3110	C₂×C₆₇	1	1	yes	characteristic, complemented
219	#3100	C₂×C₆₇	1	1	yes	characteristic
220	#2972	C₂×C₆₇	1	1	yes	characteristic, complemented
221	#1681	C₂×C₆₇	1	2	yes	complemented
222	#1679	C₂×C₆₇	1	2	yes	complemented
223	#1669	C₂×C₆₇	1	2	yes	complemented
224	#1665	C₂×C₆₇	1	2	yes	complemented
225	#1572	C₂×C₆₇	1	1	yes	characteristic, complemented
226	#1553	C₂×C₆₇	1	1	yes	characteristic, complemented
227	#1544	C₂×C₆₇	1	1	yes	characteristic, complemented
228	C₆₇	#19388	1	1	yes	characteristic, direct product factor, Sylow
229	#533	C₂²×C₆₇	1	1	yes	characteristic
230	#532	C₂²×C₆₇	1	1	yes	characteristic
231	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
232	#494	C₂²×C₆₇	1	1	yes	characteristic, complemented
233	#492	C₂²×C₆₇	1	1	yes	characteristic, complemented
234	#487	C₂²×C₆₇	1	1	yes	characteristic, complemented
235	#486	C₂²×C₆₇	1	1	yes	characteristic, complemented
236	#484	—	2	1	no	—
237	#480	—	2	1	no	—
238	#474	C₂²×C₆₇	1	2	yes	—
239	#474	C₂²×C₆₇	1	2	yes	—
240	#464	—	2	2	no	—
241	#461	C₂²×C₆₇	1	2	yes	—
242	#459	C₂²×C₆₇	1	2	yes	—
243	#452	—	2	2	no	—
244	#450	—	2	1	no	—
245	#447	—	2	1	no	—
246	#440	—	2	2	no	—
247	C₄×D₈	—	2	2	no	—
248	C₈×D₄	—	2	2	no	—
249	C₈×D₄	C₂²×C₆₇	1	2	yes	—
250	C₈×D₄	C₂²×C₆₇	1	2	yes	—
251	#432	C₂²×C₆₇	1	1	yes	characteristic
252	#431	C₂²×C₆₇	1	1	yes	characteristic
253	#431	C₂²×C₆₇	1	1	yes	characteristic
254	#431	C₂²×C₆₇	1	1	yes	characteristic
255	C₂×C₂²⋊C₈	—	2	2	no	—
256	#335	C₂²×C₆₇	1	2	yes	complemented
257	#334	C₂²×C₆₇	1	2	yes	complemented
258	#332	C₂²×C₆₇	1	2	yes	complemented
259	#332	C₂²×C₆₇	1	2	yes	complemented
260	#332	C₂²×C₆₇	1	2	yes	complemented
261	#331	C₂²×C₆₇	1	2	yes	complemented
262	C₂×#41	—	2	1	no	—
263	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
264	C₂³×C₄	—	2	1	no	—
265	C₂×Q₁₆	—	2	2	no	—
266	C₂×SD₁₆	—	2	2	no	—
267	C₂×SD₁₆	—	2	2	no	—
268	C₂×D₈	—	2	2	no	—
269	C₂²×C₈	—	2	2	no	—
270	C₂²×C₈	—	2	2	no	—
271	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
272	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
273	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
274	#127	C₂³×C₆₇	1	1	yes	characteristic
275	#125	—	2	1	no	—
276	#124	—	2	1	no	—
277	C₂²⋊Q₈	—	2	1	no	—
278	C₂²⋊Q₈	—	2	1	no	—
279	#121	—	2	1	no	—
280	#121	—	2	1	no	—
281	C₄×Q₈	—	2	1	no	—
282	C₄×Q₈	—	2	1	no	—
283	C₄×D₄	—	2	1	no	—
284	C₄×D₄	—	2	1	no	—
285	C₄×D₄	—	2	1	no	—
286	C₄×D₄	—	2	1	no	—
287	C₄×D₄	D₄×C₆₇	1	2	yes	—
288	C₄×D₄	C₂³×C₆₇	1	2	yes	—
289	#117	—	2	1	no	—
290	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
291	C₂×C₂²⋊C₄	—	2	1	no	—
292	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
293	#107	—	2	2	no	—
294	#107	—	2	2	no	—
295	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
296	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
297	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
298	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
299	#103	—	2	2	no	—
300	#103	—	2	2	no	—
301	#102	—	2	2	no	—
302	#102	—	2	2	no	—
303	#102	—	2	2	no	—
304	#102	—	2	2	no	—
305	C₂²⋊C₈	—	2	1	no	—
306	C₂²⋊C₈	—	2	1	no	—
307	C₂²⋊C₈	—	2	1	no	—
308	C₂²⋊C₈	—	2	2	no	—
309	C₂²⋊C₈	—	2	2	no	—
310	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
311	#97	—	2	1	no	—
312	C₄×C₈	—	2	1	no	—
313	C₄×C₈	—	2	1	no	—
314	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
315	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
316	C₂⁴	—	2	1	no	—
317	#41	—	8	2	no	—
318	C₂×Q₈	—	2	1	no	—
319	C₂×Q₈	—	2	1	no	—
320	C₂×Q₈	—	2	1	no	—
321	C₂×D₄	—	2	4	no	—
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	—	2	1	no	—
325	C₂×D₄	—	2	1	no	—
326	C₂×D₄	—	4	1	no	—
327	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	4	1	no	—
332	C₂²×C₄	—	8	2	no	—
333	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
334	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
335	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
336	Q₁₆	—	8	2	no	—
337	SD₁₆	—	4	4	no	—
338	SD₁₆	—	8	2	no	—
339	D₈	—	4	4	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	8	2	no	—
344	C₂×C₈	—	8	2	no	—
345	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
346	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	1	no	—
352	C₄⋊C₄	—	2	1	no	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	4	1	no	—
355	C₄⋊C₄	—	4	1	no	—
356	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
357	C₂²⋊C₄	—	2	2	no	—
358	C₂²⋊C₄	—	2	2	no	—
359	C₂²⋊C₄	—	4	1	no	—
360	C₂²⋊C₄	—	4	1	no	—
361	C₂²⋊C₄	—	4	1	no	—
362	C₄²	—	4	1	no	—
363	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
364	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
365	C₄²	#41×C₆₇	1	1	yes	characteristic
366	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
367	C₂³	—	2	4	no	—
368	C₂³	—	2	2	no	—
369	C₂³	—	4	1	no	—
370	C₂³	—	8	1	no	—
371	C₂³	—	8	1	no	—
372	C₂³	#121×C₆₇	1	1	yes	characteristic
373	Q₈	—	4	2	no	—
374	Q₈	—	4	2	no	—
375	Q₈	—	4	2	no	—
376	D₄	—	2	4	no	—
377	D₄	—	2	2	no	—
378	D₄	—	4	2	no	—
379	D₄	—	4	2	no	—
380	D₄	—	8	1	no	—
381	D₄	—	8	2	no	—
382	D₄	#135×C₆₇	1	4	yes	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	1	no	—
390	C₂×C₄	—	4	1	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	8	2	no	—
393	C₂×C₄	—	8	2	no	—
394	C₂×C₄	—	8	1	no	—
395	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
396	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
397	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
398	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
399	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
400	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
401	C₈	—	2	2	no	—
402	C₈	—	2	2	no	—
403	C₈	—	4	2	no	—
404	C₈	—	4	2	no	—
405	C₈	—	4	2	no	—
406	C₂²	—	2	1	no	—
407	C₂²	—	2	1	no	—
408	C₂²	—	2	4	no	—
409	C₂²	—	2	1	no	—
410	C₂²	—	4	2	no	—
411	C₂²	—	4	2	no	—
412	C₂²	—	8	4	no	—
413	C₂²	—	8	1	no	—
414	C₂²	—	8	1	no	—
415	C₂²	—	8	1	no	—
416	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
417	C₄	—	2	1	no	—
418	C₄	—	2	1	no	—
419	C₄	—	2	1	no	—
420	C₄	—	2	1	no	—
421	C₄	—	4	2	no	—
422	C₄	—	8	1	no	—
423	C₄	—	8	1	no	—
424	C₄	—	8	1	no	—
425	C₄	C₂×#135×C₆₇	1	2	yes	—
426	C₄	#576×C₆₇	1	2	yes	—
427	C₂	—	4	1	no	—
428	C₂	—	4	4	no	—
429	C₂	—	16	1	no	—
430	C₂	C₆₇×#3041	1	1	yes	characteristic
431	C₂	C₆₇×#3050	1	1	yes	characteristic
432	C₂	C₆₇×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₆₇×#3100	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3210	C₂	semidirect
2. C₆₇×#3110	C₂	semidirect
4. C₆₇×#2972	C₂	semidirect
5. C₆₇×#1681	C₂	semidirect
6. C₆₇×#1679	C₂	semidirect
7. C₆₇×#1669	C₂	semidirect
8. C₆₇×#1665	C₂	semidirect
9. C₆₇×#1572	C₂	semidirect
10. C₆₇×#1553	C₂	semidirect
11. C₆₇×#1544	C₂	semidirect
15. #494×C₆₇	C₂²	semidirect
16. #492×C₆₇	C₂²	semidirect
17. #487×C₆₇	C₂²	semidirect
18. #486×C₆₇	C₂²	semidirect
39. #335×C₆₇	C₂²	semidirect
40. #334×C₆₇	C₂²	semidirect
41. #332×C₆₇	C₂²	semidirect
42. #332×C₆₇	C₂²	semidirect
43. #332×C₆₇	C₂²	semidirect
44. #331×C₆₇	C₂²	semidirect
210. #19388	C₆₇	direct (N × H)
217. #3210	C₂×C₆₇	semidirect
218. #3110	C₂×C₆₇	semidirect
220. #2972	C₂×C₆₇	semidirect
221. #1681	C₂×C₆₇	semidirect
222. #1679	C₂×C₆₇	semidirect
223. #1669	C₂×C₆₇	semidirect
224. #1665	C₂×C₆₇	semidirect
225. #1572	C₂×C₆₇	semidirect
226. #1553	C₂×C₆₇	semidirect
227. #1544	C₂×C₆₇	semidirect
228. C₆₇	#19388	direct (N × H)
232. #494	C₂²×C₆₇	semidirect
233. #492	C₂²×C₆₇	semidirect
234. #487	C₂²×C₆₇	semidirect
235. #486	C₂²×C₆₇	semidirect
256. #335	C₂²×C₆₇	semidirect
257. #334	C₂²×C₆₇	semidirect
258. #332	C₂²×C₆₇	semidirect
259. #332	C₂²×C₆₇	semidirect
260. #332	C₂²×C₆₇	semidirect
261. #331	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3210	2
C₆₇×#3110	2
C₆₇×#3100	2
C₆₇×#2972	2
C₆₇×#1681	2
C₆₇×#1679	2
C₆₇×#1669	2
C₆₇×#1665	2
C₆₇×#1572	2
C₆₇×#1553	2
C₆₇×#1544	2
#19388	67

Subgroups — C67×#19388 #30698639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#19388 #30698639