Subgroups — C₆₇×#32952 #30712203

Order 17152.

1266 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3573	C₂	1	1	yes	maximal, characteristic
2	C₂×#473×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#3059	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3022	C₂	1	2	yes	maximal
5	C₆₇×#1915	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1902	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1814	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1813	C₂	1	1	yes	maximal, characteristic, complemented
9	#585×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#137×C₆₇	C₂²	1	4	yes	—
11	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
12	#568×C₆₇	—	2	4	no	—
13	#568×C₆₇	—	2	2	no	—
14	#568×C₆₇	C₂²	1	2	yes	—
15	#568×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#131×C₆₇	—	2	4	no	—
17	C₂×#131×C₆₇	—	2	2	no	—
18	C₂×#131×C₆₇	C₂²	1	2	yes	—
19	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
20	#536×C₆₇	C₂²	1	2	yes	—
21	#473×C₆₇	—	2	4	no	—
22	#473×C₆₇	—	2	4	no	—
23	#470×C₆₇	—	2	2	no	—
24	#467×C₆₇	—	2	2	no	—
25	#456×C₆₇	—	2	4	no	—
26	#455×C₆₇	—	2	4	no	—
27	C₂×#108×C₆₇	C₂²	1	2	yes	complemented
28	#427×C₆₇	C₂²	1	1	yes	characteristic, complemented
29	#421×C₆₇	C₂²	1	4	yes	—
30	#419×C₆₇	C₂²	1	2	yes	complemented
31	#413×C₆₇	C₂²	1	2	yes	—
32	C₂×#101×C₆₇	C₂²	1	4	yes	complemented
33	#344×C₆₇	C₂²	1	4	yes	complemented
34	#338×C₆₇	C₂²	1	2	yes	complemented
35	2₋¹⁺⁴×C₆₇	—	2	2	no	—
36	2₋¹⁺⁴×C₆₇	D₄	1	2	yes	—
37	H(5,𝔽₂)×C₆₇	—	2	2	no	—
38	H(5,𝔽₂)×C₆₇	—	2	2	no	—
39	H(5,𝔽₂)×C₆₇	D₄	1	2	yes	—
40	C₂×#41×C₆₇	—	2	4	no	—
41	C₂×#41×C₆₇	—	2	2	no	—
42	C₂×#41×C₆₇	C₂³	1	2	yes	—
43	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
44	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
45	#137×C₆₇	—	2	8	no	—
46	#137×C₆₇	—	2	8	no	—
47	C₂×Q₁₆×C₆₇	—	2	4	no	—
48	C₂×Q₁₆×C₆₇	—	2	1	no	—
49	C₂×Q₁₆×C₆₇	—	2	1	no	—
50	C₂×Q₁₆×C₆₇	—	2	2	no	—
51	C₂×Q₁₆×C₆₇	—	2	2	no	—
52	C₂×SD₁₆×C₆₇	—	2	4	no	—
53	#131×C₆₇	—	2	4	no	—
54	#131×C₆₇	—	2	8	no	—
55	#131×C₆₇	—	2	4	no	—
56	#131×C₆₇	—	2	4	no	—
57	#131×C₆₇	—	2	4	no	—
58	#131×C₆₇	—	2	2	no	—
59	#131×C₆₇	—	2	2	no	—
60	#131×C₆₇	—	2	4	no	—
61	#131×C₆₇	—	2	2	no	—
62	#131×C₆₇	—	2	2	no	—
63	#131×C₆₇	D₄	1	4	yes	—
64	#131×C₆₇	D₄	1	4	yes	—
65	C₂×M₁₆×C₆₇	—	2	8	no	—
66	C₂×M₁₆×C₆₇	—	2	4	no	—
67	C₂×M₁₆×C₆₇	—	2	2	no	—
68	C₂×M₁₆×C₆₇	—	2	4	no	—
69	C₂×M₁₆×C₆₇	—	2	2	no	—
70	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
71	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
72	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
73	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₆₇	—	2	4	no	—
75	C₂²×C₈×C₆₇	—	2	2	no	—
76	C₂²×C₈×C₆₇	D₄	1	2	yes	—
77	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₄⋊Q₈×C₆₇	—	2	2	no	—
79	#124×C₆₇	—	2	2	no	—
80	#123×C₆₇	—	2	4	no	—
81	C₂²⋊Q₈×C₆₇	—	2	4	no	—
82	#117×C₆₇	C₂³	1	2	yes	—
83	#108×C₆₇	—	2	2	no	—
84	#108×C₆₇	—	2	2	no	—
85	#107×C₆₇	—	2	1	no	—
86	#106×C₆₇	—	2	1	no	—
87	#104×C₆₇	—	2	8	no	—
88	#103×C₆₇	—	2	4	no	—
89	#101×C₆₇	—	2	2	no	—
90	#101×C₆₇	—	2	8	no	—
91	#100×C₆₇	—	2	2	no	—
92	#41×C₆₇	—	2	4	no	—
93	#41×C₆₇	—	2	8	no	—
94	#41×C₆₇	—	2	4	no	—
95	#41×C₆₇	—	2	4	no	—
96	#41×C₆₇	—	2	4	no	—
97	#41×C₆₇	—	2	4	no	—
98	#41×C₆₇	—	2	2	no	—
99	#41×C₆₇	—	2	2	no	—
100	#41×C₆₇	—	2	2	no	—
101	#41×C₆₇	—	2	2	no	—
102	#41×C₆₇	C₂×D₄	1	4	yes	—
103	#41×C₆₇	C₂×D₄	1	4	yes	—
104	C₂×Q₈×C₆₇	—	2	2	no	—
105	C₂×Q₈×C₆₇	—	2	4	no	—
106	C₂×Q₈×C₆₇	—	2	4	no	—
107	C₂×Q₈×C₆₇	—	2	2	no	—
108	C₂×Q₈×C₆₇	—	2	2	no	—
109	C₂×Q₈×C₆₇	—	2	4	no	—
110	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂×Q₈×C₆₇	#41	1	2	yes	—
112	C₂×D₄×C₆₇	—	2	8	no	—
113	C₂×D₄×C₆₇	—	2	2	no	—
114	C₂×D₄×C₆₇	—	2	4	no	—
115	C₂×D₄×C₆₇	—	2	2	no	—
116	C₂×D₄×C₆₇	—	2	4	no	—
117	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂×D₄×C₆₇	#41	1	2	yes	—
119	C₂×D₄×C₆₇	#41	1	2	yes	—
120	C₂²×C₄×C₆₇	—	2	4	no	—
121	C₂²×C₄×C₆₇	—	2	2	no	—
122	C₂²×C₄×C₆₇	—	2	2	no	—
123	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
124	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
125	Q₁₆×C₆₇	—	4	8	no	—
126	Q₁₆×C₆₇	—	4	1	no	—
127	Q₁₆×C₆₇	—	4	1	no	—
128	Q₁₆×C₆₇	—	4	2	no	—
129	SD₁₆×C₆₇	—	4	8	no	—
130	M₁₆×C₆₇	—	2	4	no	—
131	M₁₆×C₆₇	—	2	8	no	—
132	M₁₆×C₆₇	—	2	2	no	—
133	M₁₆×C₆₇	—	2	2	no	—
134	M₁₆×C₆₇	—	2	4	no	—
135	M₁₆×C₆₇	—	2	4	no	—
136	M₁₆×C₆₇	—	2	4	no	—
137	M₁₆×C₆₇	—	2	4	no	—
138	M₁₆×C₆₇	—	2	4	no	—
139	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
140	C₂×C₈×C₆₇	—	2	4	no	—
141	C₂×C₈×C₆₇	—	2	8	no	—
142	C₂×C₈×C₆₇	—	2	2	no	—
143	C₂×C₈×C₆₇	—	2	4	no	—
144	C₂×C₈×C₆₇	—	2	2	no	—
145	C₂×C₈×C₆₇	—	2	4	no	—
146	C₂×C₈×C₆₇	—	2	2	no	—
147	C₂×C₈×C₆₇	—	2	4	no	—
148	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
149	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
150	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
151	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
152	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
153	C₄⋊C₄×C₆₇	—	2	2	no	—
154	C₄⋊C₄×C₆₇	—	4	4	no	—
155	C₂²⋊C₄×C₆₇	—	2	2	no	—
156	C₂²⋊C₄×C₆₇	—	4	4	no	—
157	C₄²×C₆₇	—	2	2	no	—
158	C₂³×C₆₇	—	2	4	no	—
159	C₂³×C₆₇	—	2	2	no	—
160	C₂³×C₆₇	#121	1	2	yes	—
161	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
162	Q₈×C₆₇	—	2	4	no	—
163	Q₈×C₆₇	—	2	2	no	—
164	Q₈×C₆₇	—	2	2	no	—
165	Q₈×C₆₇	—	4	2	no	—
166	Q₈×C₆₇	—	4	4	no	—
167	Q₈×C₆₇	—	4	2	no	—
168	Q₈×C₆₇	#121	1	4	yes	—
169	D₄×C₆₇	—	2	8	no	—
170	D₄×C₆₇	—	2	4	no	—
171	D₄×C₆₇	—	2	4	no	—
172	D₄×C₆₇	—	2	4	no	—
173	D₄×C₆₇	—	2	4	no	—
174	D₄×C₆₇	—	2	2	no	—
175	D₄×C₆₇	—	2	2	no	—
176	D₄×C₆₇	#121	1	4	yes	—
177	C₂×C₄×C₆₇	—	2	4	no	—
178	C₂×C₄×C₆₇	—	2	8	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	2	4	no	—
181	C₂×C₄×C₆₇	—	2	2	no	—
182	C₂×C₄×C₆₇	—	2	4	no	—
183	C₂×C₄×C₆₇	—	2	2	no	—
184	C₂×C₄×C₆₇	—	4	2	no	—
185	C₂×C₄×C₆₇	—	4	2	no	—
186	C₂×C₄×C₆₇	—	4	4	no	—
187	C₂×C₄×C₆₇	#121	1	2	yes	—
188	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
189	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
192	C₈×C₆₇	—	2	4	no	—
193	C₈×C₆₇	—	2	2	no	—
194	C₈×C₆₇	—	4	4	no	—
195	C₈×C₆₇	#120	1	1	yes	characteristic
196	C₈×C₆₇	#120	1	2	yes	—
197	C₈×C₆₇	#120	1	1	yes	characteristic
198	C₂²×C₆₇	—	2	2	no	—
199	C₂²×C₆₇	—	2	4	no	—
200	C₂²×C₆₇	—	4	1	no	—
201	C₂²×C₆₇	—	4	2	no	—
202	C₂²×C₆₇	—	8	2	no	—
203	C₂²×C₆₇	—	8	4	no	—
204	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
205	C₂²×C₆₇	#525	1	2	yes	—
206	C₄×C₆₇	—	2	4	no	—
207	C₄×C₆₇	—	2	2	no	—
208	C₄×C₆₇	—	8	2	no	—
209	C₄×C₆₇	—	8	2	no	—
210	C₄×C₆₇	C₂×#120	1	1	yes	characteristic
211	C₄×C₆₇	C₂×#121	1	2	yes	—
212	C₄×C₆₇	#525	1	1	yes	characteristic
213	#32952	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
214	C₂×C₆₇	—	2	1	no	—
215	C₂×C₆₇	—	2	2	no	—
216	C₂×C₆₇	—	4	2	no	—
217	C₂×C₆₇	—	4	4	no	—
218	C₂×C₆₇	#2847	1	1	yes	characteristic
219	#3573	C₂×C₆₇	1	1	yes	characteristic
220	C₂×#473	C₂×C₆₇	1	2	yes	complemented
221	#3059	C₂×C₆₇	1	1	yes	characteristic, complemented
222	#3022	C₂×C₆₇	1	2	yes	—
223	#1915	C₂×C₆₇	1	2	yes	complemented
224	#1902	C₂×C₆₇	1	4	yes	complemented
225	#1814	C₂×C₆₇	1	2	yes	complemented
226	#1813	C₂×C₆₇	1	1	yes	characteristic, complemented
227	C₆₇	#32952	1	1	yes	characteristic, direct product factor, Sylow
228	#585	C₂²×C₆₇	1	1	yes	characteristic
229	C₂×#137	C₂²×C₆₇	1	4	yes	—
230	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
231	#568	—	2	4	no	—
232	#568	—	2	2	no	—
233	#568	C₂²×C₆₇	1	2	yes	—
234	#568	C₂²×C₆₇	1	1	yes	characteristic
235	C₂×#131	—	2	4	no	—
236	C₂×#131	—	2	2	no	—
237	C₂×#131	C₂²×C₆₇	1	2	yes	—
238	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
239	#536	C₂²×C₆₇	1	2	yes	—
240	#473	—	2	4	no	—
241	#473	—	2	4	no	—
242	#470	—	2	2	no	—
243	#467	—	2	2	no	—
244	#456	—	2	4	no	—
245	#455	—	2	4	no	—
246	C₂×#108	C₂²×C₆₇	1	2	yes	complemented
247	#427	C₂²×C₆₇	1	1	yes	characteristic, complemented
248	#421	C₂²×C₆₇	1	4	yes	—
249	#419	C₂²×C₆₇	1	2	yes	complemented
250	#413	C₂²×C₆₇	1	2	yes	—
251	C₂×#101	C₂²×C₆₇	1	4	yes	complemented
252	#344	C₂²×C₆₇	1	4	yes	complemented
253	#338	C₂²×C₆₇	1	2	yes	complemented
254	2₋¹⁺⁴	—	2	2	no	—
255	2₋¹⁺⁴	D₄×C₆₇	1	2	yes	—
256	H(5,𝔽₂)	—	2	2	no	—
257	H(5,𝔽₂)	—	2	2	no	—
258	H(5,𝔽₂)	D₄×C₆₇	1	2	yes	—
259	C₂×#41	—	2	4	no	—
260	C₂×#41	—	2	2	no	—
261	C₂×#41	C₂³×C₆₇	1	2	yes	—
262	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
263	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
264	#137	—	2	8	no	—
265	#137	—	2	8	no	—
266	C₂×Q₁₆	—	2	4	no	—
267	C₂×Q₁₆	—	2	1	no	—
268	C₂×Q₁₆	—	2	1	no	—
269	C₂×Q₁₆	—	2	2	no	—
270	C₂×Q₁₆	—	2	2	no	—
271	C₂×SD₁₆	—	2	4	no	—
272	#131	—	2	4	no	—
273	#131	—	2	8	no	—
274	#131	—	2	4	no	—
275	#131	—	2	4	no	—
276	#131	—	2	4	no	—
277	#131	—	2	2	no	—
278	#131	—	2	2	no	—
279	#131	—	2	4	no	—
280	#131	—	2	2	no	—
281	#131	—	2	2	no	—
282	#131	D₄×C₆₇	1	4	yes	—
283	#131	D₄×C₆₇	1	4	yes	—
284	C₂×M₁₆	—	2	8	no	—
285	C₂×M₁₆	—	2	4	no	—
286	C₂×M₁₆	—	2	2	no	—
287	C₂×M₁₆	—	2	4	no	—
288	C₂×M₁₆	—	2	2	no	—
289	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
290	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
291	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
292	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
293	C₂²×C₈	—	2	4	no	—
294	C₂²×C₈	—	2	2	no	—
295	C₂²×C₈	D₄×C₆₇	1	2	yes	—
296	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
297	C₄⋊Q₈	—	2	2	no	—
298	#124	—	2	2	no	—
299	#123	—	2	4	no	—
300	C₂²⋊Q₈	—	2	4	no	—
301	#117	C₂³×C₆₇	1	2	yes	—
302	#108	—	2	2	no	—
303	#108	—	2	2	no	—
304	#107	—	2	1	no	—
305	#106	—	2	1	no	—
306	#104	—	2	8	no	—
307	#103	—	2	4	no	—
308	#101	—	2	2	no	—
309	#101	—	2	8	no	—
310	#100	—	2	2	no	—
311	#41	—	2	4	no	—
312	#41	—	2	8	no	—
313	#41	—	2	4	no	—
314	#41	—	2	4	no	—
315	#41	—	2	4	no	—
316	#41	—	2	4	no	—
317	#41	—	2	2	no	—
318	#41	—	2	2	no	—
319	#41	—	2	2	no	—
320	#41	—	2	2	no	—
321	#41	C₂×D₄×C₆₇	1	4	yes	—
322	#41	C₂×D₄×C₆₇	1	4	yes	—
323	C₂×Q₈	—	2	2	no	—
324	C₂×Q₈	—	2	4	no	—
325	C₂×Q₈	—	2	4	no	—
326	C₂×Q₈	—	2	2	no	—
327	C₂×Q₈	—	2	2	no	—
328	C₂×Q₈	—	2	4	no	—
329	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
330	C₂×Q₈	#41×C₆₇	1	2	yes	—
331	C₂×D₄	—	2	8	no	—
332	C₂×D₄	—	2	2	no	—
333	C₂×D₄	—	2	4	no	—
334	C₂×D₄	—	2	2	no	—
335	C₂×D₄	—	2	4	no	—
336	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
337	C₂×D₄	#41×C₆₇	1	2	yes	—
338	C₂×D₄	#41×C₆₇	1	2	yes	—
339	C₂²×C₄	—	2	4	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
343	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
344	Q₁₆	—	4	8	no	—
345	Q₁₆	—	4	1	no	—
346	Q₁₆	—	4	1	no	—
347	Q₁₆	—	4	2	no	—
348	SD₁₆	—	4	8	no	—
349	M₁₆	—	2	4	no	—
350	M₁₆	—	2	8	no	—
351	M₁₆	—	2	2	no	—
352	M₁₆	—	2	2	no	—
353	M₁₆	—	2	4	no	—
354	M₁₆	—	2	4	no	—
355	M₁₆	—	2	4	no	—
356	M₁₆	—	2	4	no	—
357	M₁₆	—	2	4	no	—
358	M₁₆	C₂×D₄×C₆₇	1	4	yes	—
359	C₂×C₈	—	2	4	no	—
360	C₂×C₈	—	2	8	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	2	4	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	2	4	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	4	no	—
367	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
368	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
369	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
370	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
371	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	4	4	no	—
374	C₂²⋊C₄	—	2	2	no	—
375	C₂²⋊C₄	—	4	4	no	—
376	C₄²	—	2	2	no	—
377	C₂³	—	2	4	no	—
378	C₂³	—	2	2	no	—
379	C₂³	#121×C₆₇	1	2	yes	—
380	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
381	Q₈	—	2	4	no	—
382	Q₈	—	2	2	no	—
383	Q₈	—	2	2	no	—
384	Q₈	—	4	2	no	—
385	Q₈	—	4	4	no	—
386	Q₈	—	4	2	no	—
387	Q₈	#121×C₆₇	1	4	yes	—
388	D₄	—	2	8	no	—
389	D₄	—	2	4	no	—
390	D₄	—	2	4	no	—
391	D₄	—	2	4	no	—
392	D₄	—	2	4	no	—
393	D₄	—	2	2	no	—
394	D₄	—	2	2	no	—
395	D₄	#121×C₆₇	1	4	yes	—
396	C₂×C₄	—	2	4	no	—
397	C₂×C₄	—	2	8	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	4	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	4	no	—
406	C₂×C₄	#121×C₆₇	1	2	yes	—
407	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
408	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
409	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
410	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
411	C₈	—	2	4	no	—
412	C₈	—	2	2	no	—
413	C₈	—	4	4	no	—
414	C₈	#120×C₆₇	1	1	yes	characteristic
415	C₈	#120×C₆₇	1	2	yes	—
416	C₈	#120×C₆₇	1	1	yes	characteristic
417	C₂²	—	2	2	no	—
418	C₂²	—	2	4	no	—
419	C₂²	—	4	1	no	—
420	C₂²	—	4	2	no	—
421	C₂²	—	8	2	no	—
422	C₂²	—	8	4	no	—
423	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
424	C₂²	#525×C₆₇	1	2	yes	—
425	C₄	—	2	4	no	—
426	C₄	—	2	2	no	—
427	C₄	—	8	2	no	—
428	C₄	—	8	2	no	—
429	C₄	C₂×#120×C₆₇	1	1	yes	characteristic
430	C₄	C₂×#121×C₆₇	1	2	yes	—
431	C₄	#525×C₆₇	1	1	yes	characteristic
432	C₂	—	2	1	no	—
433	C₂	—	2	2	no	—
434	C₂	—	4	2	no	—
435	C₂	—	4	4	no	—
436	C₂	C₆₇×#2847	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	#2847
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	#585×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#473×C₆₇	C₂	semidirect
3. C₆₇×#3059	C₂	semidirect
5. C₆₇×#1915	C₂	semidirect
6. C₆₇×#1902	C₂	semidirect
7. C₆₇×#1814	C₂	semidirect
8. C₆₇×#1813	C₂	semidirect
11. C₂²×Q₁₆×C₆₇	C₂²	semidirect
27. C₂×#108×C₆₇	C₂²	semidirect
28. #427×C₆₇	C₂²	semidirect
30. #419×C₆₇	C₂²	semidirect
32. C₂×#101×C₆₇	C₂²	semidirect
33. #344×C₆₇	C₂²	semidirect
34. #338×C₆₇	C₂²	semidirect
213. #32952	C₆₇	direct (N × H)
220. C₂×#473	C₂×C₆₇	semidirect
221. #3059	C₂×C₆₇	semidirect
223. #1915	C₂×C₆₇	semidirect
224. #1902	C₂×C₆₇	semidirect
225. #1814	C₂×C₆₇	semidirect
226. #1813	C₂×C₆₇	semidirect
227. C₆₇	#32952	direct (N × H)
230. C₂²×Q₁₆	C₂²×C₆₇	semidirect
246. C₂×#108	C₂²×C₆₇	semidirect
247. #427	C₂²×C₆₇	semidirect
249. #419	C₂²×C₆₇	semidirect
251. C₂×#101	C₂²×C₆₇	semidirect
252. #344	C₂²×C₆₇	semidirect
253. #338	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×#3573	2
C₂×#473×C₆₇	2
C₆₇×#3059	2
C₆₇×#3022	2
C₆₇×#1915	2
C₆₇×#1902	2
C₆₇×#1814	2
C₆₇×#1813	2
#32952	67

Subgroups — C67×#32952 #30712203

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32952 #30712203